Class 10th Mathematics Tamilnadu Board Solution
Exercise 3.1- x + 2y = 7, x - 2y = 1 Solve the following system of equation by elimination…
- 3x + y = 8, 5x + y = 10 Solve the following system of equation by elimination…
- x + y/2 = 4 , x/3 + 2y = 5 Solve the following system of equation by elimination…
- 11x - 7y = xy, 9x - 4y = 6xy Solve the following system of equation by…
- 3/x + 5/y = 20/xy , 2/x + 5/y = 15/xy , x not equal 0 , y not equal 0 Solve the…
- 8x - 3y = 5xy, 6x - 5y = - 2xy Solve the following system of equation by…
- 13x + 11y = 70, 11x + 13y = 74 Solve the following system of equation by…
- 65x - 33y = 97, 33x - 65y = 1 Solve the following system of equation by…
- 15/x + 2/y = 17 , 1/x + 1/y = 36/5 , x not equal 0 , y not equal 0 Solve the…
- 2/x + 2/3y = 1/6 , 3/x + 2/y = 0 , x not equal 0 , y not equal 0 Solve the…
Exercise 3.10- Multiply the following and write your answer in lowest terms. (i) x^2 - 2x/x+2 x…
- x/x+1 / x^2/x^2 - 1 Divide the following and write your answer in lowest terms.…
- x^2 - 36/x^2 - 49 / x+6/x+7 Divide the following and write your answer in…
- x^2 - 4x-5/x^2 - 25 / x^2 - 3x-10/x^2 + 7x+10 Divide the following and write…
- x^2 + 11x+28/x^2 - 4x-77 / x^2 + 7x+12/x^2 - 2x-15 Divide the following and…
- 2x^2 + 13x+15/x^2 + 3x-10 / 2x^2 - x-6/x^2 - 4x+4 Divide the following and…
- 3x^2 - x-4/9x^2 - 16 / 4x^2 - 4/3x^2 - 2x-1 Divide the following and write your…
- 2x^2 + 5x-3/2x^2 + 9x+9 / 2x^2 + x-1/2x^2 + x-3 Divide the following and write…
Exercise 3.11- Simplify the following as a quotient of two polynomials in the simplest form.…
- Which rational expression should be added to x^3 - 1/x^2 + 2 to get 3x^3 + 2x^2…
- Which rational expression should be subtracted from 4x^3 - 7x^2 + 5/2x-1 to get…
- If p = x/x+y , q = y/x+y then find 1/p-q - 2q/p^2 - q^2
Exercise 3.12- 196a^6 b^8 c^10 Find the square root of the following
- 289 (a-b)^4 (b-c)^6 Find the square root of the following
- (x + 11)^2 -44x Find the square root of the following
- (x-y)^2 + 4xy Find the square root of the following
- 121x^8 y^6 */* 81x^4 y^8 Find the square root of the following
- 64 (a+b)^4 (x-y)^8 (b-c)^6/25 (x+y)^4 (a-b)^6 (b+c)^10 Find the square root of…
- 16x^2 -24x + 9 Find the square root of the following:
- (x^2 - 25)(x^2 + 8x + 15)(x^2 -2x-15) Find the square root of the following:…
- 4x^2 + 9y^2 + 25z^2 -12xy + 30yz-20zx Find the square root of the following:…
- x^4 + 1/x^4 + 2 Find the square root of the following:
- (6x^2 + 5x -6) (6x^2 -x-2)(4x^2 + 8x + 3) Find the square root of the…
- (2x^2 -5x + 2) (3x^2 -5x-2) (6x^2 - x -1) Find the square root of the…
Exercise 3.13- x^4 -4x^3 + 10x^2 -12x + 9 Find the square root of the following polynomials by…
- 4x^4 + 8x^3 + 8x^2 + 4x + 1 Find the square root of the following polynomials…
- 9 x^4 -6 x^3 + 7 x^2 -2x + 1 Find the square root of the following polynomials…
- 4 + 25x^2 12x-24x^3 + 16x^4 Find the square root of the following polynomials…
- 4x^4 -12 x^3 + 37x^2 + ax + b Find the values of a and b if the following…
- x^4 -4x^3 + 6x^2 -ax + b Find the values of a and b if the following…
- ax^4 + bx^3 + 109x^2 -60x + 36 Find the values of a and b if the following…
- a x^4 -bx^3 + 40 x^2 + 24x + 36 Find the values of a and b if the following…
Exercise 3.14- (2x + 3)^2 - 81 = 0 Solve the following quadratic equations by factorization…
- 3x^2 -5x -12 = 0 Solve the following quadratic equations by factorization…
- root 5x^2 + 2x-3 root 5 = 0 Solve the following quadratic equations by…
- 3(x^2 - 6) =x (x + 7)-3 Solve the following quadratic equations by…
- 3x - 8/x = 2 Solve the following quadratic equations by factorization method.…
- x + 1/x = 26/5 Solve the following quadratic equations by factorization method.…
- x/x+1 + x+1/x = 34/15 Solve the following quadratic equations by factorization…
- a^2 b^2 x^2 - (a^2 + b^2) x + 1 =0 Solve the following quadratic equations by…
- 2(x + 1)^2 -5 (x + 1) =12 Solve the following quadratic equations by…
- 3(x-4)^2 - 5 (x - 4) = 12 Solve the following quadratic equations by…
Exercise 3.15- x^2 + 6x -7 = 0 Solve the following quadratic equations by completing the…
- x^2 + 3x + 1 =0 Solve the following quadratic equations by completing the…
- 2x^2 + 5x -3 = 0 Solve the following quadratic equations by completing the…
- 4x^2 + 4bx - (a^2 - b^2) = 0 Solve the following quadratic equations by…
- x^2 - (√3 + 1) x + √3 = 0 Solve the following quadratic equations by completing…
- 5x+7/x-1 =3x + 2 Solve the following quadratic equations by completing the…
- x^2 7x + 12= 0 Solve the following quadratic equations using quadratic formula.…
- 15x^2 - 11x + 2 = 0 Solve the following quadratic equations using quadratic…
- x + 1/x = 2 1/2 Solve the following quadratic equations using quadratic…
- 3a^2 x^2 - ax -2b^2 = 0 Solve the following quadratic equations using quadratic…
- a (x^2 + 1) = x (a^2 + 1) Solve the following quadratic equations using…
- 36x^2 - 12ax + (a^2 - b^2) = 0 Solve the following quadratic equations using…
- x-1/x+1 + x-3/x-4 = 10/3 Solve the following quadratic equations using…
- a^2 x^2 + (a^2 - b^2) x - b^2 = 0 Solve the following quadratic equations using…
Exercise 3.16- The sum of a number and its reciprocal is 65/8 . Find the number.…
- The difference of the squares of two positive numbers is 45. The square of the…
- A farmer wishes to start a 100 sq. rectangular vegetable garden. Since he has…
- A rectangular field is 20 m long and 14 m wide. There is a path of equal width…
- A train covers a distance of 90 km at a uniform speed. Had the speed been 15…
- The speed of a boat in still water is 15 km/hr. It goes 30 km upstream and…
- One year ago, a man was 8 times as old as his son. Now his age is equal to the…
- A chess board contains 64 equal squares and the area of each square is 6.25 cm2.…
- A takes 6 day less than the time taken by B to finish a piece of work. If both A…
- Two trains leave a railway station at the same time. The first train travels…
Exercise 3.17- x^2 - 8x + 12 = 0 Determine the nature of the roots of the equation.…
- 2x^2 - 3x + 4 = 0 Determine the nature of the roots of the equation.…
- 9x^2 + 12x + 4 = 0 Determine the nature of the roots of the equation.…
- 3x^2 -2√6x + 2 = 0 Determine the nature of the roots of the equation.…
- 3/5 x^2 - 12/3 x+1 = 0 Determine the nature of the roots of the equation.…
- (x - 2a) (x - 2b) = 4ab Determine the nature of the roots of the equation.…
- 2x^2 - 10x + k = 0 Find the values of k for which the roots are real and equal…
- 12x^2 + 4kx + 3 = 0 Find the values of k for which the roots are real and equal…
- x^2 + 2k (x - 2) + 5 = 0 Find the values of k for which the roots are real and…
- (k + 1) x^2 - 2 (k - 1) x + 1 = 0 Find the values of k for which the roots are…
- Show that the roots of the equation x^2 + 2(a + b) x + 2 (a^2 + b^2) = 0 are…
- Show that the roots of the equation 3p^2 x^2 - 2pqx + q^2 = 0 are not real.…
- If the roots of the equation (a^2 + b^2) x^2 - 2 (ac + bd) x + c^2 + d^2 = 0,…
- Show that the roots of the equation (x - a) (x - b) + (x - b) (x - c) + (x - c)…
- If the equation (1 + m^2) x^2 + 2mcx + c^2 - a^2 = 0 has equal roots, then prove…
Exercise 3.18- x^2 - 6x + 5 = 0 Find the sum and the product of the roots of the following…
- kx^2 + ax + pk = 0 Find the sum and the product of the roots of the following…
- 3x^2 - 5x = 0 Find the sum and the product of the roots of the following…
- 8x^2 - 25 = 0 Find the sum and the product of the roots of the following…
- Form a quadratic equation whose roots are (i) 3, 4 (ii) 3 + √7, 3 - √7 (iii) 4 +…
- If α and β are the roots of the equation 3x^2 - 5x + 2= 0, then find the values…
- If α and β are the roots of the equation 3x^2 - 6x + 4 = 0, find the value of…
- If α, β are roots of 2x^2 - 3x - 5 = 0, from an equation whose roots are α^2 and…
- If α, β are roots of x^2 - 3x + 2 = 0, form a quadratic equation whose roots are…
- If α and β are roots of x^2 - 3x-1 = 0, then form a quadratic equation whose…
- If α and β are roots of 3x^2 - 6x + 1 = 0, then form a quadratic equation whose…
- Find a quadratic equation whose roots are the reciprocal of the roots of the…
- If one root of the equation 3x^2 + kx - 81 = 0 is the square of the other, find…
- If one root of the equation 2x^2 - ax + 64 = 0 is twice the other, then find…
- If α and β are roots of 5x^2 - px + 1 = 0 and α - β = 1, then find P.…
Exercise 3.19- If the system 6x - 2y = 3, kx - y = 2 has a unique solution, thenA. k = 3 B. k ≠…
- A system of two linear equations in two variables is consistent, if their…
- The system of equations x -4y = 8 , 3x -12y = 24A. has infinitely many solutions…
- If one zero of the polynomial p(x) = (k + 4)x^2 + 13x + 3k is reciprocal of the…
- The sum of two zeros of the polynomial f(x) = 2x^2 + (p + 3)x + 5 is zero, then…
- The remainder when x^2 - 2x + 7 is divided by x + 4 isA. 28 B. 29 C. 30 D. 31…
- The quotient when x^3 - 5x^2 + 7x - 4 is divided by x-1 isA. x^2 + 4x + 3 B. x^2…
- The GCD of (x^3 + 1) and x^4 - 1 isA. x^3 - 1 B. x^3 + 1 C. -(x + 1) D. x-1…
- The GCD of x^2 - 2xy + y^2 and x^4 - y^4 isA. 1 B. x + y C. x - y D. x^2 - y^2…
- The LCM of x^3 - a^3 and (x - a)^2 isA. (x^3 - a^3) (x + a) B. (x^3 - a^3) (x -…
- The LCM of ak,ak + 3, ak + 5 where k inn isA. a k + 9 B. ak C. ak + 6 D. ak + 5…
- The lowest form of the rational expression x^2 + 5x+6/x^2 - x-6 isA. x-3/x+3 B.…
- If a+b/a-b and a^3 - b^3/a^3 + b^3 are the two rational expressions, then their…
- On dividing x^2 - 25/x+3 by x+5/x^2 - 9 is equal toA. (x - 5) (x - 3) B. (x -…
- If a^3/a-b is added with b^3/b-a , then the new expression isA. a^2 + ab + b^2…
- The square root of 49 (x^2 - 2x + y^2)^2 isA. 7 |x - y| B. 7(x + y) (x - y) C.…
- The square root of x^2 + y^2 + z^2 - 2xy + 2yz - 2zxA. |x + y - z| B. |x - y +…
- The square root of 121 x^4 y^8 z^6 (l - m)^2 isA. 11x^2 y^4 z^4 |l - m| B.…
- If ax^2 + bx + c = 0 has equal roots, then c is equalA. b^2/2a B. b^2/4a C. -…
- If x^2 + 5kx + 16 = 0 has no real roots, thenA. k 8/5 B. k - 8/5 C. - 8/5 k 8/5…
- A quadratic equation whose one root is 3 isA. x^2 - 6x - 5 = 0 B. x^2 + 6x - 5…
- The common root of the equation x^2 - bx + c = 0 and x^2 + bx - a = 0 isA.…
- If α, β are the roots of ax^2 + bx + c = 0 a ≠ 0, then the wrong statement isA.…
- If α and β are the roots of ax^2 + bx + c = 0, then one of the quadratic…
- Let b = a + c. Then the equation ax^2 + bx + c = 0 has equal roots, ifA. a = c…
Exercise 3.2- 3x + 4y = 24, 20x - 11y = 47 Solve the following systems of equation using…
- 0.5x + 0.8y = 0.44, 0.8x + 0.6y = 0.5 Solve the following systems of equation…
- 3x/2 - 5y/3 = - 2 , x/3 + y/2 = 13/6 Solve the following systems of equation…
- 5/x - 4/y = - 2 , 2/x + 3/y = 13 Solve the following systems of equation using…
- One number is greater than thrice the other number by 2. If 4 times the smaller…
- The ratio of income of two persons is 9: 7 and the ratio of their expenditure…
- A two digit number is seven times the sum of its digits. The number formed by…
- Three chairs and two tables cost ₹ 700 and five chairs and three tables cost…
- In a rectangle, if the length is increased and the breadth is reduced each by 2…
- A train travelled a certain distance at a uniform speed. If the train had been…
Exercise 3.3- x^2 - 2x - 8 Find the zeros of the following quadratic polynomials and verify…
- 4x^2 - 4x + 1 Find the zeros of the following quadratic polynomials and verify…
- 6x^2 - 3 - 7x Find the zeros of the following quadratic polynomials and verify…
- 4x^2 + 8x Find the zeros of the following quadratic polynomials and verify the…
- x^2 - 15 Find the zeros of the following quadratic polynomials and verify the…
- 3x^2 - 5x + 2 Find the zeros of the following quadratic polynomials and verify…
- 2x^2 - 2√2 x + 1 Find the zeros of the following quadratic polynomials and…
- x^2 + 2x - 143 Find the zeros of the following quadratic polynomials and verify…
- 3, 1 Find a quadratic polynomial each with the given numbers as the sum and…
- 2, 4 Find a quadratic polynomial each with the given numbers as the sum and…
- 0, 4 Find a quadratic polynomial each with the given numbers as the sum and…
- root 2 , 1/5 Find a quadratic polynomial each with the given numbers as the sum…
- 1/3 , 1 Find a quadratic polynomial each with the given numbers as the sum and…
- 1/2 ,-4 Find a quadratic polynomial each with the given numbers as the sum and…
- 1/2 , - 1/3 Find a quadratic polynomial each with the given numbers as the sum…
- root 3 , 2 Find a quadratic polynomial each with the given numbers as the sum…
Exercise 3.4- x^3 + x^2 - 3x + 5) ÷ (x - 1) Find the quotient and remainder using synthetic…
- (3x^3 - 2x^2 + 7x - 5) ÷ (x + 3) Find the quotient and remainder using…
- (3x^3 + 4x^2 - 10x + 6) ÷ (3x - 2) Find the quotient and remainder using…
- (3x^3 - 4x^2 - 5) ÷ (3x + 1) Find the quotient and remainder using synthetic…
- (8x^4 - 2x^2 + 6x + 5) ÷ (4x + 1) Find the quotient and remainder using…
- (2x^4 - 7x^3 - 13x^2 + 63x - 48) ÷ (2x - 1) Find the quotient and remainder…
- If the quotient on dividing x^4 + 10x^3 + 35x^2 + 50x + 29 by x + 4 is x^3 -…
- If the quotient on dividing, 8x^4 - 2x^2 + 6x - 7 by 2x + 1 is 4x^3 + px^2 - qx…
Exercise 3.5- x3 - 2x2 - 5x + 6 Factorize each of the following polynomials.
- 4x3 - 7x + 3 Factorize each of the following polynomials.
- x3 - 23x2 + 142x - 120 Factorize each of the following polynomials.…
- 4x3 - 5x2 + 7x - 6 Factorize each of the following polynomials.
- x3 - 7x + 6 Factorize each of the following polynomials.
- x^3 + 13x2 + 32x + 20 Factorize each of the following polynomials.…
- 2x3 - 9x^2 + 7x + 6 Factorize each of the following polynomials.
- x3 - 5x + 4 Factorize each of the following polynomials.
- x3 - 10x2 - x + 10 Factorize each of the following polynomials.
- 2x3 + 11x2 - 7x - 6 Factorize each of the following polynomials.
- x3 + x^2 + x - 14 Factorize each of the following polynomials.
- x3 - 5x2 - 2x + 24 Factorize each of the following polynomials.
Exercise 3.6- 7x2 yz4, 21x2 y^5 z^3 Find the greatest common divisor of
- x2y, x3y, x2y^2 Find the greatest common divisor of
- 25bc^4 d^3 , 35b2c^5 , 45c^3 d Find the greatest common divisor of…
- 35x^5 y^3 z^4 , 49x^2 yz^3 , 14xy^2 z^2 Find the greatest common divisor of…
- x^3 - x^2 + x - 1, x^4 - 1 Find the GCD of the following
- c2 - d^2 , - c(c - d) Find the GCD of the following
- x4 - 27a3 x,(x - 3a)^2 Find the GCD of the following
- m2 - 3m - 18, m^2 + 5m + 6 Find the GCD of the following
- x2 + 14x + 33, x3 + 10x2 - 11x Find the GCD of the following
- x2 + 3xy + 2y^2 , x^2 + 5xy + 6y2 Find the GCD of the following
- 2x2 - x - 1,4x2 + 8x + 3 Find the GCD of the following
- x2 - x - 2,x2 + x - 6,3x2 - 13x + 14 Find the GCD of the following…
- 24(6x4 - x^3 - 2x2),20(2x6 + 3x5 + x^4) Find the GCD of the following…
- (a - 1)^5 (a + 3)^2 ,(a - 2)^2 (a - 1)^3 (a + 3)^4 Find the GCD of the…
- x^3 - 9x^2 + 23x - 15, 4x^2 - 16x + 12 Find the GCD of the following pairs of…
- 3x^3 + 18x^2 + 33x + 18, 3x^2 + 13x + 10 Find the GCD of the following pairs of…
- 2x^3 + 2x^2 + 2x + 2, 6x^3 + 12x^2 + 6x + 12 Find the GCD of the following…
- x^3 - 3x^2 + 4x - 12, x^4 + x^3 + 4x^2 + 4x Find the GCD of the following pairs…
Exercise 3.7- x^3 y^2 , xyz Find the LCM of the following
- 3x^2 yz, 4x^3 y^3 Find the LCM of the following
- a^2 bc, b^2 ca, c^2 ab Find the LCM of the following
- 66a^4 b^2 c^3 , 44a^3 b^4 c^2 , 24a^2 b^3 c^4 Find the LCM of the following…
- am + 1, am + 2, am + 3 Find the LCM of the following
- x^2 y + xy^2 , x^2 + xy Find the LCM of the following
- 3(a - 1), 2(a - 1)^2 , (a^2 - 1) Find the LCM of the following
- 2x^2 - 18, 5x^2 y + 15xy^2 , x^3 + 27y^3 Find the LCM of the following…
- (x + 4)^2 (x - 3)^3 , (x - 1)(x + 4)(x - 3)^2 Find the LCM of the following…
- 10(9x^2 + 6xy + y^2), 12(3x^2 - 5xy - 2y^2), 14(6x^4 + 2x^3) Find the LCM of…
Exercise 3.8- x^2 - 5x + 6, x^2 + 4x - 12 whose GCD is x - 2. Find the LCM of each pair of…
- x^4 + 3 x^3 + 6 x^2 + 5x + 3, x^4 + 2 x^2 + x + 2 whose GCD is x^2 + x + 1 Find…
- 2x^3 + 15x^2 + 2x - 35, x^3 + 8x^2 + 4x - 21 whose GCD is x + 7. Find the LCM…
- 2x^3 - 3x^2 - 9x + 5, 2x^4 - x^3 - 10x^2 - 11x + 8 whose GCD is 2x - 1 Find the…
- (x + 1)^2 (x + 2)^2 , (x + 1) (x + 2), (x + 1)^2 (x + 2) Find the other…
- (4x + 5)^3 (3x - 7)^3 , (4x + 5) (3x - 7)^2 , (4x + 5)^3 (3x - 7)^2 Find the…
- (x^4 - y^4) (x^4 + x^2 y^2 + y^4), x^2 - y^2 , x^4 - y^4 . Find the other…
- (x^3 - 4x) (5x + 1), (5 x^2 + x), (5 x^3 - 9 x^2 - 2x). Find the other…
- (x - 1) (x - 2) (x^2 - 3x + 3), (x - 1), (x^3 - 4 x^2 + 6x - 3). Find the other…
- 2(x + 1) (x^2 - 4), (x + 1), (x + 1) (x - 2). Find the other polynomial q(x) of…
Exercise 3.9- 6x^2 + 9x/3x^2 - 12x Simplify the following into their lowest forms.…
- x^2 + 1/x^4 - 1 Simplify the following into their lowest forms.
- x^3 - 1/x^2 + x+1 Simplify the following into their lowest forms.…
- x^3 - 27/x^2 - 9 Simplify the following into their lowest forms.
- x^4 + x^2 + 1/x^2 + x+1 (Hint : x^4 + x^2 + 1 =(x^2 + 1)^2 - x^2) Simplify the…
- x^3 + 8/x^4 + 4x^2 + 16 Simplify the following into their lowest forms.…
- 2x^2 + x-3/2x^2 + 5x+3 Simplify the following into their lowest forms.…
- 2x^4 - 162/(x^2 + 9) (2x-6) Simplify the following into their lowest forms.…
- (x-3) (x^2 - 5x+4)/(x-4) (x^2 - 2x-3) Simplify the following into their lowest…
- (x-8) (x^2 - 5x-50)/(x+10) (x^2 - 13x+40) Simplify the following into their…
- 4x^2 + 9x+5/8x^2 + 6x-5 Simplify the following into their lowest forms.…
- (x-1) (x-2) (x^2 - 9x+14)/(x-7) (x^2 - 3x+2) Simplify the following into their…
- x + 2y = 7, x - 2y = 1 Solve the following system of equation by elimination…
- 3x + y = 8, 5x + y = 10 Solve the following system of equation by elimination…
- x + y/2 = 4 , x/3 + 2y = 5 Solve the following system of equation by elimination…
- 11x - 7y = xy, 9x - 4y = 6xy Solve the following system of equation by…
- 3/x + 5/y = 20/xy , 2/x + 5/y = 15/xy , x not equal 0 , y not equal 0 Solve the…
- 8x - 3y = 5xy, 6x - 5y = - 2xy Solve the following system of equation by…
- 13x + 11y = 70, 11x + 13y = 74 Solve the following system of equation by…
- 65x - 33y = 97, 33x - 65y = 1 Solve the following system of equation by…
- 15/x + 2/y = 17 , 1/x + 1/y = 36/5 , x not equal 0 , y not equal 0 Solve the…
- 2/x + 2/3y = 1/6 , 3/x + 2/y = 0 , x not equal 0 , y not equal 0 Solve the…
- Multiply the following and write your answer in lowest terms. (i) x^2 - 2x/x+2 x…
- x/x+1 / x^2/x^2 - 1 Divide the following and write your answer in lowest terms.…
- x^2 - 36/x^2 - 49 / x+6/x+7 Divide the following and write your answer in…
- x^2 - 4x-5/x^2 - 25 / x^2 - 3x-10/x^2 + 7x+10 Divide the following and write…
- x^2 + 11x+28/x^2 - 4x-77 / x^2 + 7x+12/x^2 - 2x-15 Divide the following and…
- 2x^2 + 13x+15/x^2 + 3x-10 / 2x^2 - x-6/x^2 - 4x+4 Divide the following and…
- 3x^2 - x-4/9x^2 - 16 / 4x^2 - 4/3x^2 - 2x-1 Divide the following and write your…
- 2x^2 + 5x-3/2x^2 + 9x+9 / 2x^2 + x-1/2x^2 + x-3 Divide the following and write…
- Simplify the following as a quotient of two polynomials in the simplest form.…
- Which rational expression should be added to x^3 - 1/x^2 + 2 to get 3x^3 + 2x^2…
- Which rational expression should be subtracted from 4x^3 - 7x^2 + 5/2x-1 to get…
- If p = x/x+y , q = y/x+y then find 1/p-q - 2q/p^2 - q^2
- 196a^6 b^8 c^10 Find the square root of the following
- 289 (a-b)^4 (b-c)^6 Find the square root of the following
- (x + 11)^2 -44x Find the square root of the following
- (x-y)^2 + 4xy Find the square root of the following
- 121x^8 y^6 */* 81x^4 y^8 Find the square root of the following
- 64 (a+b)^4 (x-y)^8 (b-c)^6/25 (x+y)^4 (a-b)^6 (b+c)^10 Find the square root of…
- 16x^2 -24x + 9 Find the square root of the following:
- (x^2 - 25)(x^2 + 8x + 15)(x^2 -2x-15) Find the square root of the following:…
- 4x^2 + 9y^2 + 25z^2 -12xy + 30yz-20zx Find the square root of the following:…
- x^4 + 1/x^4 + 2 Find the square root of the following:
- (6x^2 + 5x -6) (6x^2 -x-2)(4x^2 + 8x + 3) Find the square root of the…
- (2x^2 -5x + 2) (3x^2 -5x-2) (6x^2 - x -1) Find the square root of the…
- x^4 -4x^3 + 10x^2 -12x + 9 Find the square root of the following polynomials by…
- 4x^4 + 8x^3 + 8x^2 + 4x + 1 Find the square root of the following polynomials…
- 9 x^4 -6 x^3 + 7 x^2 -2x + 1 Find the square root of the following polynomials…
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Exercise 3.1
Question 1.Solve the following system of equation by elimination method.
x + 2y = 7, x – 2y = 1
Answer:The given equations are
x + 2y = 7 … (1)
x – 2y = 1 … (2)
Adding (1) and (2),
⇒ x + 2y + x – 2y = 7 + 1
⇒ 2x = 8
⇒ x = 4
Substituting x = 4 in (1),
⇒ 4 + 2y = 7
⇒ 2y = 7 – 4 = 3
⇒ y = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYElEQVQYV42OORKAMAwDZQj3HUIgAf7/TYxTMECTbbawPBLwZckpaYFzmGATLVefifexuzVTKgZW1eOoNTwbi5K/WCgQG39yW0NUgHt/+2BKiVk+MybIsRzvrO463vnmAs3JA2tWswYWAAAAAElFTkSuQmCC)
∴ (4,
) is the solution to the given system.
Question 2.Solve the following system of equation by elimination method.
3x + y = 8, 5x + y = 10
Answer:The given equations are
3x + y = 8 … (1)
5x + y = 10 … (2)
Here, the coefficients of y in both equations are numerically equal.
Subtracting (2) from (1),
⇒ (3x + y) – (5x + y) = 8 – 10
⇒ 3x + y – 5x – y = – 2
⇒ – 2x = – 2
⇒ x = 1
Substituting x = 1 in (1),
⇒ 3 (1) + y = 8
⇒ y = 8 – 3
⇒ y = 5
∴ (1, 5) is the solution to the given system.
Question 3.Solve the following system of equation by elimination method.
![](data:image/png;base64,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)
Answer:The given equations are
x +
= 4 … (1)
+ 2y = 5 … (2)
(1) becomes 2x + y = 8
(2) becomes x + 6y = 15
Now, (2) × 2 – (1)
⇒ 2x + 12y – (2x + y) = 30 – 8
⇒ 2x + 12y – 2x – y = 22
⇒ 11y = 22
⇒ y = 2
Substituting y = 2 in (1),
⇒ 2x + 2 = 8
⇒ 2x = 6
⇒ x = 3
∴ (3, 2) is the solution to the given system.
Question 4.Solve the following system of equation by elimination method.
11x – 7y = xy, 9x – 4y = 6xy
Answer:The given equations are
11x – 7y = xy … (1)
9x – 4y = 6xy … (2)
Dividing both sides of the equation by xy,
⇒
–
= 1 i.e.
+
= 1 … (3)
⇒
–
= 6 i.e.
+
= 6 … (4)
Let a =
and b =
.
Equations (3) and (4) become
⇒ – 7a + 11b = 1 … (5)
⇒ – 4a + 9y = 6 … (6)
Now, (6) × 7 – (5) × 4
⇒ – 28a + 63b – ( – 28a + 44b) = 42 – 4
⇒ – 28a + 63b + 28a – 44b = 38
⇒ 19b = 38
⇒ b = 2
Substituting b = 2 in (5),
⇒ – 7a + 11(2) = 1
⇒ – 7a = 1 – 22 = – 21
⇒ a = 3
When a = 3, we have
= 3. Thus, x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bjLFCZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAW0lEQVQYV42MSQ6AIBAEG1TcV1BUlP8/U2YOHiQa61JJp2aACD9ntNmyYQPLt71Jd6pEoIrfvSxUB373d2gLIVvADxNWqXl2OfsYO5IRCRvYVI+z1nDBsIrvHlxvGQNJwgEIiAAAAABJRU5ErkJggg==)
When b = 2, we have
= 2. Thus, y = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAWUlEQVQYV2NgwACSAiwgMSE2bjDNwCCIn5bkZxYFqWIEAg5M43CIgFQDAdHq4QpFuBkZ2RkYJLh4GYSZ+MDC4qwQWhBivTBQGsSDUGJASoyHQYITZB0Pun0AG8sCbODE2DAAAAAASUVORK5CYII=)
∴ (
,
) is the solution for the given system.
Question 5.Solve the following system of equation by elimination method.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATcAAAA7CAYAAAAel839AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAaoSURBVHja7Z09btxIEIV5AB/ABqYVrQ4ww9EVRoTgA1giHK3hQDAIGI4NSpmAhS+gzJkdbLzBOt1EN3CgG/gOazUpcvg7ZLOruovDF3yJMZaq+3U/dhV7SsHt7W0AAADHBiYBAABzAwAAmBsAAMDcAAAA5gYAAHbmliaXpxsVPARB8H+G2jxcJumpj0DT9O2rMAgey1gy1K8oSdde4umYmyhOLro+m8TRRflZ/TnCmE3iWKJOSXK53u3Cz0oFP9Xuw6e5xA0YzS1NonUYRn+9v7t7Ud9E4ePbNH3lOtAPO/WpvvCeCK+++jG2aK2C4FcrnifC+OZ19bM3cfg62yTPhqONTv/f5ue441iiTtncK/VTG5uO45C5SYobMJrb3d37F7tw97kwtuZmcr1xsqfqyfl9Mx4f6LmJVsEPFcVp1fj1v+WbYm/+xWmguanyjWT3kDCJY4k6tR8w/eYmNW7AmJb2nRRcm9tVGHzNFqfa/uMz5SpPtLvkXZ/ZVFOZ3MTaqU0xj4dOEpRxLFEnE3OTGjdwaG7ZInB8VO9MvdTmwUdqPG5z5yem0mRW0Y/WCbio7zDNZTUO6HTY3Oa0vgCDuemibFZv20Z/+zq6X19f/xFHZx/3C9FP7W8oTSwMqzSwDnM7ZHzUcUCn4ZPbHNYXIDa3fYqzf6qp8OqLz9pELSZBBd/8BLB/4pcngo4Y92Og30DNOKDTOHOTvr4A08lNC57EZ2+Kp5pNrYhkA5ev7eU8Xa9C9aVai/Rlbs04oJOZuUldX4DJ3Nr1Cf+i9xXrfcXS3DiH6mqHUlbqOCTMjW+dTM1N2voCDswtPxm4L1ZLTr/0xjn41tLRC4W+OKSl63MxNynrCzg0t+yJxlAIn7RgPceRX3SO08NP//aDgPpKzVAcS9dpirlJiBs4NLfy4qiA1EfXlnynOvrlSttoLk+Lb3b03WejuMRrEseSdZpqbhLiBgzmlm0MnTZl3yVN1oWxacG7NhL35jgJz++L77QWcZzFyRtfk1Z9udJFdRM1azfF/6U4tZnEsUSdmubWVQaQHDdgMDct8Hl4cl/bKJ5ubzc3cHUh+jK2PjPp+7J1bQxEzQemxLEknarp/yHTlxg38JCWAgDmCWdnGqljgfAAHDmcnWkkjwXiA3DEcHamkT4WZ4FJvGAKoNNcyW4uKPV9KL3k7EwjfSzYNAA6MZ0wmi8x6v9u98Kn7wpQyzQILpK3xvP882rfw7W4E8g1FmwaAJ2YTiNFn7jqhsxrRuGjzRvZsScdys405fWwRvqnf8cm2Pw7dTycY1nEpulsIS3gbhjM7bh1qp5sdMF77EaunmjGzkdXQZ26eUOhT/V3jV0vPsaCEwGATk5S1PBxu1XfbN9Q6p+3VdtvYwyJ2tya6Z+pWbseS+1DY521ic0TuE9sijgomTo31LFT/nzo5Cb28tRC8F1Vk4cPR2eaPNXO64WZ4ajo+9QxcY8FJwIAnZjR5rZarf6zTadNTjpDdbWpnWmqbyZt1oqLsaCWg5obdOJMS59PN8lN8tL0D/fY1qn2c0rXmaY0GbV50Gm25LHgRACgE2O9rXo6sWn2OjUF5OhMY5tmuxoLNo1Aqn+ez3UDSuhEo1Ffe7DydPpkDCbXJ2zax4/pTGOy5mxbn3GPBeYm/Inv648rQycajco7bpUUup12j0tRx1xyHWKoM43pmpva787FWJybG5hGHIYpWl1DI0njyQr4J+f30rsUY3FKPh30/FV5AI18jmcup3ssUIHomkYcRX/6aA4KlqnR0HjqKfU8uolgoQIABslqZbrORdRJGuYGAAAwNwAA6DE3F32oTODuI2UdT+/8xBeccUvSSZpGknQCwk5unH2oJi1Upj5SUxk7P9xxS9JJmkaSdALC0lLbPlTU2PSR4to4Y+aHO25JOknTSJJOQFjNjboPFUmaQdRHijb16Z8fF3FL0UmiRpJ0AoLMbX+El1F3oOwjRcWY+XERtxSdJGokSScgzNwo+lDR1HVo+khRb5qh+XERtxSdJGokSScgJS216EPFWj8x7CPFupFHzA933JJ0kqaRJJ2AEHOj7EN1jOmX6fxwxS1RJ0mlDCk6ASHmRt2HiuNkIOENnMn8cMQtVScJGknSCQgyN8o+VBxM7SNF9/unzQ913JJ18q2RJJ2AwJqbRObSR+pY4l7aWJekE8xNGHN9i7Wkt29zHiveksLcnC+4ufWRmnPcSxvrknSCuQljjn2k5hz30sa6JJ1gbgAAAHMDAACYGwAAwNwAAADmBgAADX4D2y1NK9iOgOUAAAAASUVORK5CYII=)
Answer:The given equations are
+
=
… (1)
+
=
… (2)
Multiplying both sides of the equation with xy,
3y + 5x = 20 i.e. 5x + 3y = 20 … (3)
2y + 5x = 15 i.e. 5x + 2y = 15 … (4)
Subtracting (4) from (3),
⇒ 5x + 3y – (5x + 2y) = 20 – 15
⇒ 5x + 3y – 5x – 2y = 5
⇒ y = 5
Substituting y = 5 in (3),
⇒ 5x + 3 (5) = 20
⇒ 5x = 20 – 15 = 5
⇒ x = 1
∴ (1, 5) is the solution for the given system.
Question 6.Solve the following system of equation by elimination method.
8x – 3y = 5xy, 6x – 5y = – 2xy
Answer:The given equations are
8x – 3y = 5xy … (1)
6x – 5y = – 2xy … (2)
Dividing both sides of the equation by xy,
⇒
–
= 5 i.e.
+
= 5 … (3)
⇒
–
= – 2 i.e.
+
= – 2 … (4)
Let a =
and b =
.
Equations (3) and (4) become
⇒ – 3a + 8b = 5 … (5)
⇒ – 5a + 6y = – 2 … (6)
Now, (5) × 5 – (6) × 3
⇒ – 15a + 40b – ( – 15a + 18b) = 25 – ( – 6)
⇒ – 15a + 40b + 15a – 18b = 31
⇒ 22b = 31
⇒ b = ![](data:image/png;base64,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)
Substituting b =
in (5),
⇒ – 3a + 8(
) = 5
⇒ – 3a = 5 –
= ![](data:image/png;base64,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)
⇒ a = ![](data:image/png;base64,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)
When a =
, we have
=
. Thus, x = ![](data:image/png;base64,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)
When b =
, we have
=
. Thus, y = ![](data:image/png;base64,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)
∴ (
,
) is the solution for the given system.
Question 7.Solve the following system of equation by elimination method.
13x + 11y = 70, 11x + 13y = 74
Answer:The given equations are
13x + 11y = 70 … (1)
11x + 13y = 74 … (2)
Adding (1) and (2),
⇒ 24x + 24y = 144
Dividing by 24,
⇒ x + y = 6 … (3)
Subtracting (2) from (1),
⇒ 2x + ( – 2y) = – 4
Dividing by 2,
⇒ x – y = – 4 … (4)
Solving (3) and (4),
⇒ x + y + (x – y) = 6 – 4
⇒ 2x = 2
⇒ x = 1
Substituting x = 1 in (3),
⇒ 1 + y = 6
⇒ y = 5
∴ (1, 5) is the solution to the given system.
Question 8.Solve the following system of equation by elimination method.
65x – 33y = 97, 33x – 65y = 1
Answer:The given equations are
65x – 33y = 97 … (1)
33x – 65y = 1 … (2)
Adding (1) and (2),
⇒ 98x – 98y = 98
Dividing by 98,
⇒ x – y = 1 … (3)
Subtracting (1) and (2),
⇒ 32x + 32y = 96
Dividing by 32,
⇒ x + y = 3 … (4)
Solving (3) and (4),
⇒ x – y + (x + y) = 1 + 3
⇒ 2x = 4
⇒ x = 2
Substituting x = 2 in (4),
⇒ 2 + y = 3
⇒ y = 1
∴ (2, 1) is the solution to the given system.
Question 9.Solve the following system of equation by elimination method.
![](data:image/png;base64,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)
Answer:The given equations are
+
= 17 … (1)
+
=
… (2)
Let a =
and b =
.
⇒ 15a + 2b = 17 … (3)
⇒ a + b =
… (4)
Now, (3) – (4) × 2
⇒ 15a + 2b – (2a + 2b) = 17 – ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAgCAMAAAD68tKbAAAAAXNSR0IArs4c6QAAAFFQTFRFAAAAAAAAAAA6AGa2OgAAOgA6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kDqQkLbbkNv/tmY625A627Zm29u22////7Zm/9uQ//+2///bdKdOswAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAgUlEQVQoU62Q2xLCIAxEF2u11WJFaIH8/4eacHFQx7EPPW9h2c1kgT+QVkwHLBelTgDNK+hmEMc73MEkt+d3IfRpFjlhz1lmt+Dytyrb4iqyhPjpJcdB9vK8M5LasHP6jzjLG0tXuZiPu75m/l/aqYnh+O4hncutjbY6PXjdddNpT5RuBJPm6DidAAAAAElFTkSuQmCC)
⇒ 15a + 2b – 2a – 2b = ![](data:image/png;base64,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)
⇒ 13a = ![](data:image/png;base64,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)
⇒ a = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAEJQTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOma2OpDbZjoAZrbbZrb/kDoAkDo6kNv/tmYAtmY625A625Bm2////9uQ///b7M5aaQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAU0lEQVQYV2NgwACiAiwgMSE2TjDNwCCInxblZxYGqWIEAg5M43CIgFQDAdHq4QpB9jDxAe3jhroORgPF2SFCIqwQMVEeDgZRXj4wX1QAqI0LwzoABl4CTEIhdkMAAAAASUVORK5CYII=)
Substituting a =
in (4),
⇒
+ b = ![](data:image/png;base64,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)
⇒ b =
= 7
When a =
,
=
. Thus, x = 5.
When b = 7,
= 7. Thus, y = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAADlQTFRFAAAAAAAAAGaQAGa2OmZmOma2OpDbZjoAkDoAkDo6kDqQkNv/tmYAtmY625A625Bm2////7Zm///bwySdCgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAASklEQVQYV2NgwABCvMwgMX4WdjDNwMCHnxbiYRIAqWIEAjZM43CIgFQDAdHqYQqFOEDamBmEuAQYhDi5wcKCrGAKzoW4FwcX1UYA4/4B9peiBbYAAAAASUVORK5CYII=)
∴ (5,
) is the solution to the given system.
Question 10.Solve the following system of equation by elimination method.
![](data:image/png;base64,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)
Answer:The given equations are
+
=
… (1)
+
= 0 … (2)
Let a =
and b =
.
⇒ 2a +
b =
… (3)
⇒ 3a + 2b = 0 … (4)
Now, (3) × 3 – (4) × 2
⇒ 6a + 2b – (6a + 4b) = 1/2 – 0
⇒ 6a + 2b – 6a – 4b = 1/2
⇒ – 2b = 1/2
⇒ b = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAgCAMAAADDlWPAAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAABmADqQAGaQAGa2OgAAOjoAOmZmOma2OpDbZgAAZjoAZjpmZrbbZrb/kDoAkDo6kGaQtmYAtmY6tv//25A625Bm29uQ2////7Zm/9uQ//+2///byJLbuQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAb0lEQVQoU2NgIA3IirCiahDn4EETYWAQwyEiw80IBvxAI3CpQTJdVphFEsUyMZBmLtIcTJpqiPuQAGnaqaRamg0UQsghIciOJiLGJ4AqIsUpiyoiwysqK8AnhBSC4OBjZEYNUzRdQEsl2JhEifYIAMsrA/O8VZaYAAAAAElFTkSuQmCC)
Substituting b =
in (4),
⇒ 3a + 2(
) = 0
⇒ 3a = 1/2
⇒ a = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAADqQAGaQAGa2OgAAOjo6OmZmOmaQOma2OpC2OpDbZgA6ZjoAZjo6kDoAkNv/tmYAtmY625A625Bm27Zm29u229vb2////9uQ/9u2//+2///bmbpMdwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYklEQVQYV2NgwAAyQiwgMRF2XjDNwCCMn5YRZJYAqWIEAm5M43CIgFQDAdHqkRSK8nEALZQR4gJZC3OdDD8HHxtQXJKVU1yMn4VBmkcAKMfEIMPPKSED5DNI8TIycoJ1IAMAZUYDtd9yyncAAAAASUVORK5CYII=)
When a =
,
=
. Thus, x = 6.
When b =
,
=
. Thus, y = – 4
∴ (6, – 4) is the solution to the given system.
Solve the following system of equation by elimination method.
x + 2y = 7, x – 2y = 1
Answer:
The given equations are
x + 2y = 7 … (1)
x – 2y = 1 … (2)
Adding (1) and (2),
⇒ x + 2y + x – 2y = 7 + 1
⇒ 2x = 8
⇒ x = 4
Substituting x = 4 in (1),
⇒ 4 + 2y = 7
⇒ 2y = 7 – 4 = 3
⇒ y =
∴ (4,) is the solution to the given system.
Question 2.
Solve the following system of equation by elimination method.
3x + y = 8, 5x + y = 10
Answer:
The given equations are
3x + y = 8 … (1)
5x + y = 10 … (2)
Here, the coefficients of y in both equations are numerically equal.
Subtracting (2) from (1),
⇒ (3x + y) – (5x + y) = 8 – 10
⇒ 3x + y – 5x – y = – 2
⇒ – 2x = – 2
⇒ x = 1
Substituting x = 1 in (1),
⇒ 3 (1) + y = 8
⇒ y = 8 – 3
⇒ y = 5
∴ (1, 5) is the solution to the given system.
Question 3.
Solve the following system of equation by elimination method.
Answer:
The given equations are
x + = 4 … (1)
+ 2y = 5 … (2)
(1) becomes 2x + y = 8
(2) becomes x + 6y = 15
Now, (2) × 2 – (1)
⇒ 2x + 12y – (2x + y) = 30 – 8
⇒ 2x + 12y – 2x – y = 22
⇒ 11y = 22
⇒ y = 2
Substituting y = 2 in (1),
⇒ 2x + 2 = 8
⇒ 2x = 6
⇒ x = 3
∴ (3, 2) is the solution to the given system.
Question 4.
Solve the following system of equation by elimination method.
11x – 7y = xy, 9x – 4y = 6xy
Answer:
The given equations are
11x – 7y = xy … (1)
9x – 4y = 6xy … (2)
Dividing both sides of the equation by xy,
⇒ –
= 1 i.e.
+
= 1 … (3)
⇒ –
= 6 i.e.
+
= 6 … (4)
Let a = and b =
.
Equations (3) and (4) become
⇒ – 7a + 11b = 1 … (5)
⇒ – 4a + 9y = 6 … (6)
Now, (6) × 7 – (5) × 4
⇒ – 28a + 63b – ( – 28a + 44b) = 42 – 4
⇒ – 28a + 63b + 28a – 44b = 38
⇒ 19b = 38
⇒ b = 2
Substituting b = 2 in (5),
⇒ – 7a + 11(2) = 1
⇒ – 7a = 1 – 22 = – 21
⇒ a = 3
When a = 3, we have = 3. Thus, x =
When b = 2, we have = 2. Thus, y =
∴ (,
) is the solution for the given system.
Question 5.
Solve the following system of equation by elimination method.
Answer:
The given equations are
+
=
… (1)
+
=
… (2)
Multiplying both sides of the equation with xy,
3y + 5x = 20 i.e. 5x + 3y = 20 … (3)
2y + 5x = 15 i.e. 5x + 2y = 15 … (4)
Subtracting (4) from (3),
⇒ 5x + 3y – (5x + 2y) = 20 – 15
⇒ 5x + 3y – 5x – 2y = 5
⇒ y = 5
Substituting y = 5 in (3),
⇒ 5x + 3 (5) = 20
⇒ 5x = 20 – 15 = 5
⇒ x = 1
∴ (1, 5) is the solution for the given system.
Question 6.
Solve the following system of equation by elimination method.
8x – 3y = 5xy, 6x – 5y = – 2xy
Answer:
The given equations are
8x – 3y = 5xy … (1)
6x – 5y = – 2xy … (2)
Dividing both sides of the equation by xy,
⇒ –
= 5 i.e.
+
= 5 … (3)
⇒ –
= – 2 i.e.
+
= – 2 … (4)
Let a = and b =
.
Equations (3) and (4) become
⇒ – 3a + 8b = 5 … (5)
⇒ – 5a + 6y = – 2 … (6)
Now, (5) × 5 – (6) × 3
⇒ – 15a + 40b – ( – 15a + 18b) = 25 – ( – 6)
⇒ – 15a + 40b + 15a – 18b = 31
⇒ 22b = 31
⇒ b =
Substituting b = in (5),
⇒ – 3a + 8() = 5
⇒ – 3a = 5 – =
⇒ a =
When a =, we have
=
. Thus, x =
When b =, we have
=
. Thus, y =
∴ (,
) is the solution for the given system.
Question 7.
Solve the following system of equation by elimination method.
13x + 11y = 70, 11x + 13y = 74
Answer:
The given equations are
13x + 11y = 70 … (1)
11x + 13y = 74 … (2)
Adding (1) and (2),
⇒ 24x + 24y = 144
Dividing by 24,
⇒ x + y = 6 … (3)
Subtracting (2) from (1),
⇒ 2x + ( – 2y) = – 4
Dividing by 2,
⇒ x – y = – 4 … (4)
Solving (3) and (4),
⇒ x + y + (x – y) = 6 – 4
⇒ 2x = 2
⇒ x = 1
Substituting x = 1 in (3),
⇒ 1 + y = 6
⇒ y = 5
∴ (1, 5) is the solution to the given system.
Question 8.
Solve the following system of equation by elimination method.
65x – 33y = 97, 33x – 65y = 1
Answer:
The given equations are
65x – 33y = 97 … (1)
33x – 65y = 1 … (2)
Adding (1) and (2),
⇒ 98x – 98y = 98
Dividing by 98,
⇒ x – y = 1 … (3)
Subtracting (1) and (2),
⇒ 32x + 32y = 96
Dividing by 32,
⇒ x + y = 3 … (4)
Solving (3) and (4),
⇒ x – y + (x + y) = 1 + 3
⇒ 2x = 4
⇒ x = 2
Substituting x = 2 in (4),
⇒ 2 + y = 3
⇒ y = 1
∴ (2, 1) is the solution to the given system.
Question 9.
Solve the following system of equation by elimination method.
Answer:
The given equations are
+
= 17 … (1)
+
=
… (2)
Let a = and b =
.
⇒ 15a + 2b = 17 … (3)
⇒ a + b = … (4)
Now, (3) – (4) × 2
⇒ 15a + 2b – (2a + 2b) = 17 –
⇒ 15a + 2b – 2a – 2b =
⇒ 13a =
⇒ a =
Substituting a = in (4),
⇒ + b =
⇒ b = = 7
When a =,
=
. Thus, x = 5.
When b = 7, = 7. Thus, y =
∴ (5,) is the solution to the given system.
Question 10.
Solve the following system of equation by elimination method.
Answer:
The given equations are
+
=
… (1)
+
= 0 … (2)
Let a = and b =
.
⇒ 2a + b =
… (3)
⇒ 3a + 2b = 0 … (4)
Now, (3) × 3 – (4) × 2
⇒ 6a + 2b – (6a + 4b) = 1/2 – 0
⇒ 6a + 2b – 6a – 4b = 1/2
⇒ – 2b = 1/2
⇒ b =
Substituting b = in (4),
⇒ 3a + 2() = 0
⇒ 3a = 1/2
⇒ a =
When a =,
=
. Thus, x = 6.
When b = ,
=
. Thus, y = – 4
∴ (6, – 4) is the solution to the given system.
Exercise 3.10
Question 1.Multiply the following and write your answer in lowest terms.
(i)
(ii) ![](data:image/png;base64,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)
(iii)
(iv) ![](data:image/png;base64,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)
(v)
(vi) ![](data:image/png;base64,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)
Answer:(i)![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
=3x required solution
(ii)![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG0AAAAjCAMAAACpdHR4AAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ29u229vb2/+22////7Zm/7aQ/9uQ/9u2//+2///bpp7BswAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACJUlEQVRYR+1XWVvCMBBMEJCIWkHBoxQPDlu8wNL8/5/mJmnSnIWqL/qRt3anM7ubhB0Q+v+LTjHuPO9fZ1O8yVxcLYvRYH+1pniHubhuUBt83RRvCOY3a+N5O8L4HF7RRc9bso1HaDVmeO+SZDK4uV3ndxqSJgO0HQ5Qdjbyqtl4yOrC1qLJRDBKMrQirVlOuh9DjLG+b8UQkHMmlBpqIbyElXEurNQqsk13nUZuB2gSrbdM0VJDQfz5+JR1UotrtUkylLa1DhasRlgTxFrNszBrg2cvPifR+1vCeyHiFRM8KDKU9buezS1GMXTC7SQK4S9nIMSYKr6qk4rs84kmnaVzknJSqtH5sZFMAM86z7PT4kpNkWWkR+fYU92K8BuQsr5qxyeIL5ulx5UanEVxnQ7r0IHdHRCXeddSPLuAtfHd2RwQv96BYuifqt8Vqnc2htrPTBBPsIGzaQCtqd12NjkpZ737jQVlv94tGDr2yvq4xeed61psZ0On/ZCaDU39QDp9RCvIQnMt0ks4ziaNpY+RKYehQk23JfKb/GRmGADhJfhs153QJqomlPw0AOVzMAKQa1u292wul66F0+heQxkK2BmaxA/WEAzYGCARm8zjmimZ43YMIelamJrXm/Cx7Yx1v41hJMw1+qheyKQQrkU0KOA1NEeojlrIlkxnojYrzmTgvXItLJ2QN0Gv5Mj8cxCC0gVcgNhDlRF+AyqLV3P//nToCyFxTB7V1uLjAAAAAElFTkSuQmCC)
We know a2 – b2 = (a-b) (a+b)
So,
![](data:image/png;base64,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)
Also,
x2 + 6x + 8 = x2 + 4x + 2x +8
= x(x+4) +2(x+4)
= (x+2)(x+4)
And
x2 - 5x – 36 = x2 - 9x + 4x – 36
= x(x-9)+4(x-9)
= (x+4)(x-9)
So,
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution
(iii)![](data:image/png;base64,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)
x2 - 3x – 10 = x2 - 5x + 2x – 10
= x(x-5)+2(x-5)
= (x+2)(x-5)
And,
x2 - x - 20 = x2 - 5x + 4x - 20
= x(x-5)+4(x-5)
= (x+4)(x-5)
We know the formula a3 + b3 = (a+b)(a2 + b2 - ab)
So x3 + 8 = x3 + 23
= (x+2)(x2 + 4 - 2x)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXUAAABcCAAAAAC0QnpeAAAABGdBTUEAALGPC/xhBQAAAAFzUkdCAK7OHOkAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAAAAJiS0dEAP+Hj8y/AAAACXBIWXMAAA7DAAAOwwHHb6hkAAAHPklEQVR42u2czbWlKhCFi7WYMGNoAEZgBIRgDuZgEkZhFkZCOj7wBwHBA4oPu5s96Xvu1RI+sShwn4a56P8X5G7AP6lCPYcK9Rwq1HOoUM+hQj2HCvUcKtRzqFDPoUI9hwr1HCrUc6hQz6FCPYcK9Rz6u6l3FKDiuVtxlov6WDWxYVo6PIjA0TxX9ZS8cwMa5gHq5HEfy0GdoS7w5LEhAJjJwTTg+k4EMRYXiR/b4JPiRKrrv/c1BiCx16YwRJ6h60y9QWPguQOIsTQRoPIDJ3V8hJkiKrWAGV/BPkF/+XcGZBIPBbCoqC2kpT6Av++Dean14wSw5IYR2t8RbFFqdCX4bgWL/xrFDMlHtQMc2MhFEyJpqdOLyw/OS+23vUb8Z4TrDmEafmaYfkJXPbsoK87UaZM2w6gR626bm/o6D/brqZcRrjvUpB7sodBFz2LGeot5WurNGq1HMmeL2Ebd5aTew96o9YctAgOZMEd8nTDNDg2R2dWpFskMJ5ou8jll8qeAk5rlyrwBOSWJRhvVpk19FD1MS53CDkDmi9qc/53UCdpLPgJ6BLbcBHw98CmuZBW0feLwPMX0tWiRGOQgyHVLfRQQU+Rp+U/TcajErA7EbKQVQVZFaakDmndozdwTc4Xhot4cv6NLqlER5GhrfhTLrJrmqVIlNURMCRdBYahIVPVP0N5RChzX1rrKor7Mv4mpqytQ6LBKswPo0o7vtIJlbYmKwBHqjtvG9ABmNzje8zkkWSoPQIhrQeptQn2UuqKQ1AYK1c/YnshxKUVFX8f7D6afOkenCvA81nXoNnV5q8JGXL33KA118chE7QJo0EWbz4sqc6xXnrET1UD780GdnMKeqI9IX/nb1MfQtQdLS32MS1Sdvo5qHRsI1IE3cYZRv6ibFqzdFJs6x8bA2PL6HoGTvvqxMNwvtRebSfL6iKrAZ2w9XK90O0zOTXif+l6BzIyu86Eum3qzTkJsO8qsYWTZJlL7Zf/ZigdvlFLUMGJ13/URq+OZrNesZBNaNDbnO/Y+9b0k6WTJysHcUbGoT4DlJsq+OOZGvT7XsvzqgVyl2GUTRNQwG6T+eb0uisVatoSHbmF2QGQnMMhHTjREtGGyUruLOgl7ij2yqQ9rVmHLlC2agXrjjwZ1VRMMG7NWi0DlqZMoAvBF96dGVgnVHjXB2rRZrs4Ah0KhR2FGlu03CtRqhYO67NeDwX6avvDt1FptRe/9CCj5PsxH5dhzjNhGMU9kDyOwR7nyT9KT/XVDWj1zM0LUttmfLee7pBu9H5C2jr4Vof13oDvfmw6P35veiPDGe9PP6u/2CHxVhXoOFeo5VKjnkEkdPqCPNeeV3pWxnkOFeg4V6jn0TepDio32D+ubnl6DOq+jb8GIjdcaNyIklsXnkadXA5TW06tT7xELfPfMuwptu/UT1d6m2BG044LFwl+4OE3CBp/He44cb4De8vRGvI2jUHPltq0V9lME47gwjRBM3WMS5rrj44mnV6pRToJ3PL0cXfiYrJ6twer1Lb/aeT5HMI6zkbmvREg4dY9JWN/IfujpHVCnksErnl529brfSYJtDeq2xvoiMNeM7aHOSESG2VCdItXKYPbQ08txe6TgVzy96OqbFh7q6xjmm5/IF4GFj/URxnjqp7HXH2CeeXorqk98L3h6N9fAJOanVoYmxm1xxlZ+/tUi4o3g9P07qXPMYmbTjSM7MdBI2RhiPL29GJsa9Rc8vfurVI7kD53lpHOR6JV1ar39vgg9uIpbJ/VGnBRJfTMJmwyICv7E08sRM4q8Fzy9dM85AxAxXq0E5CAxHT4DBp0/wuT2I7ioL72OpL6bhA0GVM0wTzy9FZlX6lvp+4KnVxnJZLe1GthojjYUONGvyLwR+Mn1620fx6vDhnEccvgi5Vc1GBx+sQee3tHu9wue3oOZrLLt807jz4BpU9cicOKxpDnGensabb/7g/ThqhgEUQ/w9M5Whknu6dWY1YBsUqfwtW6rPFE/ItQ++6WvXo/KMLpJWGdwQT3G07v+zs7rKT29R0M7MlpfXDlTN5dZe14/R/B/xTIFdb10Nhj483qUp9ei/oKnV9nBBsSXCf6K+oTWInErvMwaRotgHZeaum4SNhj4a5gYT++iXo3ANzy9+2/GxeVKrJnGOryGZVNnr04IeCJYxwVRbyB0F1UzCZsMuL9ej/H0Lp1CsM9Rb3h6+Vr3iqvQ5TuNJmeLullSbGtTRwRv6eGnLniE+tnoMfeaDC7Wpl/z9NYxX7sw1O7P3O0IiVX592G+5unlqLoRaTmRPoyQWPp2x+c9vVHfdtFUYf4wQloZ3+H6vqe3R038/2Y0Ee1rvrciJJbB50/w9E6P35veiJBYv9+bFr2uQj2HCvUcKtRzqFDPoUI9hwr1HCrUc6hQz6FCPYcK9Rwq1HOoUM+hQj2HCvUcKtRzqFDPoUI9hwr1HCrUc6hQz6FCPYcK9Rwq1HPoPxQZKfqOzJ7wAAAAJXRFWHRkYXRlOmNyZWF0ZQAyMDE5LTAxLTIxVDEzOjM4OjEzKzAwOjAwLhYtTwAAACV0RVh0ZGF0ZTptb2RpZnkAMjAxOS0wMS0yMVQxMzozODoxMyswMDowMF9LlfMAAAAASUVORK5CYII=)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution
(iv)![](data:image/png;base64,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)
We know a2 – b2 = (a-b) (a+b)
x2 – 16 = (x-4) (x+4)
x2 – 4 = (x-2) (x+2)
x2 - 3x + 2 = x2 – 2x – x + 2
= x(x-2)-1(x-2)
= (x-1)(x-2)
x2 - 2x - 8 = x2 – 4x + 2x - 8
= x(x-4)+2(x-4)
= (x-4)(x+2)
We know the formula a3 + b3 = (a+b)(a2 + b2 - ab)
So x3 + 64 = x3 + 43
= (x+4)(x2 + 16 - 4x)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADsAAAAsCAMAAADsKDCiAAAAAXNSR0IArs4c6QAAAG9QTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOjqQOmY6OmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmZmZpDbZrb/kDoAkDo6kGY6kNv/tmYAttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bdD90QAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA0UlEQVRIS+2V6w6CMAyFV2/zAt6dbjrZ3N7/GS0xGBMpkGICMewXCXxrzzkdE+Jf132rmdIeGxjfeKxfXi2XxYoD29z2zrwaMecqXiQATI7NJQ5f9sgBg+G9FncAPsS896p5aKn/pz237KW3uOXn6SWbDemBYDG0aWYBTqRjJnFUXYMvoqJRN89INiqsm5BVw1oLkhVeLlYF+jV2UeG2yFqisaoIXHFEytmw29dcuLTesw7prHIuKTaaXBDQZuWXeXlGUeEPJKRQBaNm9mB1e86ecdMMQLenezYAAAAASUVORK5CYII=)
Required solution
(v) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARwAAAAwCAMAAAD0MzUrAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtpBmttv/tv//25A627Zm27aQ2////7Zm/9uQ/9u2//+2///b8AcjjQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEu0lEQVRoQ+1aaXfbIBAUaZu4V9I4Pd00SuomPvT//1+5WdgFFp5yvDzpixNbswzDsoBGw7BciwIzKjDdfvrbH257fufBDZEgrAdPEJ6TiQs/bb489GszDPuPvyy8KVKABW1eDJOgzaX6c/r3QYh3Ny0q3X81t+/fm89poyPVrhQGtDFM7q+eggloxnUAM9+90XNqPPkzHDYnOXWmjUsPF+FwJQxyGLanOvVspLI4GObvd0zEz+GwdrHTaJjI0MdkBM3YDiDm0+ZMfzeqj61IJXD3I07787ut64BR3kUqioNhIHE4TLA4nUxgh3Ops18BPUbZ3VGI04etSGQiBmzw4pj5ZCPReKBYAvO/REx2sn0yEkWkl4lMdtPNXEHYhZl0uNYLzyi/QRSK4pi0c5FIPCWOgYVZBee0GiaSSVmcNiayBZv8MRNM6bgW4lzVjmkjMyctrRVxVNFx4pB4WhxTq+wFhslmIRWpJk4LE5fsUqSIiafkk1x+c7/SQu5Xny/gkMr0tldcrwNUxw4lKOApaAKjmLg0T5jkiISmTS8JJiQ0zKaMOHC85BTUGbPFi1ZD5pD4pswJpR0zacmcChOwgnDE2a+UOMdv39FC2lBzSHxTzRnP7HQjmNTEgdWvwsQ3Y9dqvMzaNcJkmC7e0++b4zoqlboQ4UXeZ+9xDVYrGk+JY2D+8quV3nXs5G8Uk7I4LUx8MzIdYiaeks2tSe3/DrIUD9Moadn5FZiT4rjZF+1zaDwUJ4alTMyGW0UhI9HidDDxzagqa/f4KHfsvvRwvRLiQmqzEScyc4SpPoB5kjmTul+8/aFrFNghZ/AuEAHzbfgdsi7/l3QkLE4nE1uiTZGlFyv2iQhPyDAf2s5WDoeGi3k2ewImMCmWU3le8IanMESQOZ+ivBwmhfRcfloUWBRYFJhFgezRbZbory2IP3A/0x9Bz2ci4Jt9bSM7c3+WaTWzoEu4oWVz2rkpdionZid4PtJCIozZvHSIXGgyK3tNTn/k9B6p+iaI00Yi9KLTc6XpYHFYB2LCI2ThQnNZs1PfYoM1eK+dnqtpLRirecdT38kyKwmPsIDjeKTxKPG814BpcToxG47jaccsfSxKVWHsEZZMToZHGjfC9F49qMnpJMQBFm8xdbhmpcox6BGWcAyPNBaH6b1CEN9zJZ+uVh1PM6vsM9iaWSlvjTzCEo7hVySziue9kuJUnc6MOBXHMxKnZlZ6Q9x6hCWTk+F0ZcQpe68ZcSpOJy2OT9aMb6XbosxK5eMjozPxCEmcSi+eRxqLU/ZeyZAMz7XABrw/UBIn2J6E2Qm6kHqEJVx/5uiJnvVeGZlDOJ0Um7rjGU2rmlmZeoReHKY1CUYaL4+RMZ33Xhk1h+gGJU7d8dRtublHmp2BDfIISziGR0quVjXvlRSn6nQSbBiOp27L2Z6k2enZYI+whGN4pLE4TO81EoftdGI2LMfTzCv15kDFrHRlFnqEBRx+/wl7pMlyZV5fqHivHtPkdFI7ZGOs6gUp53ja1GG9CBonkU65NpwNQO9IO4OpstDnubruVM5WspcttucjHYPbSIS0eyQ6oYGWRymP9gClhQRYJ/reoXcByHfp4wm//LcosCjQpcB/NpS/nY8tPLEAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
Required solution
(vi)![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVAAAAAxCAMAAAC76QGgAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bKDkV4gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAE+klEQVR4Xu1bi1LbMBC0AgWXUmgCfdFAg1tqkhbb0f9/XCVZlhXpJJ0cpfUEe4YZSOTbu9Xp9FqybHomBiYGDsHA8wWZfT2E4Vdjk/68/NUHS+9X2Xr2GIi+vHp6NfzEBkqXH16Md+q3IUKz+t23WKB/1p5ubgl5EwxB+LP5iGuHd54u52bj5uHO9f76gsiCgCAd74TdknIkJCnm2wW5y5rFiTbsXJ40twTTLCqQyrJYkFMXoeWMfbMmogvKMzOxo3B3GtOlme/FbJU1y2DpASGLc+4eCQ+h+uqpTE0oXXJ049nksDOycSGoTJmiAKFYUly9WDFCC0LOXkriozYRoT1S3XLXf7C9eczkh5avHaGiD4BagUlRMEibUGGq2CN9xLsFS3GHbelrIkJ7pEqOqh76OXcvmypWndQaQIys+AcKEgy6edhjKdHmBF2yDLXmCN3nVIQqpI7QMLRwY5OzSUnOi+3Ij34gJIDQ7YKQq0EA+vCp8/fXykM2OOTT1+ZUhLJR3SIpgzvQ0gfLgzpn3d2V+4GEKmgXiCJgkw8d8v3MUAbmtWSEZhKpy1DGU3hK5cnVjiNR4YZlKITkqHNVu56If4pzmdvbT5/9nZKM0A5JERqEZmFtF6JsyrliYA3NACQHoWJADHjEiq5i79L7R+mzy0oqQhVSN6GHoWWNV6m5XQyLFkKyCG2XEHzpM+ARCzpazMUPM+LzEzEuMR70SLLaIKBFxSVsmyoX9gPXoSASQCgrQI2sLZiI9Day9M/pkk2gfHJzMUof2BxLTvc/CtKRxE4pCN35KzaEK/HXsJ0SjGQP+YbHej14lo/tg4TtB67Pk26UEoYzAlPAaRPCqzGfNiHcP2iT3fNQHNR0HorjaWo1MTAxMDEwFgags42x+HYcfqjDo0P90tN0KIRou+lcOo4cGFUU05AfVXdMzoyDgcDp3/9wEnIpYnOG25P9ZtKG/q4jYZg+Qg8G6vcfcCnq+ABzarC98emHYpQkXSzmOSlgww+qkUJ/5EMvNzorft0H5oBLCwB58OpuFlCSQNcFdW5cCDlsmKCQrSq//rPvmPHrPmyxio2nB4A7eXVfFweUJOCV6xeTUFh4YYICtqrcKWzC0+wNARSrmLZ1C67c29FMrAMl1K0kAUgo5v3lpeaZZcMCtW2piueXsSBELobuwyNWcXaUvFNylghNM1GGpiS3ksQmobp8AQk1bdiggC11L+aXsQRFLpbuQ71hi1VcjHZSHNf9aq+ZqBifle9Gz6MksUjgaiSIUNMGAAqItE6+8xWIvH5316WA0gTQfcSLVZTEyqkA6NQZHE5pwqAdVZfkppzD2ZYRWhodZNrYAWU5Ae7k2B0au9fciPtwHdxu7P+WvW7pPnxiFR8JPkkF8hbYqyQxs6rqTix2CUWpUYDrW7FialPC720wFutCPE6sos1dzgzFqDN4PfEqSUCVhT3kUWoUF6FCT+H3NhyLqfuIFKuoADgfcJVFqTN4ZniVJDhCcWoUqIZyJTVPCb+33m9B3UekWEUFwBVAsFIBqc5ob7zdShKQUHP8IdUotq06ZzqrNZvq/d4GvuVabkP3ESlWUQHwYg7/DwVanRFQkoC7m1612Q4OpBoFsFXz/6BYBbQkoVhs3UesWKULgAeD2ynhtx1H1xKzl9eCRu7lj46miICSnzZFYB9n0+TnocdJ0xTVxMArZuAvSESlms/bvNcAAAAASUVORK5CYII=)
We know a3 - b3 = (a-b) (a2 + ab + b2)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution.
Question 2.Divide the following and write your answer in lowest terms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution
Question 3.Divide the following and write your answer in lowest terms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution.
Question 4.Divide the following and write your answer in lowest terms.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOMAAAA7CAYAAACJ4z0OAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAX7SURBVHja7V29btRMFJ0H4AFA2klFHiDMpqTdWIgHYNdK94kiQpY+USMnXSSUF0hHBwU1DS0Nb0CRN+AdYMeLvWNnPPau5+fe3VMciQRlM3fOnJl77zg+4ubmRgAAkB6YBACAGAEAgBgB4JDFeJ2r10KIPxXU6hMmlzfAJ1MxlkV2JtXqrvm3EL/l4t17TDBPgE/GYiwuL1++vb19Un/9biHfi1n23fwewAfg84BqxirFAXmHlbKCT55i1Dsp0prDAfhkKsayvHymTi7usYseSP0IPvmKMVeqvCzLZ6l2cCHUQ6rfP7iohXhoOpQV5O+sKM8oL5CYfJbF8vSFFD+b+ZEvfi6L8pRMLV0szxYL9UFK8WsoUyjy7FUTyzqOMTx7F4PKr18nWez/un5UxbjZKEwh0r8yiMmn5k+p7GN9Am+FSYPPqm6W8pcWoubOJcbNtdB6o82LV7Uw9docmkuvgzUHqL+ORuT61NnuqPTEyDHVi8nn7e3bJwu1+NCdn3qDTbXB9wutX4x1BtT9/zFZW7hdP6IoVkreySwvs5n4TlGMKyU+VQTK+bd6t6R+Iqbkk7MYN3P3uPwYc187WNfoH25/n1adU+3ganWnd9euGCnEs02fjYW9riGi1WEMOW1tYkYqTyEWlxibNWi5BmrG6ChNrClDvZObH7oZhHqgVFDrAOdy/lkvbJsYKcVzdXX1PM/O/98KM95Jw4nTulFSlR3z7Gt3UaeOxSXGRnAWMbqE6kxTtwtb/NEpQvW1lF+o7Z46Pa3H1CdGavGYY4nZwOHAaWtu6pNvnfX0LuwEsTjFWGdBFl5d63OwZtwe/ephPpefKeXtdW5uTshQsJTiMccSsw6jzqnJZZGfv6mzCOcpFDmWJGI0f7HvR6HsDQI7eoP+9yDzLsHuE8/Usbo/N36tRpVTd71t53TXWHyMcVSaahGjK4UdLcbZbPZj14WWpts33P2jFM9mocVr5FDn1NnEcYgxdixRGzitxSKzL8V18XRz4tB+WmQwTSUWT4qHr7lx2my8tsWdKJZxVxuP1+CYa5rBLuWYdIG6GCnGYzafYtWLXDntLvyUsQxe+vfcJ+516d83AU1quN6lqLXCXWJMHY8W3Ym6uK9/x6Y1L+/O8+JN6kVNhVP96FuVwlXPohZn5jz19QZSxdLUqY50s9sPqJtROz8O19zhGOp/XKPRS2/6xJg6HrMrqGEKM27dRZdTzd2FOrlvNVB6nlZKFYv14Q3HCdnifeQD76SLdwA4JmASAABiBAAAYgQAiBGYgqaJgJdDQYwAxAhAjAAAMQIAkECMY59oB/zDRs74vzJ4fMmNOeXHOXYkAECaCgAAxMgU6KZCjMEBL0BaYgQfRypGeAESFDz4OE4xwguQFsAHasZ2igTywQfEmB7wAgQfECOFeiWyQYzNgqzPC4OrpRsXPuzzm9brw5rGB7SFI0V+XC9A+2sU6rdU204IbpZunPhodXBtIDDXoW3hSKVDsd4K3bzUKMtL0w9w+2r59i58jO69sb02q7flWbISijYEoWzhyAQX09uxMuZcFP/1ibSbfnKzdKPAxy61pl68y6VdbPX7UUNuhLvWxaFs4YLn/UOWXVS8ANvCM2zlElu6ceXDV+PH/JxQlnA+xTjlreJBUkBO9mPWibRMVkpLN458+BCjLUUNEY9PMU6xhQu7qIlbytmbOu5TL5WlGzc+fIixL0X1HY9XMU5wooqQItG2H+s2EcaMM5WlGyc+fIjR9Rk+4zl4MZqD5mA/tishqSzduPMxVjRjTrvQFn99Yw1lCxecfA72Y1W30NJdnZrSUowzFh9TT8YxXVRf8RxsA6c7mdTtx6prDpWXe50yjJ7ZjM3HVDEO/bzPeMJcbexuCxesPuFgP1bdp3VcjjbjXZ4qlX10CS22pRs3PqaIcShF9R2PbzHuawsXrHNH3VKu6w7VV2tSsHTjyMcUMdps4kPGs68YfdvCBehI0reU0xMz9m1rFCzdOPIxRYx6zH2LNkQ8Y8ca2haOxYICgGMAJgEAIEYAACBGACCIv9BBBqZRf2YEAAAAAElFTkSuQmCC)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEIAAAAsCAMAAADmdJ7wAAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjpmOmY6OmaQOma2OpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGY6kNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bJ1juAAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABIElEQVRIS+2WW2/CMAyFY8YlG3e2pbTcU/L//yKuNK2p45RaAQQSeeqD8/n4NOmpUu+FDhwWucAHZ35pdTmHj20Swo43uwiiABgedwDNrowKpWIIVfRyFeyQIZxBFVMypAyhrP6a+AQc7W+hPn9FB8EZSSluE6o4L1eB1TKEy/LzbJTihSvQyhMQP3kV4cBVZ2eghyqAMEKEW2s0uf8tOJ/v0tdxIHqHEkb4v5YdHxJa1VvvMchNhD0QspcEEdVlq3uOH4UEvVYH2SaldUDwiVY36oDgE02GYBPNR3xqGFxx00+08AC77EeV5tobYRKNmG41Dd5mAZdoBBHmVaOATTTvblb921XwieYh0IbStP0mRRKtRpQLPN/jjfREPnf9BWEZE93xey4LAAAAAElFTkSuQmCC)
Required solution.
Question 5.Divide the following and write your answer in lowest terms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution
Question 6.Divide the following and write your answer in lowest terms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASQAAAAwCAMAAACPF/fnAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAA///b//+22///tv///9u2/9uQttv/kNv//7Zm27aQ27ZmkLbbZrb/Zrbb25Bm25A6tpBmOpDbOpC2tmYAkGY6Oma2OmaQZmYAOmY6kDo6AGa2kDoAZjpmZjo6OjqQZjoAOjpmOjo6OjoAADqQADpmZgAAOgA6OgAAAABmAAA6AAAAUbPmnQAAAAF0Uk5TAEDm2GYAAAOnSURBVHja7ZnrcpswEIVlEzs4bVJKSdLW7t11Cfv+79cBcZHQarVIisMP7UwnM7XP+rDostInRIoUrxibjz9u/NXF821IJlUeZs3TCG1gTHj6lIXUeP/73jfTh1yVh1rzNEIZmIyU3Z+7nwB/c/7we/dn+P7+X65m4jmrAQCaXJFbrTmMTMaXGEFkRBxeuoFZwYPYXV5urH5n5a6a92J36p5SFOdMycSS7+vpP3q51RodqnG3EUpGDaSjFLV/CrhnF0n5vnwVQ6bFRbK9SSqhrxFCRo57xf4B7kUFcM4KmJXLeEr5S/JNyOHdZ0L1ZJFss4NKOB9z8lO3EUJGpW/y2UNXTW4+FFak3eOwMXSvZciE6ckiSbndGmoIfVtuI4SMWSTpfHM6Z0Xpmi9iewF47udy1U7qIROmR4r0qx73iQpfE6iElpI7jRAyqk2YFrlh1O3rL0/6GjdEo8/du3p4E+2PjJlmelS++fowrvs2j1hCPNk4XWxGODLWSJrWu6LJnStvP6dL8wViely+r0vOSEIN4Qu80wghYxWpOvbf3H77bOyhjqdUlwJMj8u3lyNnTUINaRtVpq1uTN1cxpmXXbNwKNuZkA/u7U8ph2q/O2wvyqaC6s0OopxqLOV2a6ghobdZh5JnhJAxxmrXK2yqsvs3ziOiSE0udic5TLX2BNUjvWgrl+/Z0SfhhvQuq/sOxwghY3XcXZSbEzR5u3M5nnL3WAM8ZWLe6OJ6U/4dAPoOgu64LYbm+0LJM0LJ3Ge3kBOux9kNkYdb8zTCOLu95S2A6xC+mluAdJ90K1KkSJFivWE9q6YgAlYS63GUxkSabilWEz6td7w+V7nWuEKj7dt6Lz966QQ28MQ0FekqRza/Q9yiszNCYKNx00Bwu5BFIyjafbFk8e0msNG4aSC4JVm0eY+MoGhqIB2FX5Eic9NQcEuyaKRIHiiXCXBNAhuNmwaC2+kTLIMFVmoomvpBinsyCGw0bhoKbqdPkAxowhmKZhSJB3BNAhuNm4aCW0GxaKRIBoqm2gQLgOUSWC5vdXLTYHBrY9H244aOohkjaQHA1QhsNG4aDm4JFm2ZvxqK5hSJD3A1AhuNmwaDW4pFW4qkoWjGBGcBXITARuOmoeCWZNHIlmSgaPfo5QFchMBG46aB4JZm0UiRDBTt7LjZANcksNG4aRC4dbBoc7ohKDrW2W3+TmNy0yuB2yWWQm4BxOtw0xXfArz1fZKnkSvfJ6VIkSJe/Ad0FNg/DCeQhgAAAABJRU5ErkJggg==)
The like terms are cancelled.
=1 required solution.
Question 7.Divide the following and write your answer in lowest terms.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN8AAAA7CAYAAAD7LF+4AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAbwSURBVHja7V29bhs5EGb6uE8CU13SW5TK64LFXn5KA5EX6gIXgrBAEANbGMHanYE7P0C6uPMV114Tl9fET+AD7CeQn8E6cVdcUSvuipJ2yeHuFB8QS0n4M/ORwyE9Hzk/PycIBMI8cBIQCCQfAoHkq7dBQp4JoAEQbfYNo42dBezDbGKfErCjK3Q4RJt9w1hDcegfUBb8mf2ZkEfqjb+i4yHa6hvGGgqHw9+OLy6ei5/HHv1K9v0b+TNEO9FW37AbZiD5EC32DWsN89UNw05Em32j+Asp81R19imOhy97Hf97W3e9pmXzqhwPBN8wZZ+CA/DgNaPkV5Z9ot3bw9Hpm6oaDRiLh3H8sq2rOiHsoSnjr3o8tn3DpH0UxPMPGPP/ECvPgojVdIgPjgVnH9pIvNPR2y7P5DWFfFWPx7ZvmLbP0g8XF8fPPeZ9y2/5Iv2768TEA/ZxFsufiFCW/9wWIvJwambZ29m4p00gX9Xjse0bNuyj1zEF+YrOhEXnxHQ7n4exGdh9nYPctI914ojRS+oHsb9PbppAvrLxuOAbEOyj5cBHjFwRFvyQVwlGyH26SqTg2anlz+nED+MDmyszlD7y1DllR5c8smgC+crG44JvQLFPKelGo8NushX3/L+Po2gvH6IGjPxIJlO6k0mfCbH7QRi/tj2pEPqYZO9o75obswnk0xmPC74BwT6FTutT8lMKA6Z8ZcifBaW/N2WD+GMUHe/5lP4FaVWz3Ucezoi2mkA+3fG44Bu27bPWccOg/2l23puI8KE4vGMPvR69rvqQnD8/FEEvBN28j7u0n78sdp18m44Hum/Yts9GCZeiTs1fpE+rfhKkPoirse5FxDZ93KX99LHw0aXsBMnqL4wbRa9cumzfdjxQfQOCfTbYnslVGfkopf8W7Y5gDtUG+1jiHCIRYTyjZ2M8UH0Dgn0266xi9UpWEPr7dXgWvkhjfPr4dnTaBbdqA+hjU7Kd2mGnA75h0z6FcbT8mdiO8ytXFA1fiUzRSng627YhTCikPraJfC74BijyJU/JCHlI33KOuiIOTi4g57/sqCDkSW6HPBExfpXvQbcjHqw+toV8LvgGOPIljfc635cOq7T3jx+E7/L/cH6Ps3SgXY2j7V6mQuvjIv3uzllvm/G44BsQ7OO08REIl4GTgEAg+RAIJB8CqrGw3imSD2EeWaoei04h+RBWyDch1P+J5EPyIRCIqsin+0ocUT3Kznh1/luEPTvnDfaEsIM8efRf7asvq3FO4dsZt38EAs98iLXGwqsGJJ8Nh2u700G8akDbNJh8qOe3Qj4wVw1omwaTD/X8gC8EaJvmkg/1/OACbdOSM99SmIMGRtsg+coP33UcwE1qtm3Tf+jJhzrtY1pPr25fM+VHlTSSlZpI63smF7/9wXhQ2RnDkGYbr9DteewbpeQ/XWfiYw/8/pdOh9ylyYdFWX04Z7T6JN9M6+mdjg7fLARNZpiNxRuM30Oab10/2rmhrO6FlIXjhXZ1amnqwoRmWxI6dTp3cxJplbkbD/qDJP1P2S9eagNi2FW35JtJPb2FhNdqqT9eFRtMCK7pR5WEHCpDpp/Tx13rdJjWbBNFXteRb14M6Alylq9uyTeTthGLPFcSkhcSUZIeWlEqHT/aOeRIqp0p7nmyS+H5d9vIddnQbNOZNN4PyAWCdXZDmXwu2CbZwb3wc2HkJS30EKTh6idfjmDKSdn3b8I4KUm4kWyULc22dZOWLTg8uxdFe6698shLvm0j6QVBT0+GXE0dikSZuZ1PkWbO10B0RTZq3aTNHW/a94MvS2UWK9atr4N0RZJvLkl6FS7yUqILwnhqJ5+85efLgKuI6YJsVNmkyePl5FskmPx3aSJg9zNubQ66RvLNFUkvdfTVvYUmUVY7+aRGeNjxwFd+UeU68GmcrDy5kBS6bFTZpJXt9GJHhPzOcZ3kmwtyb6shJ70s6qctaThj5BMr/+IeiU74uz9+95WsOooBQ5YUKyVfyRk3I6YDpeDLJN9ckHuT/09o0nDGyVccnqkP4ZAlxbR2vrIEkyM6DEWSby7IvYl+qrKfUHzNGvnEHZhq14MuG6V35iuQQZ5nd11446h6DO2KpFf6cCCIVaEiFF8zTj4+eLFl94PwU/57F2Sj1k2a+D7/okK8vjD5IGCTs8uyHVYl31yR9EruF3OKshz82Zn8ksf2eIyQTyRYxgPvffrmTn2H4oJslLgsFomTogN1msamE/6mUBg+GTuwd526km+uSHplz/mkO7z8fR6E8ej6UXUp7E7nTk6/qx0WrmxU9pvimgfq0GOf5YfkOucPG9lNHck3FyS9xHthnSpuNseziR+BPpcgEE0GTgICgeRDIJB8CATCAP4HWN1neNVuo+MAAAAASUVORK5CYII=)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution,
Question 8.Divide the following and write your answer in lowest terms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution.
Multiply the following and write your answer in lowest terms.
(i) (ii)
(iii) (iv)
(v) (vi)
Answer:
(i)
The like terms are cancelled.
=3x required solution
(ii)
We know a2 – b2 = (a-b) (a+b)
So,
Also,
x2 + 6x + 8 = x2 + 4x + 2x +8
= x(x+4) +2(x+4)
= (x+2)(x+4)
And
x2 - 5x – 36 = x2 - 9x + 4x – 36
= x(x-9)+4(x-9)
= (x+4)(x-9)
So,
The like terms are cancelled.
Required solution
(iii)
x2 - 3x – 10 = x2 - 5x + 2x – 10
= x(x-5)+2(x-5)
= (x+2)(x-5)
And,
x2 - x - 20 = x2 - 5x + 4x - 20
= x(x-5)+4(x-5)
= (x+4)(x-5)
We know the formula a3 + b3 = (a+b)(a2 + b2 - ab)
So x3 + 8 = x3 + 23
= (x+2)(x2 + 4 - 2x)
The like terms are cancelled.
Required solution
(iv)
We know a2 – b2 = (a-b) (a+b)
x2 – 16 = (x-4) (x+4)
x2 – 4 = (x-2) (x+2)
x2 - 3x + 2 = x2 – 2x – x + 2
= x(x-2)-1(x-2)
= (x-1)(x-2)
x2 - 2x - 8 = x2 – 4x + 2x - 8
= x(x-4)+2(x-4)
= (x-4)(x+2)
We know the formula a3 + b3 = (a+b)(a2 + b2 - ab)
So x3 + 64 = x3 + 43
= (x+4)(x2 + 16 - 4x)
The like terms are cancelled.
Required solution
(v)
The like terms are cancelled.
Required solution
(vi)
We know a3 - b3 = (a-b) (a2 + ab + b2)
The like terms are cancelled.
Required solution.
Question 2.
Divide the following and write your answer in lowest terms.
Answer:
The like terms are cancelled.
Required solution
Question 3.
Divide the following and write your answer in lowest terms.
Answer:
The like terms are cancelled.
Required solution.
Question 4.
Divide the following and write your answer in lowest terms.
Answer:
The like terms are cancelled.
Required solution.
Question 5.
Divide the following and write your answer in lowest terms.
Answer:
The like terms are cancelled.
Required solution
Question 6.
Divide the following and write your answer in lowest terms.
Answer:
The like terms are cancelled.
=1 required solution.
Question 7.
Divide the following and write your answer in lowest terms.
Answer:
The like terms are cancelled.
Required solution,
Question 8.
Divide the following and write your answer in lowest terms.
Answer:
The like terms are cancelled.
Required solution.
Exercise 3.11
Question 1.Simplify the following as a quotient of two polynomials in the simplest form.
(i)
(ii) ![](data:image/png;base64,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)
(iii)
(iv) ![](data:image/png;base64,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)
(v)
(vi) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOwAAAA7CAYAAAB46GaDAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAZISURBVHja7V0xbtswFNUBeoAWMLP1AAmdsasjFD1AYyFb0akQUHQulGwBilwgW7dm6Nyla5fcoENukDu0pmTSFC3KFkWTn9YbHlAHQvX5yUf+/0nxZTc3NxkAAGkATgAAEBYAABAWAEBYmoZdF/xdlmX/avDld3QW/A8QJWxV5qeML+/Uv7PsmS0+fUGHwf8gLEGjyqurNx9vb1/I358W7Es2y3/rfwPgfxCWcniGAQP/A2kQVszwCMngfyABwlbV1St+cnGP2R3+BxIgbMF5dVVVr2KuLlnGn2LaMAX/l+Xl6WLBvzKW/d21mg95NvgEV16+PmPZo6qws7PHy7J6bW1Lkb9Vz6+ezcvqNFnCCrLw4vpdPOc3FdKpEjaU/+scmbG/goBi4PaRcMizMcYL5/k3GY1syNs9fpqtM/acF+VbSV4x3vp8TrrQoXeG+B2SvCIU3MyU0yNsDP/Lvd99SDjk2RC4vf34YsEXX83UQU76pu/qVCPLnkz7d0V0ZGd2FVIohCXNkrM7lhdVPst+T42wsfxPkbBjC242wjY+Xq2uRgi8a987TNFiNZPonS+Maf992/Doqwtf3olZM2XCpub7MYQ9VFvHEnbJs+/mSTE1rjq2ypS9ltNlwcKF2nDhNM3Ixun8qS8pjzHI52z+QxA0dcKm5vuxK+wh2upKWFEYq1Oqef7TSsoOwvaROWhIvBn82T8RHtS/GXugtLLKUFjadAyETcn3PkJi320dSlj9/WqlX0VrOvlUMbNjFd015iKFaPxpPmc/YlaA9+mcYyFsCr73mcP6bKvrCivGTlmcv292Gdp2JkNY3dG+j7pttmDMYkkXLMn++sD7MRKWsu8PUXRyaWt3sa0b+04CXVuDfXlqX7gcjbCz2ewPtT20/TorbeJS9f2hCOujrT6OZTZ59WbskC86tWYblj+U1+XLZuWiVR0eGp4kFRIn4HtvIbHHtvogbNfXTrb9Vts2UHDC6tVXW6gAwk7b9z4I67utYwkrx9CWnZb9VhIHJ2xGqzB0NftQ2l44JsKm5HuVd+5xw0XXs4do676EFccQ61C2Pjtcnkp76gM4Rm2kbddm9ZdFquhHE9XemDYjbueM9EK0YyBsCr63Fa2s4a7l2UO0dV/CirFywU/uWzax+S95TtgGvZK860OBKEUnAADcAScAAAgLAAAICwAgLJwAACAsAAAgLACAsAAApEfYfb9UAPzD7BgfX8DAr8c3LjBrAQBCYgAAQFgAAGHH/yfQEp0e0OeJEhZaotMD+jxhwkJLdHpAnx9RDgst0YmGx+jzNAkLLdHpAX2eKGFjaomKL/6L/PyzVDWjVAhRV4VoByHMy6WTzWcnph/bJSdpu1ViqJRkcMLG0nJV12ysnUdp8OhXWsrrP0Sn98kxpITY+r1hyWo/fWbew+QiJRmUsLG0XOWdPVRDMtVxFpUyijfwU+/zmBOvUDTU9V83shzbF4UPvRExGGF9aIm65EGxNEKH2GqVFVx3aqoDPrZ+b4zVlS/KD9YISutjVynJwYR1kezzpSU6lLC7JA1c2+PbVl1HRbeTSmU1Zp8f0saQ0G/2H3Orv9MKG0uecChh5aA5z4vPresljasiKUgQqvevbatXJzb/ReU+4BQkKanaqAi6JuEYKUnnkDiGPOEQEuhhiCCsbLxM7M3ZNrYEYUNadke5QpyCJCVFG5sI6uxxS2nAQZluVA4bWp5wUJjZM4upcM1wWGwJQtGRQpxJyQ8SJG0KkpTUbBQTsW5DNMLqRR3fudZYWb8+p+gdajollgRhTXCWP4h3tiqLBLd1qEtSutjo8927Ju4xUpJeCBtKntBphR04i8WQIJSDSyezno9R246iLklJxca6Kt5XNQ5VdGrNSgHlCd1y2G1S2hwWS4KwIeb2u8xiBYlwMwFJSgo21ls8vKj6o7LhUpLOhI0hTzg0L+xauWxOiSlBKENq005qhE1BkpKCjXWFv0ONrlav4/m3Ju1xk5J0ImwseUK3yut69Vof/1LnPIlIELZz6o2dzblnVlEhQwqSlBRsbCnNdUC3zUVK0omwseQJXb/4KBf8w8aJ7NnMK2JKEOqzrylFeMIv7qnsw6YgSRnbRkG4ocWqoVKSo4tOAADQApwAACAsAAAgLABMHP8BzFEoEcB0sfgAAAAASUVORK5CYII=)
(vii)
(viii) ![](data:image/png;base64,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)
Answer:(i) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEwAAAAjCAMAAAAJ6xyWAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjqQZpDbZrbbZrb/kDoAkGY6kLbbkLb/kNv/tmYAttv/tv//25A625Bm27Zm27aQ29u22/+22////7Zm/7aQ/9uQ/9u2//+2///bZqA5cgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABLElEQVRIS+1VYXOCMAxNVex0DMd0uAFDnCC0//8Hri3KaKGr2fVutzvzEV5eXtL0FeBfBN8TsvjwJJU95yx+9UQmaI7ryhcZi8gjiqyNiS2DvVQ8eUIok+g2MjJKOksbGpxEHdQBsEhMOFsa5eugKkJUg4qAJ2HVSkI9ivkW0V4PlTMbizisArwwYPFOqDPbPL/zZJGjtTV0guxAlzwjv9BWUutq2JTxxOO1uJPpYybT0YOy/v8s7T+6kn7asfsBoG+guMsebwC+/DXjU1qUK/0mEADbvEE5WKlJWhvoYv+aMzYP3/tp1aiBOhaACfsvbnmhdJBiEYWV/YsXU4U8ttI5sitokHV5REz7VxVcMQJ1LKb914Krdu3VCKRYwLT/TrmDbATqWFzt/N3/L8s7F/TP9yq8AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
=x2 + 2x + 4 required solution.
(ii) ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASQAAAAvCAMAAAB9l0epAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtpBmttv/tv//25A627Zm27aQ2////7Zm/9uQ/9u2//+2///b8AcjjQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADRUlEQVRoQ+2aa3ecIBCGYdsm9rZpN73aNDY1G3X5//+vLAoOMiKaHbQ58CnnxMGXhwFh3mUstUQgEXgxBMTxlvM391saj3h8t6okkf8c4Cj4D9YcXv3dEKVi95s1+W61iUMgXUs8JR+yW5NZEU1SwfnVU8kHo3chKRpVXEi4NnteiijJXch8dZiMQIqjqKeAaoOQmrv9Q4xkFrnMpM+DN+GQ6izyakO19VJPB873TzEgsTr7eANfJJO8a4M9UeRDluT6gDZc1jGLstzkdux+ILBMEvl5o4zcEG22gopHmbnT12/ObGCQius4mQ0hYNpsSHUWA5L4dX86DHMEgVReSUZVDEU9BVSb/ne7+qN8cEUhX+XkrAupfisPberheA3X1kOS20QjPzvkgkTOdzKT+GBlYydutZvHhDSizTBp7jLOb+gZkU9CekEikAjEJTByyYkrYutvS5ACZihB+p8gjV5LAwZB/ch2M8nc5Vf5w3BHJ28VSZw6F57X/2YyKS23503k6tGbySSUhPjzYbYHVIKq8YJ4GN7f15Gy74y+CSSBckz+acEduX6vhySWxPfhRghe55yh7fKSgLYZhQ3g0KrSkWwza9rHL22YDveu8pC+gUNLJ6ny1MidyYUOrSpCylLceLybG80t14934V5IPm06EDq0VJK8hXsXErBD23nzxTvh9f6h1JACUinIVIAOLZWkziWbY9F2BeN2LfjisV3GQApYSl5tVgp2fiiVpKpzimZYtNqhVXPoi/dCYircv9p82kBk79ASSdKDDLdojUNbKGekGwgWPwFpsjjv1WYgQYeWSFK/RwRatP0yUYrG4sfO9uZ5psK9Desb7dc4tESS9GyFWrRgM7WmDYu/VCah2iy82u0ikmQgBVq0wKG1NgCGxF9qT8L6tlNQO7REkvQOE2jRAof2dABfN4bFeyG14d7m1dZFWg4tlaRu+QRatNChtQ4laDwKSf/uIvyc5LVoxfmXftqhJZOkTrWhFm23baocgMdbPN791dfZTeWvv4Pw6RP3hEULHVoySQGHOnQgCy9Kuq+ARJp9LySUtOgWzwiv3GBOZmkjlTSjZmP0kxZvIKXwWlcsSVOfm/T/RCARCCXwD1CmeRfR2J5iAAAAAElFTkSuQmCC)
The like terms are cancelled.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(iii) ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUwAAAAxCAMAAACGe2DGAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6tpBmttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bJbQQtQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFFklEQVR4Xu1a61obIRBdtNVta7VGe7Op1W1tkjabZN//5coAyyUMzKBJTOzur3xfOMyZwwC7cKpqeAYFDk+BP9dCXOw/7YOgubq6q6ZH9/uu5oHQlDIu3uy9mDDYh0FzcpkqzG76VojXfKm7mVw0mO27n/XJvGRGpGlCL5Oy6cWKHqbDijBNL5nN0V21HCdpduNvoRqNuKmWo+PfDI3a+uIvo5lrkqEJZVvzWcrmvOhBOpkIjuQks/00pzDmYk0yi43FzLf3xGvrmyIpqxzNqlqNvpSIyYzup5+O0AhxMp8IUKmVWrYptVS+jaw0D+BpEImp/mtlt3h7B12NQHbq4dNsLlsQEw0bs7TRKZp9OqCCjoA9jfxDBVmNhHwyYi5vzx+grx7g94aLCeLj7R0WBGc8XJrt2VynitGMWbroaFohMZVOZSMgrLuxrMzkvuNXkBDnsE2gAFTMRQ06EQGa4x/wfkttQEyaq6v7SouJAWKWLjodQKfjImAlsKjfx2ulrHrzeBU9q9XQrAPQtiodPUZ++6htNxYf5tWsJocTo4n0JvuRYk5k2msAlKUfPUvTpqOSMhEwNVkbvQK2QuWMAJDK7Mb9YpgLAAUBk5J+N+LQbPsSgBqKARHLIHo+gEkniBCrufr0mfUGo2oMxMQAiJjNqZm62QAmnQlJgU/TbA8IICWmik4EsOlAUaU2oO77PWND1VNWLdcoIBZzArXWShQRQK/qZGXyaOr5o9dMJC9szeyjEwFsOi5CXJddA2uAnr6Zp4P39SVMSRwQ0VRffNCWCrCoZQVPqS2d6sWnrqYrCojFtNGJADYdFSixIMgV+EhWpiDVXN7Was9NAJAvILV8XdIBFvDdeUeNJZOmmj6yLR4WWYxMdIqm2b50zakIRPUNfw8KDAoMCuxOAfzrdHfxX1SkQcwNDucg5n8nZupQYoNCbKKrA61Mezr03D/sGKDj/dzskPibKJqd9HEYlTlM850Uw14FOYzKpA6Hfp1xLmf10Ym6EdJPV4AzEB8esXoJYnZjeYXAfhbv+quwMpwJ4OBxxJyYB+E1cpc2WTk9a0fvUukvezjDgMA5MNeG9hpx7jfslOIbVWYf1bEb16rSkncHsjPf2qGOn+X5XhKHHG17zhADLxMTWme9RkWODsKo4qgtr4XOkmlVcTdgufR8b4NOKoNDxPScHo/2X+WupsscHYRRxQqxOH8w11RMhL4XJi0Yqo0eJT2/NY5poVD0AjirMIPes16jMkeHpUM7Ovw7P9o50d+5kd4GY+0AVaDODI5noZDt1+AsMX2DRtZrVOjocHTIrH0xTTVkqPdiEt4GuDFS1g4QU11B6gsRnoVCuXACOE9M13vWElXq6HB0SEeHJ6aZwznmtrXnbcC/mIy1w4jZ4zgWChU/hPPEtAYN3xL1ZEeHT4dydDgxOW8v7mqdfLno74b9ymRZKLRyAZwpJsvDWuroCOgQWVsxWRu1FZM2T2hrR7hmYpaIxMt3AGeKSZPqOypwdGhI2qjic7Ni+t6OJPl+Jch6GzxrB7gQ3W7Os1AgcJ6Y23F0UEYVTMzA25Ekb8o3721w1g4YUu89k2ehQOAsMbfk6KCMKoGYep8NvR1p8upLhvI29NYO6Mb7AuJaKBA4Q02KlN9FkaODMKrYfjswtIhXX9X7tLKqUKw5u5Tfx2O+zT38oz+AqDz24/+y058tnhrthxxPZFFyLrnN88wnpjHABwUGBQYFNqHAP1EsvSwG+dsiAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJAAAAAsCAMAAAC9kMGEAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOmY6OmaQOma2OpC2OpDbZgAAZjoAZjpmZmYAZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29v/2////7Zm/9uQ/9u2//+2///bZHAZDwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB+0lEQVRYR+2YcVfCIBTFQVMqLStbplTT3Cbf/xMGbqkH3oO7nKdOZ/zlOcLvXS6McSdE3y7mgNk8SHm16oxv3tRoi9NM9ux11nIuqtnwA2cce4Y0UajpZxsUIWhsx+fS1wlBQ1qh5txILeVom0uvEjEnByhSgkDabuamxzQ9WImgPiNIJ5cMo0UnZjLr0L2nlhZUquSKYTQ9fLWPyJTZ1KW6nZ7qsbY3zXp32kzm6yZMR2gmk3dbsVEcLvcK2zKUQyaLLPxRGkBzNtoBmnnud49PwdagBOkxcm4gtEZQTu9I87IKNz0hKHfzKVKLhtHqh4N2yGhbo5BeoVBQeW031L5zrIG0Ulm31+QZYvfXwDokPUXUSb3f6XFBMK1076EldKr2nf63A8wb5vcm3QtKed+BQ+ybM1Wb/L8DQQH38GoHfxwA5NRAyKHbj2yIDOrAob+/ZGeZ1oFDZ9UPBveCTiwx65vukmt7GHEfGixFlYW3b2QPBDTdGnbh5KojMRjMmrURyaDYhsbBsKzp5FSLyXtqhWAaD8OyphDu4j1JBiGQFoUhWbM2ZqOS2V7AtAgMyJrNSgVxiVhBmMbCkKzZFC7ZOH5QhtM4GJY16w8Nye9DAqJFYWDWNO5MrOqPBLH7CJSDYzA4a1YLxX/R+ZaI0iBY6oTp/+8dcA58AeoGNaseY0NDAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Required solution
(iv) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADsAAAAsCAMAAADsKDCiAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOmY6OmaQOma2OpDbZgAAZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bU5n7wwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA9ElEQVRIS+1VXQ+CMAxkKs5P/BxMURj0//9HO9TERArJDV6MfSW3XnvlLop+sui+U2qWQbNZdYqqZHpFwHbOqFydEWyDKQOwFuPs2zoNUyazRacl49eFlZ0XGJD1iRlaQqzdgm+KLIS1qikIi476xw22gZd8rOAE85VPJs9T6K9A9oNyDuQyMvwGieK01wATNMAzO/2WRYuLXIlB0NnX8ixkRDPvxJLhvrI9Ob3UcqY6vdq8Jf4+O0o5U42457xXAacFYvXh2Bd7ddIeM5Rm0ic/SePyQt8mAkrZy70MlWklRobvrU7kIKj2fJLry8j/y0jPPwAVFAxFXyT1EQAAAABJRU5ErkJggg==)
Required solution.
(v)![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required Solution ![](data:image/png;base64,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)
(vi) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Like terms are cancelled
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIgAAAArCAMAAACU7DAgAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjqQOmY6Oma2OpC2OpDbZgAAZgA6ZjoAZjpmZmYAZmZmZrbbZrb/kDoAkDo6kLaQkLbbkNv/tmYAtmY6tv//25A627Zm27aQ29v/2////7Zm/9uQ/9u2//+2///bTam5AAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABvUlEQVRYR+2Y21KDMBCGk9Y2KhZPFZUorQTJ+7+hC4VOZ9gTojNckKvONPny779LwmLMMv7EgXh8sPYqn8yKH25TTqF4uzff6fpzCgPWBpd8TUP4Lawv7Ms0SnB7CeCt3ZSF5XcKkhCBUqdNOMLwq9zEjI/Yi6nhKWIgjcaYgSM7Vm3lxMzwFL9+g5JPhGKt3G3S6wCLuwE+9SNmvM52HkeJmb0rzdFJmOJiU8SZmCkSDAVNUxq7AOyF57d+fGJLwG81zz9H6YQUfKXF15wt6qKJI0iu8pRTsfOORA97BEtuVF1DsbSTuCFQKgeuHtgzAOpoBY5YUklXvrwQkVI198S7dJIs/y8OLA7M3gH0RpuF6vNVq/zRih6Eo1x8njaL0E8i5puaGZm0SPm9A8L7pA48DoL2FpXjX2wHQjDKSAiGqNPn6UJoCN6fYUL8LpBC1BQGgvZniJBwU9JCjJLCQdD+bCikvs8NI0RH4SGX/Rl5+LadHggpqMZTQ5EgyAM1cCT09xXZASsoAgTrz/BPA0xqjJpCQtD+bLQQPYUSgvdnuBD6UBxBISBEf4YKgRQTB8kICg3R3RDLrH924AeGqisDFwMfQAAAAABJRU5ErkJggg==)
Required solution
![](data:image/png;base64,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)
(vii) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
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Like terms are cancelled
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(viii) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 0
Question 2.
Answer:Let p(x)=
, q(x)=![](data:image/png;base64,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)
And the rational expression be r(x),
So according to question,
p(x) = q(x) + r(x)
or, r(x) = p(x) – q(x)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANwAAAAtCAMAAADySvUPAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmY6OmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bSyFCfAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC90lEQVRoQ+1ZbXvbIAyErGu9blnS7q1z07hzk8ax+f+/bwLXdgxCgNxtT1b7U5twnE4SmBxCzM//nYGna7n4wZH4fCvlZw7Qj2HH4plS3T+I3WKbHmRzwwT6qdixUNEfPzDE6QnZQH80/CnV7lrK97aSenMXKpzaQw86QFGuQkD9fZnQGGQs+y9UEYrFg6hzi6uQF7Y4lf8cR13IO1Gv3/0af7pzlpwD1OXN4sUhsfSU9a20AxiFU1zpREorcrHPrE9ccRiwdLcTRFyz/h4vTrixdAKOy8eSFGcGFjCkkPLyUEqQ2dxsIbchcQZYwfABKCrQVoWBxaqixQViOalOUFy9WT5qhUDYpvkpc18FWHe1WRmAzVrCExRXfTwExAViiRenI1oeYLzKoXLe7QAV19Y3EagbIySOnjJeHIzcZ6Zzj9kn/A0MXfLyjLtJ5W0ukoAGBOJKe5mPtgJsSiyMYFvqpWOipDZopHIq13tKKrDqEkWKi31ZRIg7Zlpc8/Wbf+9BxBVXupsZQFO5UaGcf8hYYtuy7Sy96Ql1v23WXS0cNldceQnaKoCnAjVhQBw9Zbw4oKlhKxGq0Guh7U/kccSZQ5EGpQJDfay3KDqWU3FklupNBid50JbLBVRO+tQhJxSzeFbJQNMpQObvy9CUHVLp2OUF6wcMvSrmb+cMzBmYMzBn4Dwz4DsCn6eaUNT9ef8f/GFic9LNDSSk9My/f1tteebFmsOPy8AfuAogif8mX9xVgMeJjkvfaFQU3+vRuVcB7g903IlmaDMQ+54g0vjm0dneHyIOt7B5dM7VQ5zxHU12YigL4VwF4F5ta8rwHg4fn25wooV7FeARZ5xo5sPg49MN7i9yFYCLc24aUnSm802h69xf+yrAe9zpnOgUSSdjU/mm0RFONFa5wYlmqkvjm0ZHub+YuN6JZmpL5JtER7q/iLjeiWZqS+SbREe7v6643onmaiPdZr/xzaELuL/YCaV1ojlkgEnle9nVuHTMKGfYK2bgNwPTU4xKY5/RAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
So required rational expression r(x) ![](data:image/png;base64,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)
Question 3.Which rational expression should be subtracted from
to get 2x2 – 5x + 1 ?
Answer:Let p(x) =
, q(x) =![](data:image/png;base64,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)
And rational expression be r(x)
According to question,
p(x) – r(x)=q(x)
or, r(x) = p(x) – q(x)
r(x) =![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So required rational expression ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFMAAAAjCAMAAADfTsbzAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u229vb2/+22////7Zm/7aQ/9uQ/9u2//+2///bgz/pxgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABZ0lEQVRIS+1W7VKDMBBMaLGxWtRK/Sq1QmvAKpgm7/9uJqQlCQSZoWlnOppfDHe37O7ALQCc+2ELCIfvblXQ+zUNb91icjT64JgnAOQxt9BMx9B7bqefzSa/aCueclINs2guW9liCTIvNuaqImCrGxsPkCEvJsj/uoMQVn6qMY5GLuNdkwRQRTxqIVn4OQ5qj9Mxt68vAOhNSkQ0mV3ZteOBaRgVlPmR8hM44Jhg36QXCQo+N5GVajr2664Y2j8QR9eaqiKdcqOxDfP7jUXDtVkxxgjH1Js07UHObDxTNGIJrDFVPFMk3iWjSRW3IYSB8+/kzwMm8t3jp/bFuXSmekbfC5dkFNZJtB+H+j+qcmDTthnYqm2vd9lHp80IKmfS67AvphjnEbQPKn1dt+ZPF09RxyLtmkF1CGYmF22ZQXoAHYCJd7u7EVT9MQsOWfBYawQVSy6svwjdbkqxczODSivEfWH0eZ4fH/AhSq5LOYMAAAAASUVORK5CYII=)
Question 4.If
then find ![](data:image/png;base64,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)
Answer:P – Q ![](data:image/png;base64,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)
P2 –Q2 =(P + Q)(P – Q) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK0AAAAkCAMAAADMzkphAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOma2OpDbZgAAZgA6ZjpmZpDbZrb/kDoAkDo6kDpmkGZmkLb/kNv/tmYAtmY6trbbttv/tv//25A625Bm27Zm27a22////7Zm/7aQ/9uQ/9u2//+2///bUTDB1AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAClUlEQVRYR+1YaVeDMBCEehWPWrX1AA+qIoX//wMNV9jsbpINr/T5QT75JJOZTJZNplH0/xzdgf3z+2ycxerbnNtHRgBIWnmdziY2ivKbDzi7nwwBkLRqvZlRrJJ7DtyVkBkALC1bzio2qre3I4GEzADgOrhEdfCZLNIygYaIV5PFi7SIY7RXxYmuhXIgwyQGEgD81hbn3zn+NmSC92tl4yuwsoUBr0ZrEYmBtJtbJrRq89MHmToySpVc/WR8Vc2QfDAXkrUk1Tpun01T3SNSAzAB92J3NakO1NTVXZpja6NIi4RkiMRAcha2wjMq7Oel3p5N7MDZxT2xVjnYrwCQERKI1ADSvkhH2CXLOosnulsm1FpVuN1s1VqTURKIHAC4ENjZnUVbb13tuWKsVRvYFa6TzED2ACykwO3G+3m51L6mpCG08+WLtks6yBCyB8yrNotX7Gr9ahHSotbyb4fD7krggXm3g3KyFpB13U093daETDDomKK2LwG5WkvNDBPoZXB/aMPoatUrARSp9SMsag/7ldkKyF+3CNkCqDd/WS1du/WMs35nU+pW91vpTdrSbw99OvCL7E9cORlzH2hmrrehd/EJ3g4kYjLrQMsqvEdayADtqZTMugnF0HlD6APHag4pmXUcdzmbsNnOs0NbKiWz70GXMI2cpNXyOQs564WCnCsjcwTj/hXMSVotn7PIzQjkOA4KEreMzBXR+3dDGBujUnONYHMWluuGGk5JyNy/OXRBFOSksW75nIXVOqEowPrJXD8nND33UeVxmJPAV8bmLKTWCVWTG8O9ZBhA+8/Xh5GTgFrB8eOG7t8wnYeMAuT9ks9ZInw4NBxhCLHlLIHacGg4Asmw5SyB2nBoOEIg41hDfgHyR0xfSwodlwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
=P – Q
So
(∵ P2 –Q2=P – Q)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
=1
Simplify the following as a quotient of two polynomials in the simplest form.
(i) (ii)
(iii) (iv)
(v) (vi)
(vii) (viii)
Answer:
(i)
The like terms are cancelled.
=x2 + 2x + 4 required solution.
(ii)
The like terms are cancelled.
(iii)
The like terms are cancelled.
Required solution
(iv)
The like terms are cancelled.
Required solution.
(v)
The like terms are cancelled.
Required Solution
(vi)
Like terms are cancelled
Required solution
(vii)
Like terms are cancelled
(viii)
= 0
Question 2.
Answer:
Let p(x)= , q(x)=
And the rational expression be r(x),
So according to question,
p(x) = q(x) + r(x)
or, r(x) = p(x) – q(x)
So required rational expression r(x)
Question 3.
Which rational expression should be subtracted from to get 2x2 – 5x + 1 ?
Answer:
Let p(x) = , q(x) =
And rational expression be r(x)
According to question,
p(x) – r(x)=q(x)
or, r(x) = p(x) – q(x)
r(x) =
So required rational expression
Question 4.
If then find
Answer:
P – Q
P2 –Q2 =(P + Q)(P – Q)
=P – Q
So (∵ P2 –Q2=P – Q)
=1
Exercise 3.12
Question 1.Find the square root of the following
196a6b8c10
Answer:In Square root the power of each term is divided by 2
√(196a6b8c10) = √(142 a6b8c10)
Square Root = |14a3b4c5|
Question 2.Find the square root of the following
289 (a–b)4 (b–c)6
Answer:289 = 172
⇒Square Root = √[172(a–b)4 (b–c)6]
Square Root = |17(a–b)2(b–c)3|
Question 3.Find the square root of the following
(x + 11)2–44x
Answer:(x + 11)2–44x
⇒ x2 + 22x + 121–44x
⇒ x2–22x + 121 = (x–11)2
√[(x + 11)2–44x
Square root = |x–11|
Question 4.Find the square root of the following
(x–y)2 + 4xy
Answer:(x–y)2 + 4xy
⇒ x2 + y2–2xy + 4xy
⇒ x2 + y2 + 2xy = (x + y)2
√[(x–y)2 + 4xy]
Square Root = |x + y|
Question 5.Find the square root of the following
121x8y6
81x4y8
Answer:![](data:image/png;base64,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)
Square Root ![](data:image/png;base64,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)
⇒Square Root ![](data:image/png;base64,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)
Square Root![](data:image/png;base64,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)
Question 6.Find the square root of the following
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAeoAAAAwCAMAAADpYcn1AAAAAXNSR0IArs4c6QAAAL1QTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZmZmZpDbZrbbZrb/kDoAkDo6kGYAkGY6kGaQkLaQkLbbkNv/tmYAtmY6tmZmtpA6tpBmttv/tv/btv//25A625Bm27Zm27aQ29uQ29u229vb29v/2/+22////7Zm/9uQ/9u2//+2///bhr/hsQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAI40lEQVR4Xu1ceX/bNgyV023V2nVL1rhNdiTOurXKrqrJui2yre//sUqQAA8RpEiZdpNW+qO/1BZA8D0e4PFcVfMzIzAjMCOQgcC6vsx4231183KxOL6bbF7dnO1i/V6UfjK9cMvyoWDQ//nsbW6F2+O/0KS/eoJUp/sh6351Wm2Wp8pTsrkuu79+vkMzqbYvXlc3R28ySzdIlccgGYJqGgb9KhmwdvGIGsX6KRLcXqzUXxl+KrTeLoVp81ibW/5jbY/KbpXlLs/6a0V1evB7xEAGUQyD/rr+EntCqxo09Kzkp9VUVwhSd9IrqrP8oHW/OrnbAN1kbvmPci0J6lfHZ08nDOA3TxaLo5+V/1ZVPif4vWGAQRTCoKtP/kMM17WiujPseegii+ZzO4wW2sz2/G2/uvhV/BXzw9AmrSuYq09k01PmidWspPW6Pvn3/Sq7Z7dHF1V1s5Ac3+BUHQ7eg8CJsSgGGEQZDLpa1FI92+VPkuo+hlWUatmt2wU8AvioH4Zqab19eYGGaJ5aTWX9AgKgQSo2CtjfYUkN2LXIdCT4ONUlMaAgimCwXZou0Jx2kmrKHXsY1obpKEP1369o8KNBT72EfhrgXfDPJeX9Cr5c1wsoWFqvRdNTNUTz9pHxz7UPMQgcgWtpDcN/bgvTbRsyhE7Ut7OCrxgQOKr3gwExUQSDzjDQPbtTVKt/AfC7vqF0lFD2qT4SmXeLbjCf+qeGsRf9VODDh0c57FfQlfBNaX1T42ILP2xt/x7X0DJwapfWZvhP7dJyqljAAC7C3C5hRAKqIyAwVO8JA0KwCAbNo99wboSxz6FaYuWtDhmqYeDEwUGOgfRQoEAn5jo+/nJswrzZsSawZapoDz6OD8vEtc5huqpuRftyGzUFz4DAUb0fDAzVu2MgBlCRzN/Wp2r8E67bSyfJcKgWQzE+FixqHkGcHbj1DLOuv6V5wHMBBfRXannjkoXmrv9BBFtagXvWWVSvBQB6ZFKW9vRogcBCgC8nYuC7CGJAQZTAQI2fgHFHIF7qgbe/PRONfTDFBtIyrpqmY+AijoMfZtYOxwK+V9tNaeihDNUKBcQCi9BDkg9CKC0rj4Hu1QUwwOpR6/Hm6pQB3OnV9jpHU709/yG8fBNlN9iccKZ3wXZadJTq7DUWecPpobGDtObqIQhjVJfDIJ/qCAaqetSdnAxc/id5rpZrUruTWZn81ZvgXAs235+r/TbXWmfgaq7mN3WsbDv0SsJIbsY28zJWnAMhOFcXx0Bn4CUwWNeP78TmAe1o2utqmMA2Pw4PLvwBfCES9T/U5E17igox5KFvIAtQ+xPc09BXrjWZt5Z/37wVuXP/u1wi4n5mArXeKy1kLLiFgl9i8BwIDNV7wkCvq4tgsBYJ+BevVf3EfK1WW2okuxar6v+Xg8nap/r5K+FB5VWDzQvpR2R+R6JXL4Jc622p4dYH7pZZ/hka39UYfvbGie0MdhAIBfw8DAJD9b4woN2yvWGQs/1rAzbsWGl+OpxfvG6ZZo4B7NKp2YEgq3TjoSwGeUFMwSD9UMdhmk629IejfsQyi1Za+lzMuBw1txD2yp4ykDs2GaXH4hh1E8Vg1NoKmUEwAYT0U1LjzJzVWmSNnHuLjcxfVKfmrCecVyfULfmVCSCUxyA9CBbB5Mru/UUxkU84ctx7WActYMbgoHDPhc0IzAjMCMwI3FcE+GOK+xpt0bj0CdEn/ccYZJ905bFyYxjM388IPFQEPuMB/KFSNsddDIH0zRld5JQdIiveHc3RU9GtomwQdtvq47cKs4PgNxyZlvEOrw4myzqMD7P3qu+ajqpDOEGEllvlSBooDB1Efw07rb2+xZrQC6DullgrZ98ZvfsQJIhDDoVBfwunl+ouMORqeFKdIeuwuCbpC2q1Us5jfEGEllvlSRo01yqId9+9BKrbx3cp5zy67pZYKyV6r/1o9Q/J1VK8HAiDRpzpb5ZwoGxdNInIMUK3e2Wl8ayYtFopqg5PFELXRvVxOXfPPRYFHVjD9VNQfqWA7VyyIR1S8HJUQularnaPMJD3uOT1bVPd2GX5KNUKJNJqJd25d25kgrmWW8UkDbEoqGNJqkHl4V5VYwdyh2p1gTlL1jEc2Sy5WsJNt0NiIC/8m+rG5BhDkH1dhtZqRdUhBI4tWEBZB8qtYpIGhmq4RmPkHXLsnkY1irVU8awkZVB6BAIClRXImITWEq7sGwN5idBQHZNjeCB7ugyt1YqqQzTVQ0GEllvFJA0+1UN5B1KdP4CTWIuUJowkxW/uA2mKhgAvWPMCGUP14TBQJK/rb8T9LDkC401+To4RuBHr6jLUSxFhhBnwHNEG3FjVcquYpMGPwpN39M1X4lK3+DgtLdN1J7GWbqi+JIW/A89BYGMQ/vmEw2GAeUt/JRK0FbDMyTGAHH6ridEkKK0WI4zwPfiiEJJbsZKGUBTenX/ZsU6FoHMBQtqxR9fdiLWoeEuSEsQgCIGNAVH9ETFw8g8pZYnJMfz+FNRlRIQRVq8O6n9i99y9KErIO2Td9aNB8CUpzCQ2Ik1hBTLWXB3SQBXGoBE3wPUj7+TH5BhMQhTSZUSEESGqOUEEJ+uIU52Q9HLd3NUjEAiMJIWdxGLSFJirfdVEgOr9YSCXoZ1oz3AjX8Qjtc34EyaMHIOhOqTLiAgjvLk6IojAmczZ1GH7FTqdIu8wddeRYfTiHqcnSfExGJGm8AIZKy0zwhVeGFMEA5m0SOEFyJ83K3U7X7YsVo7BrWgDuoyIMMKiekwQIRb9RjZCdpywYhd5h6m7jkxFn4pBXJrCC2QM1YfBAJMEQfXmTCQy6oeoInIMjuqQLiOmDqF6tiOCCJHdMZIGJoqd5B1W3c1kLVo9L0lhSh+RprACGUP1R8QgZS/RzmACuoxMN9MEEcNZN2VtNZaQy++zoi8jTfEVZ1lBmMknQ7eWfqhTTJNAP1hm0ZAehTGaJm1guU8tvpg0RTCdL4xhQs/DIPmUNK7LSHZz6LPatH6d9luMpaQp9xIDp+d99rqMWZaR1HHmlw6FwAcz+PWFgTtfIAAAAABJRU5ErkJggg==)
Square Root ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPgAAAAyCAMAAABGfObrAAAAAXNSR0IArs4c6QAAAK5QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kGaQkLbbkNv/tmYAtmY6tmZmtpA6tpBmttv/tv/btv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bKEiTpAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFwklEQVRoQ+1aa3fbNgyV3HXzske9zl2zLZOXNWr3qJJsq2Tr//+xAQQfIAlSpJ2c+bTRhx7XMh4XAEGCN03z9DxFACMwd59/eOhI/PND225OUDqtfzla+vaiXf1spHPgEu/mP755X2t8ePGnEjm8etPcrW7U53I9RhpkdhcGeLG4kZ53znY+qzLwuXtZWgdD+8yEaPrKZmr6UgGv0NNY6eGq03pQnOnPZSK2vVDODPjdRavLZO6+L0/3YIE3Gi7IDkpBlR4jPW5mDZzEmf4scgp10+yvr+pKfViBwF2LtkaHJbRlnHLfc8cG3TDuaIln9EgYlPTh8v3cXf2GBUfihcAbbbtvP6sDPndfoDM9GNcfxfjmgeuUD4Q7p0dSrqSHFh8MgRYvBW7L7d72xqLmZoADfNNVZyx+wmAfCfhf16aTUnGOIDPCMtV6esQBeKRWPXf4clq32AzNytA2tPjwzOkXwjXBFrJC1Ur68OrGul/a3MYWSx0dGKklQyg+zL1uz27ZhP4PK+jmg4bVQ9wOW8wZ/EzraVBHHDDSSEnRv0RpeP5eqxrXX3r6I+TT+qrZb5VPSvoWYli7nd2vIXYI2TiM2sI9Vcg4yhy25DOuFPsYPQiOup3wqEoeBGkHnOmPFDCDnm0WVdmuWwbTGnxTmeOrigOHmtUPLwP6tbbqGbd6pvW3GzIfq0AD8446su+6Fvf1By4cti6gRwKnCKh/babuX0MR+KWdam4ScFc5Q7BgWBKwt4y6TnzXTanzwIbZewDgulh7MMPWeEmpexmnmqXHAj9c/pjeICGjvQ6uXuO+uJfxLHBuu7zUdc1j0HV5K7+L17gqOZ4AtjvsbkwPENbbYfvdJZ39fGljWlVL8MqqYTum8JOi7QyWNxxT1QFGa8NFv/8pGBiEUm+h+f9OteyObvg/rWfuQeeoTkbi05tXvrTdx5n+WH6AvWh+h8YDaXIgPYEFR9bnb1SNUmG+hV38362/yAXgL6/b9jl1J3NyM8WKemCrXkHG2yRye74LpM3JjekXAgf7FzkdSpcDd1rrzthOLgx6mZ5Rr80oZWXi2ryQ8NKMs3DWTFUMt5vO9JeLemAjM3sZm6+MykXxnO1jMl4zRzvbbqJmtbMw18/d5ldKuCRdPs6L0kdkXO5Cj/AtNIAXpZN/vfnC5lav+NwlnoALGXqMy8azKQQCJ48YnwBwOQ8KuB15PqYPx21nZ1OtJznyVOqfZnPLrHH/VTF9cdqhjRk97dRmFMVaSvfx22r2wzpvDtqONls+ZwscDGO+KlgU44VxAm+o1Zi8DBzud92PK1gUNiQoIsPRZiWTVcTBMOarikWxyIlNGSz1tjiPs3uWKhaFl6uZ+Wk+LCFRJA7GEklpFiV1U42QlRNHAa9jURhwOw+ri+QiEsW7zdXUl2a+cixKBjg5gaW+QddKSt1UR479CC2KNAjRZjkyxkaLcyR6aRjmK8eixMB9NsXZrwGeYz8iiwINommzHBnjgIccDL4h5ivHokRuhGyKyjXV3dKd27T+Gq6uOIsisR+JO3VOgxjaLEfGMOABB+OYLwtcYFEiNwI2Ba90qGKWgc87YKA63AIE9kM5Ko84IQ1iaTOJjIlUxByMZb4kFiXlRkgqzHBJipR3CXCFTXFIOfYjzniSBsmRMWyNp8gnW+oSixK6kWZTljOufFGX/jn2Q2inKRokR8YkgHs8SI4+ygL3tCwDxyt/yrhtxwL7IQBP0SA5MiZc4wIHk2VR4s3Fog3YlALgsLz3HV3/Ky0i+yFtoAkaJEfGOOBJDibLokRuJNmUZeD717Dn019rqSOXzH5IwFM0SIaMccDTHEyORYndSLEpy8CtL7V/qdSkaJCSszqzehqLYhTFTM7SPs5cqPjbtCwNsjydcdwRB1PhhtUTcTFVGa9gUfI0SMVY/3/P4zzvJZ8flwYp8WDhN5UZfwCLZ6Lio747P5MYn58b/wFrqAkeTypyvQAAAABJRU5ErkJggg==)
Question 7.Find the square root of the following:
16x2 –24x + 9
Answer:16x2 –24x + 9
The above expression can be rewritten as
(4x)2–2×3×4x + 32
It is in the form of (a–b)2
= (4x–3)2
Square root = √(4x–3)2
|4x–3|
Question 8.Find the square root of the following:
(x2 – 25)( x2 + 8x + 15)( x2 –2x–15)
Answer:We factorize each of the above polynomials
x2 – 25 = x2 – 52
Since it is in the form of a2–b2 = (a–b)(a + b)
⇒ x2 – 25 = (x–5)(x + 5) …(i)
x2 + 8x + 15 = x2 + 5x + 3x + 15
⇒ x2 + 8x + 15 = x(x + 5) + 3(x + 5)
⇒ x2 + 8x + 15 = (x + 3)(x + 5) …(ii)
x2 –2x–15 = x2 –5x + 3x–15
⇒ x2 –2x–15 = x(x–5) + 3(x–5)
⇒ x2 –2x–15 = (x + 3)(x–5) … (iii)
Combining (i), (ii) & (iii) we get
(x2 – 25)( x2 + 8x + 15)( x2 –2x–15) = = (x–5)2(x + 5)2(x + 3)2
Square Root = √[(x–5)2(x + 5)2(x + 3)2]
|(x–5)(x + 5)(x + 3)|
Question 9.Find the square root of the following:
4x2 + 9y2 + 25z2–12xy + 30yz–20zx
Answer:The above expression can be rewritten as
(2x)2 + (–3y)2 + (–5z)2 + 2((–3y)×(2x) + (–5z)×(–3y) + (2x)×(–5z))
The above expression is in the form of
(a–b–c)2 = a2 + b2 + c2 + 2(–ab + bc – ca)
So the expression becomes (2x–3y–5z)2
Square Root = √(2x–3y–5z)2
|2x–3y–5z|
Question 10.Find the square root of the following:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGUAAAA3CAYAAAAVOoQzAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAKISURBVHja7Zk/TvNAEMXnABzgQ/KUHCCZcAXL+sQBSKx0iAptQ4026SIhDkEHd6ClyQ0ocgPuAKwj56/teJN4s86+4hWObGW9P8+8mR0aj8cE+SVsAqBAgAIoEKAASrukVL8Tx/LETF8cPzwCyok1SuWGmL8MECL6ARTf4AAKoAAKoAAKoAAKoJwtFK2Hl13iaaJ0B1A8gTIQeiXib0DxBIrZABkO7xApnkAxaUsk1UhfHkFJRfRQ60tfoZAMXoOCkqWtdHTjk9FrlXSY6DsDsqI2RczeD04m9xdJRB+bL2+Ug4JWP5b+VZdputgn7k6TVP0PqiT2C0hx9JZ9wIDSsPKMwkmq7yeTizxqlllGZsaTvYPyEPNjm3K+zXpNlEis7srT/3Zvt17aEs02zXH992aaw3OGsrvh3hEpht78xj8AUfKRh9u8vJRZX+krn1+yTVAWkVJQsldWVcaEsmvm9ya9IkQoc/PvTjejpNRTlilLZr0evzVd4oYIZSD8UravuzvilTR2rBcqKg1rl4s1n91Xx17vPlAroURR9OmiGw4pUrJTkIJqbHf6MvmOk3c1Uv/KyjZAsV9vVh5Lqq2PWYyf9Lj3lhvQshvdLt0AxS5CWAYvRccvIsnzqkUUd58bf7jIq3/+0kRZ7BrKoacPtutV6fVt2TFLkT0UNDPrN24b3fFTmWsoh05JbdZrgFQXCBUdfSjycUpau/o6R/k6JQ0aiq9T0mCh+DglDRpK26akwRk9IgVQwoFy6EAOUBr0iVMM5ADFwsBdDeQAxSqVuRnIAYpFD9LEQA5QDoTiaiAHKHXSl+OBHKDU8BPXAzlAqVF5uR7IAUqFTjWQA5RAhU0AFAhQWqpfm3dLTyouCJYAAAAASUVORK5CYII=)
Answer:The equation can be written as
![](data:image/png;base64,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)
The above equation is in the form of (a + b)2 = a2 + b2 + 2ab
So it becomes
![](data:image/png;base64,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)
Square root![](data:image/png;base64,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)
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Question 11.Find the square root of the following:
(6x2 + 5x –6) (6x2–x–2)(4x2 + 8x + 3)
Answer:We factorize each of the above polynomials
6x2 + 5x –6 = 6x2 + 9x –4x –6
⇒ 6x2 + 5x –6 = 3x(2x + 3)–2(2x + 3)
⇒ 6x2 + 5x –6 = (3x–2)(2x + 3) …(i)
6x2–x–2 = 6x2–4x + 3x–2
⇒ 6x2–x–2 = 2x(3x–2) + 1(3x–2)
⇒ 6x2–x–2 = (2x + 1)(3x–2) …(ii)
4x2 + 8x + 3 = 4x2 + 6x + 2x + 3
⇒ 4x2 + 8x + 3 = 2x(2x + 3) + 1(2x + 3)
⇒ 4x2 + 8x + 3 = (2x + 1)(2x + 3) …(iii)
Combining (i), (ii) & (iii) we get
(6x2 + 5x –6) (6x2–x–2)(4x2 + 8x + 3) = (3x–2)2(2x + 3)2(2x + 1)2
Square Root = √ (3x–2)2(2x + 3)2(2x + 1)2
| (3x–2)(2x + 3)(2x + 1)|
Question 12.Find the square root of the following:
(2x2 –5x + 2) (3x2–5x–2) (6x2 – x –1)
Answer:We factorize each of the above polynomials
2x2 –5x + 2 = 2x2–4x–x + 2
⇒ 2x2 –5x + 2 = 2x(x–2)–1(x–2)
⇒ 2x2 –5x + 2 = (2x–1)(x–2) …(i)
3x2–5x–2 = 3x2–6x + x–2
⇒ 3x2–5x–2 = 3x(x–2) + 1(x–2)
⇒ 3x2–5x–2 = (3x + 1)(x–2) …(ii)
6x2 – x –1 = 6x2– 3x + 2x–1
⇒ 6x2 – x –1 = 3x(2x–1) + 1(2x–1)
⇒ 6x2 – x –1 = (3x + 1)(2x–1) …(iii)
Combining (i), (ii) & (iii) we get
(2x2 –5x + 2) (3x2–5x–2) (6x2 – x –1) = (2x–1)2(x–2)2(3x + 1)2
Square Root = √(2x–1)2(x–2)2(3x + 1)2
|(2x–1)(x–2)(3x + 1)|
Find the square root of the following
196a6b8c10
Answer:
In Square root the power of each term is divided by 2
√(196a6b8c10) = √(142 a6b8c10)
Square Root = |14a3b4c5|
Question 2.
Find the square root of the following
289 (a–b)4 (b–c)6
Answer:
289 = 172
⇒Square Root = √[172(a–b)4 (b–c)6]
Square Root = |17(a–b)2(b–c)3|
Question 3.
Find the square root of the following
(x + 11)2–44x
Answer:
(x + 11)2–44x
⇒ x2 + 22x + 121–44x
⇒ x2–22x + 121 = (x–11)2
√[(x + 11)2–44x
Square root = |x–11|
Question 4.
Find the square root of the following
(x–y)2 + 4xy
Answer:
(x–y)2 + 4xy
⇒ x2 + y2–2xy + 4xy
⇒ x2 + y2 + 2xy = (x + y)2
√[(x–y)2 + 4xy]
Square Root = |x + y|
Question 5.
Find the square root of the following
121x8y681x4y8
Answer:
Square Root
⇒Square Root
Square Root
Question 6.
Find the square root of the following
Answer:
Square Root
Question 7.
Find the square root of the following:
16x2 –24x + 9
Answer:
16x2 –24x + 9
The above expression can be rewritten as
(4x)2–2×3×4x + 32
It is in the form of (a–b)2
= (4x–3)2
Square root = √(4x–3)2
|4x–3|
Question 8.
Find the square root of the following:
(x2 – 25)( x2 + 8x + 15)( x2 –2x–15)
Answer:
We factorize each of the above polynomials
x2 – 25 = x2 – 52
Since it is in the form of a2–b2 = (a–b)(a + b)
⇒ x2 – 25 = (x–5)(x + 5) …(i)
x2 + 8x + 15 = x2 + 5x + 3x + 15
⇒ x2 + 8x + 15 = x(x + 5) + 3(x + 5)
⇒ x2 + 8x + 15 = (x + 3)(x + 5) …(ii)
x2 –2x–15 = x2 –5x + 3x–15
⇒ x2 –2x–15 = x(x–5) + 3(x–5)
⇒ x2 –2x–15 = (x + 3)(x–5) … (iii)
Combining (i), (ii) & (iii) we get
(x2 – 25)( x2 + 8x + 15)( x2 –2x–15) = = (x–5)2(x + 5)2(x + 3)2
Square Root = √[(x–5)2(x + 5)2(x + 3)2]
|(x–5)(x + 5)(x + 3)|
Question 9.
Find the square root of the following:
4x2 + 9y2 + 25z2–12xy + 30yz–20zx
Answer:
The above expression can be rewritten as
(2x)2 + (–3y)2 + (–5z)2 + 2((–3y)×(2x) + (–5z)×(–3y) + (2x)×(–5z))
The above expression is in the form of
(a–b–c)2 = a2 + b2 + c2 + 2(–ab + bc – ca)
So the expression becomes (2x–3y–5z)2
Square Root = √(2x–3y–5z)2
|2x–3y–5z|
Question 10.
Find the square root of the following:
Answer:
The equation can be written as
The above equation is in the form of (a + b)2 = a2 + b2 + 2ab
So it becomes
Square root
Question 11.
Find the square root of the following:
(6x2 + 5x –6) (6x2–x–2)(4x2 + 8x + 3)
Answer:
We factorize each of the above polynomials
6x2 + 5x –6 = 6x2 + 9x –4x –6
⇒ 6x2 + 5x –6 = 3x(2x + 3)–2(2x + 3)
⇒ 6x2 + 5x –6 = (3x–2)(2x + 3) …(i)
6x2–x–2 = 6x2–4x + 3x–2
⇒ 6x2–x–2 = 2x(3x–2) + 1(3x–2)
⇒ 6x2–x–2 = (2x + 1)(3x–2) …(ii)
4x2 + 8x + 3 = 4x2 + 6x + 2x + 3
⇒ 4x2 + 8x + 3 = 2x(2x + 3) + 1(2x + 3)
⇒ 4x2 + 8x + 3 = (2x + 1)(2x + 3) …(iii)
Combining (i), (ii) & (iii) we get
(6x2 + 5x –6) (6x2–x–2)(4x2 + 8x + 3) = (3x–2)2(2x + 3)2(2x + 1)2
Square Root = √ (3x–2)2(2x + 3)2(2x + 1)2
| (3x–2)(2x + 3)(2x + 1)|
Question 12.
Find the square root of the following:
(2x2 –5x + 2) (3x2–5x–2) (6x2 – x –1)
Answer:
We factorize each of the above polynomials
2x2 –5x + 2 = 2x2–4x–x + 2
⇒ 2x2 –5x + 2 = 2x(x–2)–1(x–2)
⇒ 2x2 –5x + 2 = (2x–1)(x–2) …(i)
3x2–5x–2 = 3x2–6x + x–2
⇒ 3x2–5x–2 = 3x(x–2) + 1(x–2)
⇒ 3x2–5x–2 = (3x + 1)(x–2) …(ii)
6x2 – x –1 = 6x2– 3x + 2x–1
⇒ 6x2 – x –1 = 3x(2x–1) + 1(2x–1)
⇒ 6x2 – x –1 = (3x + 1)(2x–1) …(iii)
Combining (i), (ii) & (iii) we get
(2x2 –5x + 2) (3x2–5x–2) (6x2 – x –1) = (2x–1)2(x–2)2(3x + 1)2
Square Root = √(2x–1)2(x–2)2(3x + 1)2
|(2x–1)(x–2)(3x + 1)|
Exercise 3.13
Question 1.Find the square root of the following polynomials by division method.
x4–4x3 + 10x2–12x + 9
Answer:Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
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|x2–2x + 3|
Question 2.Find the square root of the following polynomials by division method.
4x4 + 8x3 + 8x2 + 4x + 1
Answer:Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
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|2x2 + 2x + 1|
Question 3.Find the square root of the following polynomials by division method.
9 x4–6 x3 + 7 x2–2x + 1
Answer:Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient. To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
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|3x2–x + 1|
Question 4.Find the square root of the following polynomials by division method.
4 + 25x212x–24x3 + 16x4
Answer:Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
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)
|4x2–3x + 2|
Question 5.Find the values of a and b if the following polynomials are perfect squares.
4x4–12 x3 + 37x2 + ax + b
Answer:Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
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Since it is a perfect square the remainder is 0
a = –42, b = 49
Question 6.Find the values of a and b if the following polynomials are perfect squares.
x4–4x3 + 6x2–ax + b
Answer:Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
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Since it is a perfect square the remainder is 0
a = 12, b = 9
Question 7.Find the values of a and b if the following polynomials are perfect squares.
ax4 + bx3 + 109x2–60x + 36
Answer:Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
We rearrange the equation in the increasing order of power of x.
The polynomial becomes 36–60x + 109x2 + bx3 + ax4
![](data:image/jpeg;base64,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Since it is a perfect square the remainder is 0
a = 49, b = –70
Question 8.Find the values of a and b if the following polynomials are perfect squares.
a x4–bx3 + 40 x2 + 24x + 36
Answer:Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
We rearrange the equation in the increasing order of power of x.
The polynomial becomes 36 + 24x + 40x2–bx3 + ax4
![](data:image/jpeg;base64,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Since it is a perfect square the remainder is 0
a = 9, b = –12
Find the square root of the following polynomials by division method.
x4–4x3 + 10x2–12x + 9
Answer:
Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
|x2–2x + 3|
Question 2.
Find the square root of the following polynomials by division method.
4x4 + 8x3 + 8x2 + 4x + 1
Answer:
Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
|2x2 + 2x + 1|
Question 3.
Find the square root of the following polynomials by division method.
9 x4–6 x3 + 7 x2–2x + 1
Answer:
Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient. To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
|3x2–x + 1|
Question 4.
Find the square root of the following polynomials by division method.
4 + 25x212x–24x3 + 16x4
Answer:
Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
|4x2–3x + 2|
Question 5.
Find the values of a and b if the following polynomials are perfect squares.
4x4–12 x3 + 37x2 + ax + b
Answer:
Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
Since it is a perfect square the remainder is 0
a = –42, b = 49
Question 6.
Find the values of a and b if the following polynomials are perfect squares.
x4–4x3 + 6x2–ax + b
Answer:
Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
Since it is a perfect square the remainder is 0
a = 12, b = 9
Question 7.
Find the values of a and b if the following polynomials are perfect squares.
ax4 + bx3 + 109x2–60x + 36
Answer:
Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
We rearrange the equation in the increasing order of power of x.
The polynomial becomes 36–60x + 109x2 + bx3 + ax4
Since it is a perfect square the remainder is 0
a = 49, b = –70
Question 8.
Find the values of a and b if the following polynomials are perfect squares.
a x4–bx3 + 40 x2 + 24x + 36
Answer:
Step 1: Find the algebraic expression whose square gives the first term.
Step 2: Add the divisor and quotient .To this sum add a suitable expression and add the same expression in the quotient such that the product gives the next term.
Step 3: Continue the process until all the terms are divided.
We rearrange the equation in the increasing order of power of x.
The polynomial becomes 36 + 24x + 40x2–bx3 + ax4
Since it is a perfect square the remainder is 0
a = 9, b = –12
Exercise 3.14
Question 1.Solve the following quadratic equations by factorization method.
(2x + 3)2 – 81 = 0
Answer:(2x + 3)2 – 81 = 0
= (2x)2 + 2(2x)(3) + 32 – 81 = 0
= 4x2 + 12x + 9 –81 = 0
= 4x2 + 12x – 72 = 0
Divide by 4 both sides
⇒ ![](data:image/png;base64,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)
= x2 + 3x – 18 = 0
= x2 + 6x – 3x – 18 = 0
= x (x + 6) – 3 (x + 6) = 0
= (x + 6) (x – 3) = 0
x + 6 = 0 or x – 3 = 0
x = –6 or x = 3
Question 2.Solve the following quadratic equations by factorization method.
3x2 –5x –12 = 0
Answer:3x2 – 5x –12 = 0
= 3x2 – 9x + 4x – 12 = 0
= 3x (x – 3) + 4(x – 3) = 0
= (3x + 4) (x – 3) = 0
3x + 4 = 0 or x –3 = 0
3x = –4 or x = 3
or x = 3
Question 3.Solve the following quadratic equations by factorization method.
![](data:image/png;base64,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)
Answer:√5x2 + 2x – 3√5 = 0
= √5x2 + 5x – 3x – 3√5 = 0
= √5x (x + √5) – 3(x + √5) = 0
= (√5x – 3) (x + √5) = 0
√5x – 3 = 0 or x + √5 = 0
√5x = 3 or x = –√5
or x = –√5
Question 4.Solve the following quadratic equations by factorization method.
3(x2 – 6) =x (x + 7)–3
Answer:3(x2 – 6) =x (x + 7)–3
= 3x2 – 18 = x2 + 7x – 3
= 3x2 – 18 – x2 – 7x + 3 = 0
= 2x2 – 7x – 15 = 0
= 2x2 – 10x + 3x – 15 = 0
= 2x(x – 5) + 3(x – 5) = 0
= (2x + 3)(x – 5) = 0
2x + 3 = 0 or x – 5 = 0
2x = –3 or x = 5
or x = 5
Question 5.Solve the following quadratic equations by factorization method.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAF0AAAA1CAYAAAAj1uf0AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAN9SURBVHja7ZqxUttAEIY3PfSBYelMj068AeO5JFB6JrLGXcaFhlEDMyo8GeHOMwkPQAedKdKmgZImfgb7CexnwNFJOnNWziCDJeviLbbRGGS+2/v3312g2+0CRblBEAg6Qacg6AS9Wn8EwIdsEPQCIwja202GVwgwjmA/iUDWvGoHwTZBLyB6vfYW34MHQH7v+GFNPAt9p8YR7mGPP7R7vS2CvuK4dNkJAE64Hx6qz0OfH0aZP2Hu5QlBX3Gc1fFcCz1s7TCAEUEvIGRGA2veqhoe34BIckheCgqXwU1UPKeA1qDhdQ4EcET7t9R4gl5QCPcSg4+i6s7lv4EeywziY2obKw/eeOBRMb1A5HdCv2d2UWQ9WcYiLSNMVZcivLvUeayfnRP0lWs53Oos46xpilwNQS/Ep0eZ7oSneujujTHQTRkepU3QUGR73Tn7Ir6rmMW4HEMANmyF4U6pMHNy0zgBp8YQ/sjhkfS/1XUuUfG0969n3zeKfZtfl+3TO17jwEIYSOsquIlEeBW6sF6M8R+y6j8fABuVnTUmRcc7tuLOWDl4CT8rfXPQhQ7WWf171mZVfXi07pD1A7kbqsk6s66ahM0935DQF+mWKQuEIhozVve/LToMnbt6tTDEtix1Ac+FK9Wt1AvPP8dx9iWbGomlzZnpArbnNay4MNj8l9pSq82H2vUljQobVnnQlMdxLIq3yo7Otuo/nOjRrCDEswxF65XPxIVC2DSOeGdahqc+/ylPLNvdJrJsDXQG5MWT8t2jr7MhUualz5LCRraN/byFNm9mmV4bxORzEZPchVSnTXL2kXe4lP6ucb7s0teGZQ5tXQcobtBLN+NdBSFZGOBjmcOlvHKQjbI0XTDRuZmlocfal8nmZIb9qe9f+h8TfcfJsdexNlnTk+bSDXUHuxC67lTjIinMv/LSIGjt2mj3ZebPSVAQ7G6iPQwddioMR/aGiPGA2uXPQY9bfoBRMmvxLDk8iv+Rh7k/NYdwkcmaC6nvVZ7VFHJjnCMnHQNMdZG9JfNWMTM4Egte7vqf1R9IPfrclfv3mm5OgyQc3rKGgOYn62jICAJBJ+gUBJ2gU2wgdJOXKWZ2f4YvU4zNdJOXKUZro6nLFOOL0luXKQT9nbHsMoWgrwh62cuUzZYXA5cpRgM3dZliMHBzlynGQjd5mUKzEIJO0CkKir/5cRspW3OeuAAAAABJRU5ErkJggg==)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 3x2 – 8 = 2x
= 3x2 – 2x – 8 = 0
= 3x2 – 6x + 4x – 8 =0
= 3x (x – 2) + 2(x – 2) = 0
= (3x + 2) (x – 2) = 0
3x + 2 = 0 or x – 2 = 0
3x = –2 or x = 2
or x = 2
Question 6.Solve the following quadratic equations by factorization method.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= 5(x2 + 1) = 26x
= 5x2 + 5 = 26x
= 5x2 –26x + 5 = 0
= 5x2 – 25x – x + 5 = 0
= 5x (x – 5) – (x – 5) = 0
= (5x – 1) (x – 5) = 0
5x – 1 = 0 or x – 5 = 0
5x = 1 or x = 5
or x = 5
Question 7.Solve the following quadratic equations by factorization method.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= 15(x2 + x2 + 2x + 1) = 34(x2 + x)
= 15(2x2 + 2x + 1) = 34(x2 + x)
= 30x2 + 30x + 15 = 34x2 + 34x
= 34x2 + 34x – 30x2 –30x – 15 =0
= 4x2 – 4x – 15 = 0
= 4x2 – 10x + 6x – 15 = 0
= 2x(2x – 5) + 3(2x – 5) = 0
= (2x + 3) (2x – 5) = 0
2x + 3 =0 or 2x – 5 = 0
2x = –3 or 2x = 5
or x =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAEhQTFRFAAAAAAAAAAA6OgAAOgA6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kLbbkNv/tmY625A627Zm29u2/7Zm/9uQ//+2///bhAIcygAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYklEQVQYV42PWQ6AMAhEp+5atVJre/+bCsUl0ZjIzwvDNgDPIGNM4QCyWrnIeq3SVmotjQ3S5HKeFh4bXts+Bd4m8X/g7Fz77CN2M7z4FD+VkpoMrzZJERjBIrZy7vjpvroDSpYClwPClH8AAAAASUVORK5CYII=)
Question 8.Solve the following quadratic equations by factorization method.
a2b2x2 – (a2 + b2) x + 1 =0
Answer:a2b2x2 – (a2 + b2) x + 1 = 0
= a2b2x2 – a2x – b2x + 1 = 0
= a2x (b2x – 1) – (b2x – 1) = 0
= (a2x – 1) (b2x – 1) =0
a2x – 1 = 0 or b2x – 1 = 0
a2x = 1 or b2x = 1
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Question 9.Solve the following quadratic equations by factorization method.
2(x + 1)2 –5 (x + 1) =12
Answer:2(x + 1)2 – 5(x + 1) = 12
= 2(x2 + 2x + 1) – 5x – 5 = 12
= 2x2 + 4x + 2 – 5x – 5 –12 = 0
= 2x2 – x – 15 = 0
= 2x2 – 6x + 5x – 15 = 0
= 2x (x – 3) + 5(x –3) = 0
= (2x + 5) (x – 3) = 0
2x + 5 = 0 or x – 3 =0
2x = –5 or x = 3
x =
or x = 3
Question 10.Solve the following quadratic equations by factorization method.
3(x–4)2 – 5 (x – 4) = 12
Answer:3(x – 4)2 – 5(x – 4) = 12
= 3(x2 – 8x + 16) – 5x + 20 = 12
= 3x2 – 24x + 48 –5x + 20 – 12 = 0
= 3x2 – 29x + 56 = 0
= 3x2 – 21x – 8x + 56 = 0
= 3x(x – 7) – 8(x – 7) =0
= (x – 7) (3x – 8) = 0
x – 7 = 0 or 3x – 8 = 0
x = 7 or 3x = 8
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Solve the following quadratic equations by factorization method.
(2x + 3)2 – 81 = 0
Answer:
(2x + 3)2 – 81 = 0
= (2x)2 + 2(2x)(3) + 32 – 81 = 0
= 4x2 + 12x + 9 –81 = 0
= 4x2 + 12x – 72 = 0
Divide by 4 both sides
⇒
= x2 + 3x – 18 = 0
= x2 + 6x – 3x – 18 = 0
= x (x + 6) – 3 (x + 6) = 0
= (x + 6) (x – 3) = 0
x + 6 = 0 or x – 3 = 0
x = –6 or x = 3
Question 2.
Solve the following quadratic equations by factorization method.
3x2 –5x –12 = 0
Answer:
3x2 – 5x –12 = 0
= 3x2 – 9x + 4x – 12 = 0
= 3x (x – 3) + 4(x – 3) = 0
= (3x + 4) (x – 3) = 0
3x + 4 = 0 or x –3 = 0
3x = –4 or x = 3
or x = 3
Question 3.
Solve the following quadratic equations by factorization method.
Answer:
√5x2 + 2x – 3√5 = 0
= √5x2 + 5x – 3x – 3√5 = 0
= √5x (x + √5) – 3(x + √5) = 0
= (√5x – 3) (x + √5) = 0
√5x – 3 = 0 or x + √5 = 0
√5x = 3 or x = –√5
or x = –√5
Question 4.
Solve the following quadratic equations by factorization method.
3(x2 – 6) =x (x + 7)–3
Answer:
3(x2 – 6) =x (x + 7)–3
= 3x2 – 18 = x2 + 7x – 3
= 3x2 – 18 – x2 – 7x + 3 = 0
= 2x2 – 7x – 15 = 0
= 2x2 – 10x + 3x – 15 = 0
= 2x(x – 5) + 3(x – 5) = 0
= (2x + 3)(x – 5) = 0
2x + 3 = 0 or x – 5 = 0
2x = –3 or x = 5
or x = 5
Question 5.
Solve the following quadratic equations by factorization method.
Answer:
= 3x2 – 8 = 2x
= 3x2 – 2x – 8 = 0
= 3x2 – 6x + 4x – 8 =0
= 3x (x – 2) + 2(x – 2) = 0
= (3x + 2) (x – 2) = 0
3x + 2 = 0 or x – 2 = 0
3x = –2 or x = 2
or x = 2
Question 6.
Solve the following quadratic equations by factorization method.
Answer:
=
= 5(x2 + 1) = 26x
= 5x2 + 5 = 26x
= 5x2 –26x + 5 = 0
= 5x2 – 25x – x + 5 = 0
= 5x (x – 5) – (x – 5) = 0
= (5x – 1) (x – 5) = 0
5x – 1 = 0 or x – 5 = 0
5x = 1 or x = 5
or x = 5
Question 7.
Solve the following quadratic equations by factorization method.
Answer:
=
=
= 15(x2 + x2 + 2x + 1) = 34(x2 + x)
= 15(2x2 + 2x + 1) = 34(x2 + x)
= 30x2 + 30x + 15 = 34x2 + 34x
= 34x2 + 34x – 30x2 –30x – 15 =0
= 4x2 – 4x – 15 = 0
= 4x2 – 10x + 6x – 15 = 0
= 2x(2x – 5) + 3(2x – 5) = 0
= (2x + 3) (2x – 5) = 0
2x + 3 =0 or 2x – 5 = 0
2x = –3 or 2x = 5
or x =
Question 8.
Solve the following quadratic equations by factorization method.
a2b2x2 – (a2 + b2) x + 1 =0
Answer:
a2b2x2 – (a2 + b2) x + 1 = 0
= a2b2x2 – a2x – b2x + 1 = 0
= a2x (b2x – 1) – (b2x – 1) = 0
= (a2x – 1) (b2x – 1) =0
a2x – 1 = 0 or b2x – 1 = 0
a2x = 1 or b2x = 1
Question 9.
Solve the following quadratic equations by factorization method.
2(x + 1)2 –5 (x + 1) =12
Answer:
2(x + 1)2 – 5(x + 1) = 12
= 2(x2 + 2x + 1) – 5x – 5 = 12
= 2x2 + 4x + 2 – 5x – 5 –12 = 0
= 2x2 – x – 15 = 0
= 2x2 – 6x + 5x – 15 = 0
= 2x (x – 3) + 5(x –3) = 0
= (2x + 5) (x – 3) = 0
2x + 5 = 0 or x – 3 =0
2x = –5 or x = 3
x = or x = 3
Question 10.
Solve the following quadratic equations by factorization method.
3(x–4)2 – 5 (x – 4) = 12
Answer:
3(x – 4)2 – 5(x – 4) = 12
= 3(x2 – 8x + 16) – 5x + 20 = 12
= 3x2 – 24x + 48 –5x + 20 – 12 = 0
= 3x2 – 29x + 56 = 0
= 3x2 – 21x – 8x + 56 = 0
= 3x(x – 7) – 8(x – 7) =0
= (x – 7) (3x – 8) = 0
x – 7 = 0 or 3x – 8 = 0
x = 7 or 3x = 8
Exercise 3.15
Question 1.Solve the following quadratic equations by completing the square.
x2 + 6x –7 = 0
Answer:x2 + 6x – 7 = 0
= x2 + 6x = 7
Add 9 on both sides
= x2 + 6x + 9 = 7 + 9
= x2 + 2(3)(x) + 32 = 16
= (x + 3)2 = 16
= x + 3 = √16
= x + 3 = ± 4
x + 3 = 4 or x + 3 = –4
x = 4 – 3 or x = – 4 – 3
x = 1 or x = – 7
Question 2.Solve the following quadratic equations by completing the square.
x2 + 3x + 1 =0
Answer:x2 + 3x + 1 =0
= x2 + 3x = –1
Add
on both sides
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Question 3.Solve the following quadratic equations by completing the square.
2x2 + 5x –3 = 0
Answer:2x2 + 5x – 3 = 0
= 2x2 + 5x = 3
Add
on both sides
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Question 4.Solve the following quadratic equations by completing the square.
4x2 + 4bx – (a2 – b2) = 0
Answer:4x2 + 4bx – (a2 – b2) = 0
Divide the whole equation by 4
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Question 5.Solve the following quadratic equations by completing the square.
x2 – (√3 + 1) x + √3 = 0
Answer:x2 – (√3 + 1)x + √3 = 0
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Question 6.Solve the following quadratic equations by completing the square.
=3x + 2
Answer:![](data:image/png;base64,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)
= 5x + 7 = (3x + 2)(x – 1)
= 5x + 7 = 3x(x – 1) + 2 (x – 1)
= 5x + 7 = 3x2 – 3x + 2x –2
= 5x + 7 = 3x2 – x – 2
= 3x2 – x – 2 – 5x – 7 =0
= 3x2 – 6x – 9 = 0
Divide whole equation by 3
= x2 – 2x – 3 = 0
= x2 – 3x + x – 3 = 0
= x (x – 3) + (x – 3) = 0
= (x – 3) (x + 1) = 0
x – 3 = 0 or x + 1 = 0
x = 3 or x = –1
Question 7.Solve the following quadratic equations using quadratic formula.
x27x + 12= 0
Answer:x2 – 7x + 12 = 0
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN4AAAAXCAMAAABTXgH5AAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOmY6OmaQOma2OpC2OpDbZgAAZjoAZjo6ZmY6ZpDbZrbbZrb/kDoAkGYAkGY6kLbbkNv/tmYAtmY6tmZmtpA6ttv/tv/btv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///bVjVi7QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACGklEQVRYR+1Wf1PCMAxdp8gUQQaKCBRFqyh06/f/djbd+nNb1yl34t36B3cdWfJe8pIsivrTZ6DPwP/PAEEXbydncUTo8eROgxzu5xvbjnSjR5A4VwdvtGM4vRIQ288QGjjYghgZRtmsUqyO9DCUhSYtxWk1kJgUIIwWUZZ2S7XLno53FTYd6QmXre+E0tOA8BD8WkWno51NgH3cIDTxlrSG3usKxQ9ddJCn09JcBoRme0+K0JSr7FaVF26xr9YmIEfTx9gSK1sODwzbz1zYVXrxeOekrY0qkSF0QJqMNoUXmiwi9iTLALcsDaWHQZx5WrQ3nCoXpQtca1RDj6coT0EbcOrfsghzUuZdBCx/eFUxTB1VBnGTp9a3AagqaVIpfIvsG3rPguFUz0XlSEjTA9EWwpUgtIwbFaEBsaXUvDK26bH9PGlZOT+g5yIzUqECCj6/oydF0ShOaAU9tEPFKYaxgtwuTpqoJOuAQfT84sQcvH2c0SKuncUJLwWISHeQXk46oKJX1EDq12nTOoVKORHRs6Y83cUAec3u/StXTb2oDE0QH7cv/nFrwjIzoQMKPqKuJF5H2VxOTsK3NXv2fYyUgOg1N2K40n1m6C3fel9p8wcRW/HWRJfFkpP07ladvoYsxciAFFoemmYasS2KJ5+JBMG34WDdOFc0oFK4XnqNbur+yL3rqJOrczTeLs4RVY+pz0CfgT/KwDc0XzVBbCRDMAAAAABJRU5ErkJggg==)
(–7)2– 4 (1) (12)
=49 – 48
=1
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 8.Solve the following quadratic equations using quadratic formula.
15x2 – 11x + 2 = 0
Answer:15x2 – 11x + 2 = 0
![](data:image/png;base64,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)
⇒ a = 15 , b = –11 and c = 2
(–11)2– 4 (15) (2)
=121 – 120
=1
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 9.Solve the following quadratic equations using quadratic formula.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGQAAAA1CAYAAAC3ME4GAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAOVSURBVHja7ZnPTttAEMY35Qo9F8HkhuiRZMMbINcIisTBAgflVClCFvKBgHxAxOEWqWpfAbUXeIa2nHqquLeqKA9A4REK6a6TQOom8Z+s4zFMpE8CS+DM/HZmdvdjR0dHjIRHlAQCQiIgBIT0KIGIT66jCcZauUwmOGIMqAOxLINrGq8DsJ+g7dSyCCNqDGiDaVT4Csvnv+fz7IcIrJVFIHFiwB/UFl/NKpA4MRAQAkJACAgBISAEhIAQEAJCQAgIARkczN4jALKXWSDdC7lDs7Amg2HFzY9pXjKKz7P2+z3Jn3NJxIB3VdX0hTxjVyKAP70C3dod98JwLGO+COzcS6oUFM+1DWs1CEqcGDLZAsapA2uJA8td9yZULPs7CaZgHq4pXwBJBtPySjabPoZUs1md1CH3GXSzXm02J+Uz1zbnXgH71IbCLyuuO50YEOOhR0709jrf89AJ3tGglundkWg5i/rOm76gZtkXxuBGt91CIkBctzLNGbvolqPI+53sdd3n4nfxDH7rtcbCUwEyTGXOPogK+ZVohUjy4kXHHpRZ7azqOFMeLJO/Fi+/MG13Lso/TxNI6942DaNobfW+Qnj5OPEZIl+23O6R3tCqVo3nywCnUSoDAxBLh13/7maQou7cxKAXmy7+TXV1DBzqD22qeFkqwSmvNFbirMhtDfZB297vsyozO+hlnGUO78LkROkuq3O67LSu5lScFSkI3Yrvf+t/3i+Y8O3lX437TOJV/bBzxIgxDHyxnBsA8FXOE3nkj7Oqo7SssO3Fr3HNkG5O+u26Ro0jEIjXI2H5xG7YL+Q8kdu7JeuAP9UZImHInHBu1vtdqSTashzHmClB6aQ7sOReHBi7lvPEcJyZrABR2abkXRQUN9/7K6tWXX/Juf62e2hUDsQROyptlp21B3G7jGXpy+HsnU3EPDEsZz5s+8o6EG9mmLwsFuTN/T2WT6Dbu4lViDyD+Mv4/9LPX4XdAsu/HfdloErZW4sbw9td+FyguMsiERACQgqcQ5GMLUpaghuCOMYWJS8hxTW2KHkJaBRjC10wqk2yNDSKsYUqkCRMMmwKMrZQlrtKkwxdKwswttB+cVUmGbJBH2hsof3ycUwyzAprbKEOIqpJhvlMEmRsoQeiyiTDEkeQsYUeiCqTLG0YUY0tlIGoNMnShBHH2MIHQ7FJltrMiGlsYTw4KTXJ0tAoxhbdPWGrLkoCASEN0V9tGutg3a+p3AAAAABJRU5ErkJggg==)
Answer:![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEMAAAAjCAMAAAD44EcbAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOmZmOma2OpDbZgAAZgA6ZgBmZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kLbbkLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm29u22/+22////7Zm/7aQ/9uQ//+2///b+UoRsgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABKElEQVRIS+VVaxOCIBCEyrJ39rTSCi0N+P//LxB8REBNzDRN8QmHu729PVwA+K5FtxB2Tm6cyOxIgrEbBssmc0ceAOBFpqNBDz0jOwQhbEX1cb7M8Kr8pOFGbpNhYMGQUanfirDvXSYMs9KjxgAAPccAuZehkdLGqxis7qggjNpVD8U34ZTYkkQtPFgw9ouwZOCpar7IgyXTkAtw3dOwc7zXvoFB4652Xjx/Gwkeid+jMVSY1Bh8fLXWSqUDG+3a+V7+M0Asro3yj3BFqgPLxqCczPi0ruZePs3kl+udg9KD3u6STHcgbXqyQJLuanKMx3K43/B1eaxzVxtRpHvXVHe1d5oKS1aWxl3NMEgLoXNXI0bOIPLyNauitO5qwhCPyAPG23N2TbwBA6cSzTSnPhMAAAAASUVORK5CYII=)
⇒ 2(x2 + 1) = 5x
⇒ 2x2 + 2 = 5x
⇒ 2x2 – 5x + 2 = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(–5)2– 4 (2) (2)
= 25 – 16
= 9
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Question 10.Solve the following quadratic equations using quadratic formula.
3a2x2 – ax –2b2 = 0
Answer:3a2x2 – abx – 2b2 = 0
![](data:image/png;base64,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)
⇒ a = 3a2 , b = –ab and c = –2b2
(–ab)2– 4 (3a2) (–2b2)
= a2b2 + 24a2b2
=25a2b2
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Question 11.Solve the following quadratic equations using quadratic formula.
a (x2 + 1) = x (a2 + 1)
Answer:a(x2 + 1) = x(a2 + 1)
⇒ ax2 + a = a2x + x
⇒ ax2 + a – a2x – x = 0
⇒ ax2 – x(a2 + 1) + a = 0
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⇒ a = a , b = – a2 – 1 and c = a
( –a2 – 1 )2– 4 (a)(a)
= a4 + 2a2 + 1 – 4a2
= (a2)2 – 2a2 + 1
= (a2 – 1 )2
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Question 12.Solve the following quadratic equations using quadratic formula.
36x2 – 12ax + (a2 – b2) = 0
Answer:36x2 – 12ax + (a2 –b2) = 0
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⇒ a = 36, b = –12a and c = a2 – b2
(–12a)2– 4 (36)(a2 – b2)
= 144a2 – 144(a2 – b2)
= 144a2 – 144a2 + 144b2
= 144 b2
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![](data:image/png;base64,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)
Question 13.Solve the following quadratic equations using quadratic formula.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJsAAAA1CAYAAABWd5kSAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAWlSURBVHja7VxPc9NGFF/FV6DXAl7fUq5Ea75BRpWHlJsGFI2vHkaT0QxxGB1SsHPztLTn3oBeYHrorZcGTlxIucO0qT8AST4CkO6upMS1JVmK19Kz9Q5vHGkc7763v33/VvqRvb09goJShKARUBBsKAg2FBQEGwqCLf+ECNFCqRFyqlVuQQhZCXSXsgJ9DcLvRXMWn9pCgE1M1HUtZhjsMaXkb2psdSsEMm3Xtb5hlLzlf3+SQvW/LNe/AXUN5HfvGht8zgfBfNnBurvLkgAHyuD9NrtNGo33jQb5wCd8WiWw7brrerNp/tAZDC6J655nrwbA04dt378GcQ0e2Wt3CKHHpvOgJa4fOGaLEnK8Zj+6szA5W99hG1UC22DQuWQy8/sIaJH0PHONL94Jc/ob0NbA99vXdEKG1PS2R+97Jt1O2iCxA1nnOUNtNG6P3deWCWxl65zk7SjRjlm7fxsa2LYMuiO9mtdbi9sg/P92poKt12tf5UH3kGd9X8Rg3L5fqOluR/f5Nb9Hj8xu/+aygA2CznFJt81WnhO2+RRadJGeuE5ekbr5asIbBzYb8nk/z+TZxI9tMvJUGr9uvO74/mX5Qzb7jhB2aHu91WULo2XrHMlvHGQiQecu7R1hxu9Wx78CDWxngIoBWxoQU43Ps70/ufFPRcLX6VhXWpS+LGJ3l5WzlalzNP63fPywGv0sbaDf+7njDy6DAlsYKuO81xnYCBu2e72rmQuE8zCiD5tN+jJL7nB61p/JIvE9nDILhIvorErv0QXznFt3+YIehXbYWXqwjQwahpbpO8yV1UjYJ5oiIi+CWI3m1VmV3hPz6Jo3RSshbuFAhNEYsKWF2KmJqshZKKVvRC4jdlgRFVmZYCtL5yRxGHkGDWzSe1Gyn14gOM9y9dlk6U1bL7y+93VL5hL0RHSIlxlsZemcJLLFQM398UUtew2S+mlhu+YkLv1IHMz3retN2nwR7ahzl64PLd+/XpCiheYqZepsxZwt+rxAMam2nzXsFrkGkW1GPb/4DPpv7DDOE8cbnStp1Mlratx/GCWzIgG+b9CHsg/FcxlxZqc6vESHv8ExCB9Hv/drUQfyZekcJNz2Kg89/xKqv7NcV5wt1kQl7DD6hLLNH4tOI7KuQZCn0qPwPLS2Zd+yRVGTVFTFDij6TePJ7GQC3PiouiUgdkuDkI8XTahnkbJ0jnIgo9n4ZXSsRqP5R3TmWKhHy7kGomo++z5lb9P6kaXkISjVlKVUKuh5Ve9ZOARbORVct0qPJyHYEGwoCLblFJXHZQg2BFuqzOO4DMGWYUeL3ljQL5vY1Vg0QAmj2d1wOZJ1R2vy8Rzt8/j92CMU4DpnsYEyMMx5vuODfYIs8wij0HXOagMVOdu854s5G+ZsmLMh2LD1gWBDQbCNh5SyQgVKxcCGgmBDQbABm1DFWYwm2xlWDdDazMSwBA5oVWUxirNF+Ih14S+7xM1FBcMSKANXmcVoXHbddSbfzQQANlUMSwWGg+whsWosRuMiGIIY1Q4C7pFywaaSYWnixjzYfPL2varMYiTG2GT0J2o6j5PeLAfieXMzLP3vYl5sPtDBBonFSL4gzZwnaTQGAAq4CzEsxbpN1Ww+i+DZILAYifAZvbcKEWyzMiwlGl4lm88igG0eeufNa0X4jMaCBjYVDEtTQ4sKNp+8DzMuGovRrI/3hG2O7ugRG9QwOgvDUhYPMzObT56HGSFUo0UzN8lkm3uJUVByr/qVeENfUj94Xh1Sc1fa6AIMS6mJoCo2n0UJo6r1zipJYA0KFlGciGv9H2hVaV6GpfTSVhGbzyKBDQqLEdQwOrauuRiWEqoitWw+M4CtMixGkMGmimFp0uBzYPPJCraqshhBBptKhqWJG/Ng88n6MGNVWYzSWw3l5moqGZbA5QEoyytoBBQEG8ryyX9xAZFXsqGiFwAAAABJRU5ErkJggg==)
Answer:![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMIAAAAkCAMAAADsIhWcAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjqQOmZmOma2OpDbZgAAZgA6ZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/7aQ/9uQ/9u2//+2///bGAFyvQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACpElEQVRYR+1XaVPbMBB1AqQxhKYcbXNAGxdsINS1/v+vq6SVZB27kYQzDMxYn5Lx2/f2kGW9ohjXx+jA8/cHIpHmx14+iSM8gngAjQAqLZ3UovobDWsvRQ1xhMcQDziEADKQTlo4lP2Zy2ghRZC15VojPCE0oFlMpj8jlJLHlqYqUBh2JznRfjRfb6CE9uKRGALbLmQJAgHSG/lf1e2Ls+2v4mm6oyj7YEeaKEFj6vm+PQfhp3K6a8uZPbkaSmDbHYGoV5AyR3glUJRFey6gKGVfAm+qkaaGoDDd1Zrnr3r3OtvXS2fvaZ7NGth9xOtSqRpx70dI+e9upaYVUmIlmKkilcj0umvekkqPvz7le6q7msglUvZLKFxEd/vINqt7XjWI26EqHZ+ympy6JQClF+xIx6bglNAsnG1kT1Nl5CJqWauIITYSPxZ9yuJZHgB6IzmIIRsJXta/v9nmzP46sOqL3FfmdQ4Qen7U6+wFiKHDGYZSWiXY0tQYFKaaiddZMjblnFWyp2rJJova5KGKIYripTwRZ5E50fos0ICmtA9Vn9I6zmxpogSdXnczmYrNWS/pLac+bVGExzCMEsgyPm0p14f4fcGrIR5w1AsGPYPxydgB3YEKPlZ8yftJuMzzD/iDGqOb6jjs9+hAdCO9RxKjxjE7EPscGcMb/7IdM60MrrjfVbeCuHXOUD0m9OClZZDXflOWxotnRIe9JQwvPoTAa2dII9Dei6fzII4Y9ympXjtdmkKCF09fiIXGDW+q106XJpDKi6fz2I5YRNlumf/N9NrpuiRSe/F0KsRCH9hIvnVGvHa6NIUEL56xQpNNGF7cjQdeO0MagfZePIMndMSE4U302hnSGNR48RyeQ45Y8bzRa+dkMQz7+S8Yw+ofo+kO/Afrl2wFBxoNGwAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ 3(2x2 – 7x + 1) = 10(x2 – 3x – 4)
⇒ 6x2 – 21x + 3 = 10x2 – 30x – 40
⇒ 10x2 – 6x2 – 30x + 21x – 40 –3 = 0
⇒ 4x2 – 9x – 43 = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(–9)2– 4 (4) (–43)
= 91 + 688
= 769
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Question 14.Solve the following quadratic equations using quadratic formula.
a2x2 + (a2 – b2) x – b2 = 0
Answer:a2x2 + (a2 – b2)x – b2 =0
![](data:image/png;base64,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)
⇒ a = a2 , b = a2 – b2 and c = –b2
(a2 – b2)2– 4 (a2)(–b2)
= (a2)2 – 2a2b2 + (b2)2 + 4a2b2
= (a2)2 + 2a2b2 + (b2)2
= (a2 + b2)2
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Solve the following quadratic equations by completing the square.
x2 + 6x –7 = 0
Answer:
x2 + 6x – 7 = 0
= x2 + 6x = 7
Add 9 on both sides
= x2 + 6x + 9 = 7 + 9
= x2 + 2(3)(x) + 32 = 16
= (x + 3)2 = 16
= x + 3 = √16
= x + 3 = ± 4
x + 3 = 4 or x + 3 = –4
x = 4 – 3 or x = – 4 – 3
x = 1 or x = – 7
Question 2.
Solve the following quadratic equations by completing the square.
x2 + 3x + 1 =0
Answer:
x2 + 3x + 1 =0
= x2 + 3x = –1
Add on both sides
Question 3.
Solve the following quadratic equations by completing the square.
2x2 + 5x –3 = 0
Answer:
2x2 + 5x – 3 = 0
= 2x2 + 5x = 3
Add on both sides
Question 4.
Solve the following quadratic equations by completing the square.
4x2 + 4bx – (a2 – b2) = 0
Answer:
4x2 + 4bx – (a2 – b2) = 0
Divide the whole equation by 4
Question 5.
Solve the following quadratic equations by completing the square.
x2 – (√3 + 1) x + √3 = 0
Answer:
x2 – (√3 + 1)x + √3 = 0
Question 6.
Solve the following quadratic equations by completing the square.=3x + 2
Answer:
= 5x + 7 = (3x + 2)(x – 1)
= 5x + 7 = 3x(x – 1) + 2 (x – 1)
= 5x + 7 = 3x2 – 3x + 2x –2
= 5x + 7 = 3x2 – x – 2
= 3x2 – x – 2 – 5x – 7 =0
= 3x2 – 6x – 9 = 0
Divide whole equation by 3
= x2 – 2x – 3 = 0
= x2 – 3x + x – 3 = 0
= x (x – 3) + (x – 3) = 0
= (x – 3) (x + 1) = 0
x – 3 = 0 or x + 1 = 0
x = 3 or x = –1
Question 7.
Solve the following quadratic equations using quadratic formula.
x27x + 12= 0
Answer:
x2 – 7x + 12 = 0
(–7)2– 4 (1) (12)
=49 – 48
=1
Question 8.
Solve the following quadratic equations using quadratic formula.
15x2 – 11x + 2 = 0
Answer:
15x2 – 11x + 2 = 0
⇒ a = 15 , b = –11 and c = 2
(–11)2– 4 (15) (2)
=121 – 120
=1
Question 9.
Solve the following quadratic equations using quadratic formula.
Answer:
=
⇒ 2(x2 + 1) = 5x
⇒ 2x2 + 2 = 5x
⇒ 2x2 – 5x + 2 = 0
(–5)2– 4 (2) (2)
= 25 – 16
= 9
Question 10.
Solve the following quadratic equations using quadratic formula.
3a2x2 – ax –2b2 = 0
Answer:
3a2x2 – abx – 2b2 = 0
⇒ a = 3a2 , b = –ab and c = –2b2
(–ab)2– 4 (3a2) (–2b2)
= a2b2 + 24a2b2
=25a2b2
Question 11.
Solve the following quadratic equations using quadratic formula.
a (x2 + 1) = x (a2 + 1)
Answer:
a(x2 + 1) = x(a2 + 1)
⇒ ax2 + a = a2x + x
⇒ ax2 + a – a2x – x = 0
⇒ ax2 – x(a2 + 1) + a = 0
⇒ a = a , b = – a2 – 1 and c = a
( –a2 – 1 )2– 4 (a)(a)
= a4 + 2a2 + 1 – 4a2
= (a2)2 – 2a2 + 1
= (a2 – 1 )2
Question 12.
Solve the following quadratic equations using quadratic formula.
36x2 – 12ax + (a2 – b2) = 0
Answer:
36x2 – 12ax + (a2 –b2) = 0
⇒ a = 36, b = –12a and c = a2 – b2
(–12a)2– 4 (36)(a2 – b2)
= 144a2 – 144(a2 – b2)
= 144a2 – 144a2 + 144b2
= 144 b2
Question 13.
Solve the following quadratic equations using quadratic formula.
Answer:
⇒
⇒
⇒ 3(2x2 – 7x + 1) = 10(x2 – 3x – 4)
⇒ 6x2 – 21x + 3 = 10x2 – 30x – 40
⇒ 10x2 – 6x2 – 30x + 21x – 40 –3 = 0
⇒ 4x2 – 9x – 43 = 0
(–9)2– 4 (4) (–43)
= 91 + 688
= 769
Question 14.
Solve the following quadratic equations using quadratic formula.
a2x2 + (a2 – b2) x – b2 = 0
Answer:
a2x2 + (a2 – b2)x – b2 =0
⇒ a = a2 , b = a2 – b2 and c = –b2
(a2 – b2)2– 4 (a2)(–b2)
= (a2)2 – 2a2b2 + (b2)2 + 4a2b2
= (a2)2 + 2a2b2 + (b2)2
= (a2 + b2)2
Exercise 3.16
Question 1.The sum of a number and its reciprocal is
. Find the number.
Answer:Let x be the required number. Then, the reciprocal is
.
⇒ sum of a number and its reciprocal is ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 8(x2 + 1) = 65x
⇒ 8x2 + 8 = 65x
⇒ 8x2 – 65x + 8 = 0
⇒ 8x2 – x – 64x + 8 = 0
⇒ x(8x – 1) – 8(8x – 1) = 0
⇒ (x – 8) (8x – 1) = 0
x – 8 = 0 or 8x – 1 = 0
x = 8 or 8x = 1
![](data:image/png;base64,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)
Therefore, the two required numbers are 8 and
.
Question 2.The difference of the squares of two positive numbers is 45. The square of the smaller number is four times the larger number. Find the numbers.
Answer:Let ‘x’ be the larger number and ‘y’ be the smaller number.
x2 – y2 = 45 …(1)
y2 = 4x …(2)
Now, put the value of y2 in equation (1).
x2 – 4x = 45
⇒ x2 – 4x – 45 = 0
⇒ x2– 9x + 5x – 45 = 0
⇒ x(x – 9) + 5(x – 9) = 0
⇒ (x + 5)(x – 9) =0
x + 5 = 0 or x – 9 = 0
x = –5 or x = 9
Here positive 9 only admissible. From this we need to find the value of y for that we are going to aplly this value in the second equation.
y2 = 4 x
⇒ y2 = 4 × 9
⇒ y2 = 36
⇒ y = √36
⇒ y = ± 6
Here, positive 6 only admissible.
Therefore, the required numbers are 6 and 9.
Question 3.A farmer wishes to start a 100 sq. rectangular vegetable garden. Since he has only 30 m barbed wire, he fences the sides of the rectangular garden letting his house compound wall act as the fourth side fence. Find the dimension of the garden.
Answer:Let ‘x’ and ‘y’ are the dimension of the vegetable garden.
Area of rectangle = Length × Width
x × y = 100
![](data:image/png;base64,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)
we are going to cover the barbed wire for fencing only. So, it must be the perimeter of vegetable garden. Usually perimeter always covers all the four side. Bute here we are going to cover only three sides, because one side of the vegetable garden will act as the compound wall.
x + x + y = 30
⇒ 2x + y = 30
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 200 + y2 = 30y
⇒ y2 – 30y + 200 = 0
⇒ y2 – 10y –20y + 200 = 0
⇒ y(y – 10) – 20(y – 10) = 0
⇒ (y – 10)(y – 20) =0
y – 10 = 0 or y – 20 = 0
y = 10 or y = 20
Now we are going to apply these values in
to get the values of x.
If y = 10 if y = 20
![](data:image/png;base64,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)
x = 10 x = 5
Therefore, the required dimensions are 10m and 10m or 20m and 5m.
Question 4.A rectangular field is 20 m long and 14 m wide. There is a path of equal width all around it having an area of 111 sq. meters. Find the width of the path on the outside.
Answer:Length of the rectangular field = 20m
Breadth of the rectangular field = 14m
Let x be the uniform width all around the path.
Length of the rectangular field including the path
= 20 + x + x
= 20 + 2x
Width of the rectangular field including path
= 14 + x + x
= 14 + 2x
Area of path = area of rectangular field including path – area of rectangular field
⇒ 111 = (20 + 2x)(14 + 2x) – (20 × 14)
⇒ 111 = 280 + 40x + 28x + 4x2 – 280
⇒ 111 = 68x + 4x2
⇒ 4x2 + 68x – 111 = 0
⇒ 4x2 + 74x – 6x – 111 = 0
⇒ 2x(2x + 37) – 3(2x + 37) = 0
⇒ (2x + 37)(2x – 3) = 0
2x + 37 = 0 or 2x – 3 = 0
2x = –37 or 2x = 3
![](data:image/png;base64,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)
x = –18.5 or x = 1.5
Therefore, width of the path = 1.5m
Question 5.A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hr more, it would have taken 30 minutes less for the journey. Find the original speed of the train.
Answer:Let x be the usual speed of the train.
Let T1 be the time taken to cover the distance 90 km in the speed x km/hr.
Let T2 be the time taken to cover the distance 90 km in the speed x + 15 km/hr.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
By using the given condition
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Taking 90commonly from two fractions
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 15 × 180 = x2 + 15x
⇒ x2 + 15x = 2700
⇒ x2 + 15x – 2700 = 0
⇒ x2 + 60x –45x – 2700 = 0
⇒ x(x + 60) –45(x + 60) = 0
⇒ (x – 45)(x + 60) = 0
x – 45 = 0 or x + 60 = 0
x = 45 or x= –60
therefore speed of the train is 45 km/hr.
Question 6.The speed of a boat in still water is 15 km/hr. It goes 30 km upstream and return downstream to the original point in 4 hrs 30 minutes. Find the speed of the stream.
Answer:Let x km/hr be the speed of water
Speed of boat is 15km/hr.
So, speed in upstream = (15 + x) km/hr.
speed in downstream = (15 – x) km/hr.
Let T1 be the time taken to cover the distance 30 km in upstream.
Let T2 be the time taken to cover the distance 30 km in downstream.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
T1 + T2 = 4hours 30minutes
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 900 × 2 = 9(225 – x2)
Now, let us divide the entire equation by 9.
So, that we will get,
200 = 225 – x2
200 + x2 = 225
x2 = 225 – 200
x2 = 25
x = √25
x = ± 5
Speed must be positive so x =5 is the requied speed.
Speed of water = 5km/hr.
Question 7.One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Find their present ages.
Answer:Let x be the present age of son
Let ‘y’ be the present age of father
So, x – 1 be the age of son one year age
y – 1 be the age of father one year ago.
By using the given information
y = x2
y – 1 = 8(x – 1)
⇒ y = 8x – 8 + 1
⇒ y = 8x – 7
⇒ x2 = 8x – 7
⇒ x2 – 8x + 7 = 0
⇒ x2 –x – 7x + 7 = 0
⇒ x(x – 1) –7(x – 1) = 0
⇒ (x – 1)(x – 7) = 0
X – 1 = 0 or x – 7 = 0
x = 1 or x = 7
Therefore, age of father is 49.
Question 8.A chess board contains 64 equal squares and the area of each square is 6.25 cm2. An order around the board is 2 cm wide. Find the length of the side of the chess board
Answer:Let x be the side length of the square board
Area of one square in the chess board = 6.25cm2
Area of 64square = 64×6.25
(x – 4)2 = 400
⇒ x – 4 = √400
⇒ x – 4 = ± 20
x – 4 = 20 or x – 4 = –20
x = 20 + 4 or x = –20 + 4
x = 24 or x = –16
Therefore, side length of square shaped chess board is 24cm.
Question 9.A takes 6 day less than the time taken by B to finish a piece of work. If both A and B together can finish it in 4 days, find the time that B would take to finish this work by himself.
Answer:Let x be the tune taken by A to finish the work.
So, x – 6 be the time taken by B to finish the work.
Work done by A in one day = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAA///b2////9uQttv/kNv//7aQ/7Zm27ZmkLb/25Bm25A6ZpDbOpDbtmY6tmYAOma2OmZmAGa2kDoAAGaQZjoAOjoAADqQADpmADo6OgAAAAA6AAAAlcDvPwAAAAF0Uk5TAEDm2GYAAABVSURBVHjaY2DAAIx8QiCKS1BUCCLAjZ9m5BdhAvFkZGRkeBmIBTIQwEAp4JCSZmWWEmdgYBdn4hYAO0SSByzDJSYO4rJxMgpLsDBwSQkx8suIo2sHAOCIBB5cLN4+AAAAAElFTkSuQmCC)
Work done by B in one day = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAAgCAMAAADOixOHAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAA///b//+22////9u2/9uQ29vb29u2ttv/kNv//7aQ/7Zm27ZmkLb/Zrb/25Bm25A6ZpDbOpDbOpC2tmY6tmYAOma2OmaQOmZmAGa2kDoAAGaQZjo6ZjoAOjo6OjoAADqQADpmADo6ZgA6ZgAAOgAAAAA6AAAA78+ejgAAAAF0Uk5TAEDm2GYAAACLSURBVHja1ZHBDoIwEERnWVFRtEWl2ApSlbr//4cGPdjEXjQxxnd82ZlkssDH0M6mtNp3Nh3Q3/VUHzh1LSKywa+RNPgDyrbh1Pu3oy3DtciD55fvLD1r89TkmvZ479HnCsiGx/51Hsx05saM6qMWZJcC0B5YrMid5lGPYXIWKliqJUpMOhHD7267AbmFDEugSbyeAAAAAElFTkSuQmCC)
Number of days taken by both to finish the work = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAXCAMAAAA1KD/rAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAA///b//+22///tv///9uQkNv/kNvb/7ZmZrb/Zrbb25Bm25A6tpBmZpC2OpDbtmY6kGaQtmYAkDo6AGa2kDoAZjo6OjpmOjoAADqQZgA6ZgAAOgBmOgAAAABmAAAAPeV/JQAAAAF0Uk5TAEDm2GYAAABvSURBVHjatZBZDoAgDAUfICruoiiu3P+WBlkMB7A/bSbt5KXAL0Va3btBuT50fmg25nfkC+ieIwFyRgL4mSUgGAHU2pg1t0Z+ibhEDwG66A/IGWQqVATWOAiiqpHFjKWx5bP5jCScWOMrMnf2z78e+C0FgCFtSwUAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 4(2x – 6) = x2 – 6x
⇒ 8x – 24 = x2 – 6x
⇒ x2 – 6x – 8x + 24 = 0
⇒ x2 – 14x + 24 = 0
⇒ x2 – 12x – 2x + 24 = 0
⇒ x(x – 12) –2(x – 12) = 0
⇒ (x – 12)(x – 2) = 0
x – 12 = 0 or x – 2 = 0
x = 12 or x = 2
Here, 2 is not admissible.
So, B is taking 12 days to finish the work.
Question 10.Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels 5 km/hr faster than the second train. If after two hours, they are 50 km apart, find the average speed of each train.
Answer:Let x km/hr. be the speed of second train.
So, speed of first train will be (x + 15) km/hr.
Distance covered by first train in 2 hours = 2(x + 5)
Distance covered by the second train in 2 hours = 2x
By using Pythagoras theorem
[2(x + 5)]2 + (2x)2 = 502
⇒ (2x + 10)2 + (2x)2 = 502
⇒ (4x2 + 100 + 40x) + 4x2 = 2500
⇒ 8x2 + 40x + 100 = 2500
⇒ 8x2 + 40x + 100 –2500 = 0
⇒ 8x2 + 40x – 2400 = 0
Divide by 8 both sides
⇒ x2 + 5x – 300 = 0
⇒ x2 + 20x – 15x – 300 = 0
⇒ x(x + 20) – 15 (x + 20) = 0
⇒ (x – 15)(x + 20) = 0
x – 15 = 0 or x + 20 = 0
x = 15 or x = –20
Therefore, speed of the second train is 15 km/hr.
The sum of a number and its reciprocal is . Find the number.
Answer:
Let x be the required number. Then, the reciprocal is .
⇒ sum of a number and its reciprocal is
⇒
⇒
⇒ 8(x2 + 1) = 65x
⇒ 8x2 + 8 = 65x
⇒ 8x2 – 65x + 8 = 0
⇒ 8x2 – x – 64x + 8 = 0
⇒ x(8x – 1) – 8(8x – 1) = 0
⇒ (x – 8) (8x – 1) = 0
x – 8 = 0 or 8x – 1 = 0
x = 8 or 8x = 1
Therefore, the two required numbers are 8 and .
Question 2.
The difference of the squares of two positive numbers is 45. The square of the smaller number is four times the larger number. Find the numbers.
Answer:
Let ‘x’ be the larger number and ‘y’ be the smaller number.
x2 – y2 = 45 …(1)
y2 = 4x …(2)
Now, put the value of y2 in equation (1).
x2 – 4x = 45
⇒ x2 – 4x – 45 = 0
⇒ x2– 9x + 5x – 45 = 0
⇒ x(x – 9) + 5(x – 9) = 0
⇒ (x + 5)(x – 9) =0
x + 5 = 0 or x – 9 = 0
x = –5 or x = 9
Here positive 9 only admissible. From this we need to find the value of y for that we are going to aplly this value in the second equation.
y2 = 4 x
⇒ y2 = 4 × 9
⇒ y2 = 36
⇒ y = √36
⇒ y = ± 6
Here, positive 6 only admissible.
Therefore, the required numbers are 6 and 9.
Question 3.
A farmer wishes to start a 100 sq. rectangular vegetable garden. Since he has only 30 m barbed wire, he fences the sides of the rectangular garden letting his house compound wall act as the fourth side fence. Find the dimension of the garden.
Answer:
Let ‘x’ and ‘y’ are the dimension of the vegetable garden.
Area of rectangle = Length × Width
x × y = 100
we are going to cover the barbed wire for fencing only. So, it must be the perimeter of vegetable garden. Usually perimeter always covers all the four side. Bute here we are going to cover only three sides, because one side of the vegetable garden will act as the compound wall.
x + x + y = 30
⇒ 2x + y = 30
⇒
⇒
⇒ 200 + y2 = 30y
⇒ y2 – 30y + 200 = 0
⇒ y2 – 10y –20y + 200 = 0
⇒ y(y – 10) – 20(y – 10) = 0
⇒ (y – 10)(y – 20) =0
y – 10 = 0 or y – 20 = 0
y = 10 or y = 20
Now we are going to apply these values in to get the values of x.
If y = 10 if y = 20
x = 10 x = 5
Therefore, the required dimensions are 10m and 10m or 20m and 5m.
Question 4.
A rectangular field is 20 m long and 14 m wide. There is a path of equal width all around it having an area of 111 sq. meters. Find the width of the path on the outside.
Answer:
Length of the rectangular field = 20m
Breadth of the rectangular field = 14m
Let x be the uniform width all around the path.
Length of the rectangular field including the path
= 20 + x + x
= 20 + 2x
Width of the rectangular field including path
= 14 + x + x
= 14 + 2x
Area of path = area of rectangular field including path – area of rectangular field
⇒ 111 = (20 + 2x)(14 + 2x) – (20 × 14)
⇒ 111 = 280 + 40x + 28x + 4x2 – 280
⇒ 111 = 68x + 4x2
⇒ 4x2 + 68x – 111 = 0
⇒ 4x2 + 74x – 6x – 111 = 0
⇒ 2x(2x + 37) – 3(2x + 37) = 0
⇒ (2x + 37)(2x – 3) = 0
2x + 37 = 0 or 2x – 3 = 0
2x = –37 or 2x = 3
x = –18.5 or x = 1.5
Therefore, width of the path = 1.5m
Question 5.
A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hr more, it would have taken 30 minutes less for the journey. Find the original speed of the train.
Answer:
Let x be the usual speed of the train.
Let T1 be the time taken to cover the distance 90 km in the speed x km/hr.
Let T2 be the time taken to cover the distance 90 km in the speed x + 15 km/hr.
By using the given condition
Taking 90commonly from two fractions
⇒
⇒
⇒
⇒ 15 × 180 = x2 + 15x
⇒ x2 + 15x = 2700
⇒ x2 + 15x – 2700 = 0
⇒ x2 + 60x –45x – 2700 = 0
⇒ x(x + 60) –45(x + 60) = 0
⇒ (x – 45)(x + 60) = 0
x – 45 = 0 or x + 60 = 0
x = 45 or x= –60
therefore speed of the train is 45 km/hr.
Question 6.
The speed of a boat in still water is 15 km/hr. It goes 30 km upstream and return downstream to the original point in 4 hrs 30 minutes. Find the speed of the stream.
Answer:
Let x km/hr be the speed of water
Speed of boat is 15km/hr.
So, speed in upstream = (15 + x) km/hr.
speed in downstream = (15 – x) km/hr.
Let T1 be the time taken to cover the distance 30 km in upstream.
Let T2 be the time taken to cover the distance 30 km in downstream.
T1 + T2 = 4hours 30minutes
⇒
⇒
⇒
⇒
⇒
⇒
⇒ 900 × 2 = 9(225 – x2)
Now, let us divide the entire equation by 9.
So, that we will get,
200 = 225 – x2
200 + x2 = 225
x2 = 225 – 200
x2 = 25
x = √25
x = ± 5
Speed must be positive so x =5 is the requied speed.
Speed of water = 5km/hr.
Question 7.
One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Find their present ages.
Answer:
Let x be the present age of son
Let ‘y’ be the present age of father
So, x – 1 be the age of son one year age
y – 1 be the age of father one year ago.
By using the given information
y = x2
y – 1 = 8(x – 1)
⇒ y = 8x – 8 + 1
⇒ y = 8x – 7
⇒ x2 = 8x – 7
⇒ x2 – 8x + 7 = 0
⇒ x2 –x – 7x + 7 = 0
⇒ x(x – 1) –7(x – 1) = 0
⇒ (x – 1)(x – 7) = 0
X – 1 = 0 or x – 7 = 0
x = 1 or x = 7
Therefore, age of father is 49.
Question 8.
A chess board contains 64 equal squares and the area of each square is 6.25 cm2. An order around the board is 2 cm wide. Find the length of the side of the chess board
Answer:
Let x be the side length of the square board
Area of one square in the chess board = 6.25cm2
Area of 64square = 64×6.25
(x – 4)2 = 400
⇒ x – 4 = √400
⇒ x – 4 = ± 20
x – 4 = 20 or x – 4 = –20
x = 20 + 4 or x = –20 + 4
x = 24 or x = –16
Therefore, side length of square shaped chess board is 24cm.
Question 9.
A takes 6 day less than the time taken by B to finish a piece of work. If both A and B together can finish it in 4 days, find the time that B would take to finish this work by himself.
Answer:
Let x be the tune taken by A to finish the work.
So, x – 6 be the time taken by B to finish the work.
Work done by A in one day =
Work done by B in one day =
Number of days taken by both to finish the work =
⇒
⇒
⇒ 4(2x – 6) = x2 – 6x
⇒ 8x – 24 = x2 – 6x
⇒ x2 – 6x – 8x + 24 = 0
⇒ x2 – 14x + 24 = 0
⇒ x2 – 12x – 2x + 24 = 0
⇒ x(x – 12) –2(x – 12) = 0
⇒ (x – 12)(x – 2) = 0
x – 12 = 0 or x – 2 = 0
x = 12 or x = 2
Here, 2 is not admissible.
So, B is taking 12 days to finish the work.
Question 10.
Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels 5 km/hr faster than the second train. If after two hours, they are 50 km apart, find the average speed of each train.
Answer:
Let x km/hr. be the speed of second train.
So, speed of first train will be (x + 15) km/hr.
Distance covered by first train in 2 hours = 2(x + 5)
Distance covered by the second train in 2 hours = 2x
By using Pythagoras theorem
[2(x + 5)]2 + (2x)2 = 502
⇒ (2x + 10)2 + (2x)2 = 502
⇒ (4x2 + 100 + 40x) + 4x2 = 2500
⇒ 8x2 + 40x + 100 = 2500
⇒ 8x2 + 40x + 100 –2500 = 0
⇒ 8x2 + 40x – 2400 = 0
Divide by 8 both sides
⇒ x2 + 5x – 300 = 0
⇒ x2 + 20x – 15x – 300 = 0
⇒ x(x + 20) – 15 (x + 20) = 0
⇒ (x – 15)(x + 20) = 0
x – 15 = 0 or x + 20 = 0
x = 15 or x = –20
Therefore, speed of the second train is 15 km/hr.
Exercise 3.17
Question 1.Determine the nature of the roots of the equation.
x2 – 8x + 12 = 0
Answer:x2 – 8x + 12 = 0
![](data:image/png;base64,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)
⇒ a = 1 , b = –8 and c = 12
= (–8)2– 4(1)(12)
64 – 48
= 16
∴ b2 – 4ac > 0. hence, roots are real.
Question 2.Determine the nature of the roots of the equation.
2x2 – 3x + 4 = 0
Answer:2x2 – 3x + 4 = 0
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAagAAAAXCAMAAABphyvmAAAAAXNSR0IArs4c6QAAALdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZmYAZmY6ZmZmZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGY6kJBmkLbbkNvbkNv/tmYAtmY6tmZmtpA6ttv/tv+2tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///b9TDP+AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEcklEQVRoQ+1Y61abQBBmsdpgq9Um2nuTtGkMalsxaQOE93+uzgV2gZ1d8kON6WHPEUyYnZ35vrmRIOhXj0CPwM4Q+HOp1PnOTt/xwXvk/OZiHizDxY4B29Hx++Z8dvyfEpWo7hDsdD5RB3dPHUjFCurcocVKMnQYUixfSfJ+sxMvOMVMqdOfj+p4qiaV/rSbKKfz2sbkgYnaAoJYjYN81D536WxRcTgP8qnT2WKqETHIZ5EPnGJ6tM6nD+x5i3YiihOlmyjLedspP1EiCL5A3AaCeAAaEhNwpC9xjxKivDFCsnEz+uwjipCrhfyjpRZT1EmU7fyjEyVAkJ1INQZhipU6WicKKUuBp1TIDA1hDPFfk69BKxEVD73goLJgM8KAwZVBKQ4x/uGOBTEFk+4jmELpAh/Dq1v6L6AqTF+hBLAAW8KvpKSYKgUKMfCzSB0t0a1rBevgLg2v4L+qslc6jDeC8xJRv2d0loiCAELllRyFLQhISMSMBGNwlY7YjNAlN1H57AzZ1vL1wwUb09O1jyhMezwTr8hThKV4wncqiFl0soCcP5kH1OpSBbG2BKSL6WBdoBEoMV4dL3DLKmK7SSvQhWEH/2X4dZlRsJ+3kVilo/JGcl4gKgQIqA5JKNjylVcyTzUI+HhedhkiN8i37jZKNJ4hqKK8bePmYlGLDojAlhlsJV+R/uqOGUaGlRfIAYYb4dHi5iFvBeQZDCR18/4S00rv5NJH5NXjkHT4vJeIAlVUBaR9tnzlFbvYxqAFQcVmgoWlvoopF4IsemP1JhtYEFxF1Prb8qIsKYdYTlqHagOaVm5GbAvf6VpnS2NceW4ellvqD9JhcrQuvlHO4U7Tozg0y8UfJO8lVGkTDxN0VmufCELlVRN3BwT66zZROgi5tGyzUq7xgrwVTNBAePmJqnpUkygyTSQKkSpW7yLSy0iXVaMcH3Hr9QK4yTD4nERpHaI3JRauYYKDwkbBkt+KKILAV/riARcdqBQftxySs4jiXZCXJ1PTo4SAKztpPZO2yijsLzUWy4wy8Xjw68N6M5pQKruIMjp83nuJElDoIMrGoAFB6YHV16E6QFUAmKBGmNnLlVdcJrlPSPJdRAl6qSdXTaNKby6IrWSq96gBV7Ja6TOVgQ5Jwx9gavzy+52HKKPD572zRzlQs+RbplkgNCDgp9Z4Tm+BRQxDFPwBXq5fJHQVAHhyGqlEeZkob03FgU8PBzBKjYPiBuYPsITOYQ4ph+mSwltEcI/DOHzKP+n5Ap/A1nuezzEnkcgEp/SajptFY5jQOrzeC0QpmBZvcUyWUJDk2SvHakIgC5V5OIRRNoSMMm8YDp35DPrCOY2+krxIFPQpX/fL4Y2J5kha8Lp0OIf7ks/B9yA1MReg43WkXqAEvA6d/x3xQ4ov2Bp+0YrKVg8ssw7cEE7IlvKz0TH2ei8A/xZ+9oIf3mQUBBAqr1yoNiBw8blf3z/Fbxj7hcgztbYn6pkS0zarJ2o/iDLtZT/s7a3sEegR2B0C/wAkQbrZys2SZQAAAABJRU5ErkJggg==)
⇒ a = 2, b = –3 and c = 4
= (–3)2– 4(2)(4)
9 – 32
= –23
∴ b2 – 4ac < 0. hence, roots are not real.
Question 3.Determine the nature of the roots of the equation.
9x2 + 12x + 4 = 0
Answer:9x2 – 12x + 4 = 0
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbIAAAAXCAMAAABZCzrHAAAAAXNSR0IArs4c6QAAALpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZmYAZmY6ZmZmZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGY6kJBmkLaQkLbbkNvbkNv/tmYAtmY6tmZmtpA6ttv/tv+2tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///bwFEJlQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEoUlEQVRoQ+1Y2VbbMBC1zFJMC4USlq4kLW2IWVpM0saO/f+/1VksW5bGcjgnhaTHenDiSDMa3TubEgT96BHoEVgrBH6fKXW8Vha9iDEbBEN+Og6m4eRFYFqjTTcNhmz/v6csUd1u2QlDorYent3NZpAFd+7tbZOTdkNm50RnQZJPYzbxo1RcKXXo2LJSSFI11PrSbsp8MLCaZMWULQFBEo6DIrb3nbaXssVZ6VixugwWg1aLi1GFTYV5FnlRKka788VoxRhYhBNlHDzdlDkwuIfyUyaB4PXAJSDIBxhO/KxH0s5YdnRfWhnvoZPVTmuZIlibDz55KSMMjTBYaXiZypisTspcGP45ZQIE2UEz72QRxgJyGyu1O08UUpACY6kbI/rQpmMhwIakgYtAWXziR4mCPR+gK+DIIPGGGBPwiekyBeMeI+hl6QGv4fUdfQuK6WvqcXkF8AEi4RdSUoyUAoVocxap3Ske8EbB2HpIw2v4pp1V66hPI8AgUfbrivYSURBA0KeS/dGCgBZZoHF8IWVBDEelLfIBHmk5ymiLStI0w7U2PZx7KSMrYHd8ImMRJt4hf1K6zKKDCcT1wTigmpgq8L8pYF6M9uYFGoErLmf7ExSZkTOWZwPi0BVBM/loGWUgz2K0TOvQp5FgECgLj+4510gouOv1qWTGDAh4ex7N3BRvAVF3EGDEW3e5ha2MKKuD1JF0rM1PJ6bDgFdaBrG9/IQR60+MOtqofICTMfAIVLW8nmRR4IBhQXrzizMMtUqSEyPRaPom6fDhIFEGqigzSHLuen0qPqKNgQWB5jXBZFMN7FB2flAyyqK3Tg1zgTUpK0acVmxJSYrWgnsn5u4NX2vaq+urUW1N3iq0NQb1ZFmazYn0JNmdF18pDlGyrmXsdOXgFwkHCV92COqWaC9LTgLB6RoaAGgHrNxQpqyEnKDvaMErBbqnqxxZkrQdDOoMjw7KdC1rUkZbiZQhZsXsPCLFjHmZU0ozUfRmAixl6JCtlFU6fDi0tR/sHi5+bqqxGj2RMoKgPTGyq5DX5Rcflmqwq8QY73EKEyXF/taoZYITlrWXI/cJUYZ1yODTduVk6+f7eT4YUny3UVbr8OHgpUzAr4MyF4MGBCWfUgPAjg15o+7WLPYbr5oyyDWQYwBgUbKLMmEHquK6uOgQ5ixhBZhZy/Y4zxmJsY5+2iQNv4OR8atvmMLaKKt1+HBorWUtKDjrLdMcEBoQ8Kzd5ONvs+gYy36MtabqeT2clfFP91GUkiVFyvypF5vFOo8ncFUvbqFlAZsWxBtfgyNqIshY+PERW3p4W3ysOhKcAdFH7vIxXpHSBHt9Q8ftpNF+VDq8OAiUKeg077DZlvCT1vOpWkYTgpZF0BePMcZG0Epi/vT8U4UaiiuoGmqbLyI4TlokJcqgnnnv0vDPijoqk22A168dNG0KO6JTwb1KDesHEPMmUtu4Aq5Xx38GPEn2g2j4uVJUNgfAN+tAgXBIxpTvtY5LLw4CBe+wf8PrkYSfAII+VQsdTQg8gbORU8/xP8lGArO+RveUrS83LZb1lG0aZXUZ2jTLe3t7BHoE1gGBvx2aw5XMIDD0AAAAAElFTkSuQmCC)
⇒ a = 9, b = –12 and c = 4
= (–12)2– 4(9)(4)
144 – 144
= 0
∴ b2 – 4ac > 0. hence, roots are real and equal.
Question 4.Determine the nature of the roots of the equation.
3x2 –2√6x + 2 = 0
Answer:3x2 – 2√6 x + 2 =
![](data:image/png;base64,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)
⇒ a = 3, b = –2√6 and c = 2
= (–2√6 )2– 4(3)(2)
24 – 24
= 0
∴ b2 – 4ac = 0. hence, roots are real and equal.
Question 5.Determine the nature of the roots of the equation.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 9x2 – 60x + 15 = 0
![](data:image/png;base64,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)
⇒ a = 9, b = –60 and c = 15
= (–60)2– 4(9)(15)
3600 – 540
= 3060
∴ b2 – 4ac > 0. hence, roots are real.
Question 6.Determine the nature of the roots of the equation.
(x – 2a) (x – 2b) = 4ab
Answer:(x – 2a)(x – 2b) = 4ab
⇒ x(x –2b) – 2a(x – 2b) = 4ab
⇒ x2 – 2bx – 2ax + 4ab – 4ab = 0
⇒ x2 – 2x(b + a) = 0
![](data:image/png;base64,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)
⇒ a = 1, b = –b – a and c = 0
= (–b – a )2– 4(1)(0)
b2 + a2 + 2ab
∴ b2 – 4ac > 0. hence, roots are real.
Question 7.Find the values of k for which the roots are real and equal in each of the following equations
2x2 – 10x + k = 0
Answer:2x2 – 10x + k = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
⇒ 100 – 8k = 0
⇒ 100 = 8k
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAgCAMAAAD68tKbAAAAAXNSR0IArs4c6QAAAEhQTFRFAAAAAAAAAAA6OgAAOgA6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kLbbkNv/tmY625A627Zm29u2/7Zm/9uQ//+2///bhAIcygAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAe0lEQVQoU52QSRaAIAxDU+cZwen+N7XMuELNpi/QtH0fyGnriWpukkRUCJzdAsUVcgzJo3p62bh+nQOULcBR6oz0FtfEjTvbfcQ1C/N/tryH2K+8bsgd9/1fT0/0fcCfhGfosoFhMsswTGQYRgWG9ikyNNYx9O2e4YtTb6uGA+F6dzZkAAAAAElFTkSuQmCC)
∴ ![](data:image/png;base64,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)
Question 8.Find the values of k for which the roots are real and equal in each of the following equations
12x2 + 4kx + 3 = 0
Answer:12x2 + 4kx + 3 = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPIAAAAXCAMAAAAGP+KnAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmY6ZmZmZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAtmY6tmZmtpA6tpBmttv/tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///bqKHrZgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADIElEQVRYR+1Xa1fbMAyNy2DZRoEAe5CW7MHmtVvz8P//c5Ml2VYax00OhZ2zkQ85ObZl3Stdy0qWvTz/fQR+3Sh18bxR+AsuJcHuep1tFvfPyXmGS/Pt/EcMml4++OHeGjnRN9TqRGzVvDku5T5QrTiiEs40l6a83MWT0by744m9NWFi31BLyvoqmWQ9UwQBBFnWzl7ASbtkj6Z0wLa3mBSzhVN4ip8uZrTGbN7uTwwoScqb9FFu8nmUA1C29JQ9zizt0nmsOS/tDYuyUp+ytqBRfYYCoDXVYp21JeHkiRRlnWbcFR/mUXZAM2cZKDs4aZfOzpSvEXezfOAMVXZAK9Q0pZnXDCeGutUn31dq8dGGCRjX7mBEBF5dEWTUDkanAXUtRi0cUAg9W1p7DbqDikM4D7h0dk3unUhR1kSZxCTWZBWln1VmOOmekl5AzbPx6goFzzjl+nyHlIHJzlT2q8mtukYtPAhniWfZmjs4B1z27BiwpMzMMsxsUFC7clUcJ8DjnjixQHQFKSfxdNf3YlukU9Ehwqey8cLH7+9ABEs7AjzIYIZHeSBCtfUhRRzeG2RuybiqsyghCptEH2WAIoFtNSXVuusKGEo8DEJYwsiGGc/yKFIbPkNxRPBizTbnuMyiPGBSuyTewQVxm9sTcIgygxCW9tMpYz/IKY+xLIdKQdETa7JaUTIsZW2/20IIkqNzEABu4c/yMMsJYXtLa9/kLOgZHiUdn8wKS0I4I5JykzNl8BWjTGc5rVHeG7fFlxW2iHNU3r24+1aEy+yEsxxCFanYeOfWiJrA0xpSO/sYY6UVFOCv0y5cLHU2hO1760BDP2C+jLao8trg7steCFjspwaZ7ERwuR3DO85USFney1id25JkPNbL6suV69CiuRKDcBQtgs9wK/8uLPqfuTpdj1oJoGSJb1Mqe7qmtdbskTurzKygiKhX0ETwOULKve6rtUsuSPNj3dchno+Y991XZI95cEJ1Hm7V77HF/LSoPoJfxDQBdC6co/5JHZdlf7dRoOM/dmNwjvi//JSMocDEf+zHf9+fFs7L7v9mBP4ARidfLNCwq5kAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
⇒ 16k2 – 144 = 0
⇒ 16k2 = 144
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
∴ k = 3 and k = –3
Question 9.Find the values of k for which the roots are real and equal in each of the following equations
x2 + 2k (x – 2) + 5 = 0
Answer:x2 + 2k (x – 2) + 5 = 0
⇒ x2 + 2kx – 4k + 5 = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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∴ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGMAAAAXCAMAAAAoVgFwAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmY6ZmZmZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAtmY6tmZmtpA6ttv/tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///bwKViDwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABW0lEQVRIS+2U3VLCMBCFNxWIgjb4BwWioLFqkybv/3juJq3WSkN70RuHveh0IJsvZ89JAf5ffdwydj2uLLvcQp7sRoAodvH6s625HIMBqslQaUyGGqjSbRibH3DHJiOP2mH4MIbLpkWZkYQGQ0URVjwMY2hartmaGC8bljyiIo0Ijb90lEx9E7j8qsqfwSAmkQaSYMUMGcniAAppVjCszhY9LzzDZbPCSXozfAWl6GygURFjSgxc7mnRsssdBB1UhuPWkvaoS9IBfdWLAgOftR+/1mNfu8VlmDhkqHBsYlgRDSHtHme0Nen6lGtw73ecZtqPEfzw96Ot4+jcvv04ouPvrEAGz1Ft5Udcd0CGMAa7UU92wkMKUpVdhjnZ90q+P47hKZT35LliK3DP3R8fjFQRTFE3eOMnfT5T6AlBnvB2fAo64xtnk20kjSXen0Ujeyeie/77PIEhE/gCGigd9lBmJxgAAAAASUVORK5CYII=)
⇒ 4k2 + 16k – 20 =0
Divide by 4
⇒ k2 + 4k – 5 = 0
⇒ k2 + 5k – k – 5 = 0
⇒ k(k + 5) – (k + 5) =0
⇒ (k + 5)(k – 1) = 0
k + 5 = 0 or k – 1 = 0
k = –5 or k = 1
∴ k = –5 and k = 1
Question 10.Find the values of k for which the roots are real and equal in each of the following equations
(k + 1) x2 – 2 (k – 1) x + 1 = 0
Answer:(k + 1) x2 – 2 (k – 1) x + 1 = 0
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∴ ![](data:image/png;base64,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)
⇒ 4k2 – 8k + 4 – 4k – 4 = 0
⇒ 4k2 – 12k = 0
Divide by 4
⇒ k2 – 3k = 0
⇒ k(k – 3) = 0
⇒ k = 0 or k– 3 = 0
k = 0 or k = 3
∴ k = 0 and k = 3
Question 11.Show that the roots of the equation x2 + 2(a + b) x + 2 (a2 + b2) = 0 are unreal.
Answer:x2 + 2(a + b) x + 2 (a2 + b2) = 0
Compare this equation with ax2 + bx + c = 0
∴ a = 1, b = 2(a + b) and c = 2(a2 + b2)
b2 – 4ac = (2a + 2b)2 – 4(1)[2(a2 + b2)]
= 4a2 + 8ab + 4b2 – 8a2 – 8b2
= 8ab – 4a2 – 4b2
= – 4a2 + 8ab – 4b2
= –4(a2 – 2ab + b2)
= –4 (a – b)2
Since squared quantity is always positive.
Hence, (a – b)2 ≥ 0
Now, it is given a ≠ b, so (a – b)2 > 0
So, D = –4(a – b)2 will be negative.
Hence the equation has no real roots.
Question 12.Show that the roots of the equation 3p2x2 – 2pqx + q2 = 0 are not real.
Answer:3p2x2 – 2pqx + q2 = 0
Compare this equation with ax2 + bx + c = 0
∴ a = 3p2 , b = 2pq and c = q2
b2 – 4ac = (2pq)2 – 4(3p2)[q2]
= 4p2q2 – 12p2q2
= – 8p2q2
Since squared quantity is always positive.
Hence, p2q2 ≥ 0
Now, it is given p ≠ q, so p2q2 > 0
So, D = –8p2q2 will be negative.
Hence the equation has no real roots.
Question 13.If the roots of the equation (a2 + b2) x2 – 2 (ac + bd) x + c2 + d2 = 0, where a, b, c and d ≠ 0, are equal, prove that
.
Answer:Given: (a2 + b2) x2 – 2 (ac + bd) x + c2 + d2 = 0
To prove: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACcAAAAeCAMAAACYCPdiAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OgBmOjo6OmaQOma2OpDbZgAAZjoAZpDbZrbbZrb/kDoAkGYAkGY6kNv/tmYAtmY6tmaQtpA6tpBmttv/tv//25A627Zm2////7Zm/9uQ/9u2//+2///bEJciPgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAyklEQVQ4T7VT2xaCIBBkvdFdK0vNklDh/z+xDaRzOrXog+4DD8wwsywMY3NXfwU4eEQ14nHDVHpiVSxJos657DaNwVsPr1sXVuO5RV1az/HaoGAisrr/SqVc6ptkKofoDpy+SJdBNFjPPcLl9EpwhWP4KrdPeI/AYx3TvmMnl8KF722tqaWUnj8wNGcoE3jC8JIKgjN5rT7DURte+GAipD7qOx764nw/IfiRNYj1xWUSL5a6Jn1Vupd9nmDOSsybJwDtCnY1HOd+vheliQuAdsePmwAAAABJRU5ErkJggg==)
Proof:
We know that,
D = b2 – 4ac
If roots are equal, then b2 = 4ac
⇒ {–2(ac + bd)}2 = 4{(a2 + b2)( c2 + d2)}
⇒ 4(a2c2 + b2d2 + 2acbd) = 4 (a2c2 + a2d2 + b2c2 + b2d2)
⇒ 2acbd = a2d2 + b2c2
⇒ a2d2 + b2c2 – 2acbd = 0
⇒ (ad – bc)2 = 0
⇒ ad – bc = 0
⇒ ad = bc
⇒ ![](data:image/png;base64,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)
Hence proved.
Question 14.Show that the roots of the equation
(x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are always real and they cannot be unless a = b = c.
Answer:(x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0
⇒ x(x – b) – a(x – b) + x(x – c) – b(x – c) + x(x – a) – c(x – a) = 0
⇒ x2 – bx – ax + ab + x2 – cx – bx + bc + x2 – ax – cx + ac = 0
⇒ 3x2 – 2x(a + b + c) + ab + bc + ac = 0
D = b2 – 4ac
D = (a + b + c)2 – 4(3)(ab + bc + ac) = 0
D = 4(a2 + b2 + c2 + 2ab + 2bc + 2ca – 3ab – 3bc – 3ca)
D = 4(a2 + b2 + c2 – ab – bc – ca)
D = 2[(a –b)2 + (b – c)2 + (c – a)2]
Which is always greater than zero so the roots are real.
Roots are equal if D = 0
i.e. (a – b)2 + (b + c)2 + (c – a)2 = 0
since sum of three perfect square is equal to zero so each of them separately equal to zero.
So, a – b = 0, b – c = 0, c – a = 0
a = b , b = c, c = a
so, a = b = c.
Question 15.If the equation (1 + m2) x2 + 2mcx + c2 – a2 = 0 has equal roots, then prove that c2 = a2 (1 + m2)
Answer:Given: (1 + m2) x2 + 2mcx + c2 – a2 = 0
To prove: c2 = a (1 + m2)
Proof: it is being that equation has equal roots, therefore
D = b2 – 4ac = 0 …(1)
From the equation, we have
a = (1 + m2), b = 2mc, c = c2 – a2
putting values of a, b and c in (1), we get
D = (2mc)2 – 4(1 + m2)(c2 – a2)
⇒ 4m2c2 – 4(c2 + c2m2 – a2 – a2m2) = 0
⇒ 4m2c2 – 4c2 – 4c2m2 + 4a2 + 4a2m2 = 0
⇒ –4c2 + 4a2 + 4a2m2 = 0
⇒ 4c2 = 4a2 + 4a2m2
⇒ c2 = a2 + a2m2
⇒ c2 = a2 (1 + m2)
Hence proved.
Determine the nature of the roots of the equation.
x2 – 8x + 12 = 0
Answer:
x2 – 8x + 12 = 0
⇒ a = 1 , b = –8 and c = 12
= (–8)2– 4(1)(12)
64 – 48
= 16
∴ b2 – 4ac > 0. hence, roots are real.
Question 2.
Determine the nature of the roots of the equation.
2x2 – 3x + 4 = 0
Answer:
2x2 – 3x + 4 = 0
⇒ a = 2, b = –3 and c = 4
= (–3)2– 4(2)(4)
9 – 32
= –23
∴ b2 – 4ac < 0. hence, roots are not real.
Question 3.
Determine the nature of the roots of the equation.
9x2 + 12x + 4 = 0
Answer:
9x2 – 12x + 4 = 0
⇒ a = 9, b = –12 and c = 4
= (–12)2– 4(9)(4)
144 – 144
= 0
∴ b2 – 4ac > 0. hence, roots are real and equal.
Question 4.
Determine the nature of the roots of the equation.
3x2 –2√6x + 2 = 0
Answer:
3x2 – 2√6 x + 2 =
⇒ a = 3, b = –2√6 and c = 2
= (–2√6 )2– 4(3)(2)
24 – 24
= 0
∴ b2 – 4ac = 0. hence, roots are real and equal.
Question 5.
Determine the nature of the roots of the equation.
Answer:
⇒
⇒ 9x2 – 60x + 15 = 0
⇒ a = 9, b = –60 and c = 15
= (–60)2– 4(9)(15)
3600 – 540
= 3060
∴ b2 – 4ac > 0. hence, roots are real.
Question 6.
Determine the nature of the roots of the equation.
(x – 2a) (x – 2b) = 4ab
Answer:
(x – 2a)(x – 2b) = 4ab
⇒ x(x –2b) – 2a(x – 2b) = 4ab
⇒ x2 – 2bx – 2ax + 4ab – 4ab = 0
⇒ x2 – 2x(b + a) = 0
⇒ a = 1, b = –b – a and c = 0
= (–b – a )2– 4(1)(0)
b2 + a2 + 2ab
∴ b2 – 4ac > 0. hence, roots are real.
Question 7.
Find the values of k for which the roots are real and equal in each of the following equations
2x2 – 10x + k = 0
Answer:
2x2 – 10x + k = 0
∴
⇒ 100 – 8k = 0
⇒ 100 = 8k
=
∴
Question 8.
Find the values of k for which the roots are real and equal in each of the following equations
12x2 + 4kx + 3 = 0
Answer:
12x2 + 4kx + 3 = 0
∴
⇒ 16k2 – 144 = 0
⇒ 16k2 = 144
⇒
⇒
∴ k = 3 and k = –3
Question 9.
Find the values of k for which the roots are real and equal in each of the following equations
x2 + 2k (x – 2) + 5 = 0
Answer:
x2 + 2k (x – 2) + 5 = 0
⇒ x2 + 2kx – 4k + 5 = 0
∴
⇒ 4k2 + 16k – 20 =0
Divide by 4
⇒ k2 + 4k – 5 = 0
⇒ k2 + 5k – k – 5 = 0
⇒ k(k + 5) – (k + 5) =0
⇒ (k + 5)(k – 1) = 0
k + 5 = 0 or k – 1 = 0
k = –5 or k = 1
∴ k = –5 and k = 1
Question 10.
Find the values of k for which the roots are real and equal in each of the following equations
(k + 1) x2 – 2 (k – 1) x + 1 = 0
Answer:
(k + 1) x2 – 2 (k – 1) x + 1 = 0
∴
⇒ 4k2 – 8k + 4 – 4k – 4 = 0
⇒ 4k2 – 12k = 0
Divide by 4
⇒ k2 – 3k = 0
⇒ k(k – 3) = 0
⇒ k = 0 or k– 3 = 0
k = 0 or k = 3
∴ k = 0 and k = 3
Question 11.
Show that the roots of the equation x2 + 2(a + b) x + 2 (a2 + b2) = 0 are unreal.
Answer:
x2 + 2(a + b) x + 2 (a2 + b2) = 0
Compare this equation with ax2 + bx + c = 0
∴ a = 1, b = 2(a + b) and c = 2(a2 + b2)
b2 – 4ac = (2a + 2b)2 – 4(1)[2(a2 + b2)]
= 4a2 + 8ab + 4b2 – 8a2 – 8b2
= 8ab – 4a2 – 4b2
= – 4a2 + 8ab – 4b2
= –4(a2 – 2ab + b2)
= –4 (a – b)2
Since squared quantity is always positive.
Hence, (a – b)2 ≥ 0
Now, it is given a ≠ b, so (a – b)2 > 0
So, D = –4(a – b)2 will be negative.
Hence the equation has no real roots.
Question 12.
Show that the roots of the equation 3p2x2 – 2pqx + q2 = 0 are not real.
Answer:
3p2x2 – 2pqx + q2 = 0
Compare this equation with ax2 + bx + c = 0
∴ a = 3p2 , b = 2pq and c = q2
b2 – 4ac = (2pq)2 – 4(3p2)[q2]
= 4p2q2 – 12p2q2
= – 8p2q2
Since squared quantity is always positive.
Hence, p2q2 ≥ 0
Now, it is given p ≠ q, so p2q2 > 0
So, D = –8p2q2 will be negative.
Hence the equation has no real roots.
Question 13.
If the roots of the equation (a2 + b2) x2 – 2 (ac + bd) x + c2 + d2 = 0, where a, b, c and d ≠ 0, are equal, prove that .
Answer:
Given: (a2 + b2) x2 – 2 (ac + bd) x + c2 + d2 = 0
To prove:
Proof:
We know that,
D = b2 – 4ac
If roots are equal, then b2 = 4ac
⇒ {–2(ac + bd)}2 = 4{(a2 + b2)( c2 + d2)}
⇒ 4(a2c2 + b2d2 + 2acbd) = 4 (a2c2 + a2d2 + b2c2 + b2d2)
⇒ 2acbd = a2d2 + b2c2
⇒ a2d2 + b2c2 – 2acbd = 0
⇒ (ad – bc)2 = 0
⇒ ad – bc = 0
⇒ ad = bc
⇒
Hence proved.
Question 14.
Show that the roots of the equation
(x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are always real and they cannot be unless a = b = c.
Answer:
(x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0
⇒ x(x – b) – a(x – b) + x(x – c) – b(x – c) + x(x – a) – c(x – a) = 0
⇒ x2 – bx – ax + ab + x2 – cx – bx + bc + x2 – ax – cx + ac = 0
⇒ 3x2 – 2x(a + b + c) + ab + bc + ac = 0
D = b2 – 4ac
D = (a + b + c)2 – 4(3)(ab + bc + ac) = 0
D = 4(a2 + b2 + c2 + 2ab + 2bc + 2ca – 3ab – 3bc – 3ca)
D = 4(a2 + b2 + c2 – ab – bc – ca)
D = 2[(a –b)2 + (b – c)2 + (c – a)2]
Which is always greater than zero so the roots are real.
Roots are equal if D = 0
i.e. (a – b)2 + (b + c)2 + (c – a)2 = 0
since sum of three perfect square is equal to zero so each of them separately equal to zero.
So, a – b = 0, b – c = 0, c – a = 0
a = b , b = c, c = a
so, a = b = c.
Question 15.
If the equation (1 + m2) x2 + 2mcx + c2 – a2 = 0 has equal roots, then prove that c2 = a2 (1 + m2)
Answer:
Given: (1 + m2) x2 + 2mcx + c2 – a2 = 0
To prove: c2 = a (1 + m2)
Proof: it is being that equation has equal roots, therefore
D = b2 – 4ac = 0 …(1)
From the equation, we have
a = (1 + m2), b = 2mc, c = c2 – a2
putting values of a, b and c in (1), we get
D = (2mc)2 – 4(1 + m2)(c2 – a2)
⇒ 4m2c2 – 4(c2 + c2m2 – a2 – a2m2) = 0
⇒ 4m2c2 – 4c2 – 4c2m2 + 4a2 + 4a2m2 = 0
⇒ –4c2 + 4a2 + 4a2m2 = 0
⇒ 4c2 = 4a2 + 4a2m2
⇒ c2 = a2 + a2m2
⇒ c2 = a2 (1 + m2)
Hence proved.
Exercise 3.18
Question 1.Find the sum and the product of the roots of the following equation.
x2 – 6x + 5 = 0
Answer:x2 – 6x + 5 = 0
Compare this equation with ax2 + bx + c = 0
a = 1, b = –6 and c = 5
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Question 2.Find the sum and the product of the roots of the following equation.
kx2 + ax + pk = 0
Answer:kx2 + ax + pk = 0
Compare this equation with ax2 + bx + c = 0
a = k, b = –a and c = pk
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Question 3.Find the sum and the product of the roots of the following equation.
3x2 – 5x = 0
Answer:3x2 – 5x = 0
Compare this equation with ax2 + bx + c = 0
a = 3, b = –5 and c = 0
![](data:image/png;base64,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)
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Question 4.Find the sum and the product of the roots of the following equation.
8x2 – 25 = 0
Answer:8x2 – 25 = 0
Compare this equation with ax2 + bx + c = 0
a = 8, b = 0 and c = –25
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![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADgAAAArCAMAAAAaGrsZAAAAAXNSR0IArs4c6QAAAG9QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOmaQOpC2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bwbKb6wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABCUlEQVRIS+1V2Q7CMAxrucZ9FiiMsYP+/zfidGUgIVCycQgJP1R7iOPU7VylfgOHLXNOd5xp3aHqPNJAi0u0eqmKaXtPxBVTzJfZHpZYgyMkenZWl2hrjEqC5ZB5NIxKmx7AknkewUFnJlTq1rDJsF0FwZA/AXnkm7Bge+m17jS96fKcHnfBy6BjSYuvmPdhhyOSxfYKQ/6yEKwCsZjDr9GOxfoXfdKBu+vcSLz6Nx59NOn+2lGbTPJNbjJA8izkE8StpVKJ5sdMkAgxZSk8RLgQ2flUdc/w4qhEEKUV84inrQ7Ph6F/qmRwhmwpVxFCbPu3SoSgJT8ObG+c1rkAOApcuc5GNOY7is/tgw1lD6EFkgAAAABJRU5ErkJggg==)
Question 5.Form a quadratic equation whose roots are
(i) 3, 4 (ii) 3 + √7, 3 – √7 (iii) ![](data:image/png;base64,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)
Answer:i: 3 and 4
Let ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
∴![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
ii: 3 + √7 and 3 – √7
Let α=3 + √7 and β=3–√7
∴ α + β=3 + √7 + 3–√7=6 and αβ=(3 + √7)(3–√7)
=9–7=2
∴![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
iii: ![](data:image/png;base64,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)
![](data:image/png;base64,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)
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⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Question 6.If α and β are the roots of the equation 3x2 – 5x + 2= 0, then find the values of
(i)
(ii) α – β (iii) ![](data:image/png;base64,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)
Answer:3x2 – 5x + 2 = 0 compare this with ax2 – bx + c = 0
∴ a = 3 , b = –5 and c = 2
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
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![](data:image/png;base64,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)
i). ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAgCAMAAAD68tKbAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6ADo6ADqQAGaQAGa2OgAAOgA6OjoAOjo6OmZmOmaQOma2OpC2OpDbZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ29u229vb2////9uQ/9u2//+2///bvhObJAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAk0lEQVQoU61QyxKCMBDbxUd9UAXUqkBFpbD//4d2qTjbA+OFXDppMkkbgL+ges2exxaTE5/7nDld79AkJSt20D3cJuLd7Txcf/UKFxEHeKoLJ1WrFqDPSnDMLXocfY8KfTOD0wVmTp+Ma4rUf3IE1QfBxILBQCYtdsLvlH6/zLhy2Ars8hdBRrckdOhyRB01TD3sA2dsCC28k/qnAAAAAElFTkSuQmCC)
ii). α – β
α – β = √(α + β)2 – 4α β
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iii). ![](data:image/png;base64,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)
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![](data:image/png;base64,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)
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Question 7.If α and β are the roots of the equation 3x2 – 6x + 4 = 0, find the value of α2 – β2.
Answer:3x2 – 6x + 4 = 0 compare this with ax2 – bx + c = 0
∴ a = 3 , b = –6 and c = 4
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Question 8.If α, β are roots of 2x2 – 3x – 5 = 0, from an equation whose roots are α2 and β2.
Answer:2x2 – 3x – 5 = 0 compare this with ax2 – bx + c = 0
∴ a = 2 , b = –3 and c = –5
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![](data:image/png;base64,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)
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![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADEAAAArCAMAAADmD5DTAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOmaQOpDbZgAAZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bRdydOgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAvklEQVRIS+1Vyw4CMQgsPrY+1kcVXXVb+f+/lDYbE40HpjFxD3JmZoDC1LnRxeVoLyl60phAiL2dvmRGP07E0tMMaNzJYedSQGY1tL8GB3ZvGzuCM3v0gAZrCylMz3aNtNUlWZ3sgH9m3QQ4X2MJeM1eBN95nrRUU9e3qqrR/gFGrhvCnIFJnaFFLorzhXeEGmMPIxipqsqtJQDekxUkAPZW3p+bG7YH3VwBPVBXXOhPIMVLjTEcBYAwEn9KewB05wc3HLLENgAAAABJRU5ErkJggg==)
Here α = α2 and β = β2
General form of quadratic equation whose roots are α2 and β2
⇒ x2 – (α2 + β2) x + α2β2 = 0
⇒ x2 – (α2 + β2) x + (αβ)2 = 0
α2 + β2 = (α + β)2 – 2(αβ)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
x2 – (α2 + β2) x + (αβ)2 = 0
![](data:image/png;base64,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)
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4x2 – 29x + 25 = 0
Therefore the required equation is 4x2 – 29x + 25 = 0
Question 9.If α, β are roots of x2 – 3x + 2 = 0, form a quadratic equation whose roots are –α and –β
Answer:x2 – 3x + 2 = 0 compare this with ax2 – bx + c = 0
∴ a = 1 , b = –3 and c = 2
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Here α = –αand β = – β
General form of quadratic equation whose roots are α2 and β2
⇒ x2 – (–α – β) x + (–α) (–β) = 0
⇒ x2 + (α + β) x + (αβ) = 0
⇒ x2 + (3)x + (2) = 0
Therefore, the required quadratic equation is x2 – 3x + 2 = 0
Question 10.If α and β are roots of x2 – 3x–1 = 0, then form a quadratic equation whose roots are
![](data:image/png;base64,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)
Answer:x2 – 3x – 1 = 0 compare this with ax2 – bx + c = 0
∴ a = 1 , b = –3 and c = –1
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAXCAMAAABd273TAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjoAOmaQOma2OpDbZgAAZrb/kDoAkGY6kLbbkNv/tmYAttv/tv//25A627Zm27aQ2////7Zm/9uQ/9u2//+2///b4yF4bQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbElEQVQoU9VSSRKAIAxr3HBFxRWR/39Te/DgjMUzuSZtmkyJIoZfSyAb5QQmGcjpRFaY4h5e0IVLMOkcEri+mpg3ePByPFug2sMOmwpbEFk0sgUvPxQLPuE1U1aO6bkjp3P5StcroP5JEc8rXYEsA8D4EB54AAAAAElFTkSuQmCC)
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Here ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJEAAAAjCAMAAACjBBhvAAAAAXNSR0IArs4c6QAAALFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjpmOmY6OmZmOma2OpDbZgAAZgBmZjoAZjo6ZjpmZmY6ZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLaQkLbbkNu2kNv/tmYAtmY6tmZmtpA6tpBmtrZmttv/tv//25A625Bm27Zm27aQ27a229uQ29v/2/+22////7Zm/9uQ/9u2//+2///bPR0DJgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACUklEQVRYR+1XbVPiQAxOyvmyigpcPRXwfGPr3WmBO6VL9///MJNt6YuMNNsRx5kzH5jCJM8+SZ5kC8CXvWsFbLT/rnhVsFbY05Nwa4zaYsdbYwTQDrtdlKzT7bDbRf1XjKzeW0gSThDHuZ/9rTAYCoKk2DWoGMn6AnRICka68wAzQZAcW3L+uo9ReY3SAWVgL7c4okKCBaMkmLRgZELcmbhkvM3Ouog9oC7heKr4CYDg8HhVo5gZ+WIbNYRZ5w+H+hq1Y2E1Rxp1NIGY1WPUCOzVSkdx5xGWYU80D8XpendBne5WW61Zws4EPF2H8g+qM8OVyiZGMe49VVNtBM9K6tLbYAXF7KHqWTJiqAyu0BHXyEbBRvDX2BlEwpn5m52fKrd7HIU3GOV1E6NnjDTlUlpjYXNX1tGqYZsYcaVKlTRJwi2LNBQIZj1JN9v1rmW7p9iQbtZe5dtYLAqy93eD/t9qlRqjnINRfVieccMcBf4KcXALy9Ni1pCqOOffPcxe0zqCCGUjan8i7hTcI9pGzwMcG1ZTOuA7xkYY9P6pnFLcuenK7jVapDVocQbp94c0lKdcU9Cbh8yJ9Qg8oSto6Q95f0WMzMEEpk78PtAlI3PusShEjPg9kFmBF/QyxGCsKTS5WBjJ+06WgoiRZvX1PKF585iDw0fagtJ3JLE0QfcTnis/aC6qvZQrWk6H1xXV6FdtRQvCt82ICrT5Yl3jyHfN9GzfeohakGfuwq8LM+9LI1HB0KhyN8rPa/bU3+jfwajZ7+M8SEefzL4YfbKGtKDzAq+7Pvwwx/78AAAAAElFTkSuQmCC)
General form of quadratic equation whose roots are α2 and β2
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 32– 2 × (–1)
=9 + 2
= 11
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ x2 – 11x + 1 = 0
Therefore, the required equation is x2 + 11x + 1 = 0
Question 11.If α and β are roots of 3x2 – 6x + 1 = 0, then form a quadratic equation whose roots are
(i)
(ii) α2β, β2α (iii) 2α + β, 2β + α
Answer:3x2 – 6x + 1 = 0 compare this with ax2 – bx + c = 0
∴ a = 3 , b = –6 and c = 1
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![](data:image/png;base64,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)
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i). Here ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIMAAAAjCAMAAACAX0m6AAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjpmOmY6OmZmOma2OpDbZgAAZjoAZjo6ZmY6ZpDbZrbbZrb/kDoAkGYAkGY6kLaQkLbbkNu2kNv/tmYAtmY6tmZmtpA6tpBmtrZmttv/tv//25A625Bm27Zm27aQ27a229uQ29v/2/+22////7Zm/9uQ/9u2//+2///bnhC+MwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACE0lEQVRYR+2W2XLCMAxF5VAKprQFwtotTtckdEkM/v9fq+UskM4kljswvKAHhnbk6+NrSQbgHE4OqLDvlE9KdhONb/3DMziLRodnAHAUdUwn3cWZobDJzVwleqnV34yxZZGkXjnz5pYVJNFKI2I6xgSIkkF03iGxrCCKWnetJ0heMGwnGlitjtFLNqSKIfMCFwbps25guOmhkiFjIwAsgJjjNwCtw25KHyJkIItKPoek84ZryKFNTpXAJZJfBxBhJUq+AHVX1mTU+YCNP7KXsdlTXKb64ob7NyewAE20kRnfiw9tIupoX4p60AwR633vn6pZNffLnKQlSqb9lB0DauQ6VT2gDyr0WlUr0XxthodwCLWecjMLzKYNDIU3dtmcQWjsXVjvAuuhvIY2BnSjimZV08Nb36kkwXRe/S7yWbCrByNYP1qzIbqL1PPTZPy574TFP8nHsJnhNZhN8U+IvEfYTKuaZNqpNf6fEupejwcIGbWPjGaop8PPhC0lVsV2gmNchcwbffECIuo8DO3vBYXv/zm1SmiUWWvOxf83sawkMchBAHGtGw7KQ2LAXw/IQYyNz7yloD+DJAaBVTQiEuArnMrBlUPrUJTFOHNoBXRMrYidRtne5KAPL+R6OBqDnvPtL9XuQDjR41lfpeQzUhLxqU3oEzrj3lzy7mELQlzo373Hmw8UG7AeTh5nhpNfwV+AX8UoMo71g4jrAAAAAElFTkSuQmCC)
General form of quadratic equation whose roots are α2 and β2
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ x2 – 6x + 3 = 0
Therefore, required equation is x2 – 6x + 3 = 0
ii). Here, α = α2β and β = β2 α
General form of quadratic equation whose roots are α2β and β2 α
x2 – (α2β + β2 α) x + (α2β)(β2 α) = 0
⇒ x2 – αβ (α + β) x + (α3β3) = 0
⇒ x2 – αβ (α + β) x + (αβ)3 = 0
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMoAAAAmCAMAAACyPfRjAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpDbZgAAZgA6ZjoAZjpmZmYAZmaQZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLbbkNv/tmYAtmY6trbbttv/tv//25A625Bm27Zm27aQ27a229u22////7Zm/9uQ/9u2//+2///bCDZoSQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADOElEQVRYR+1YaXPTMBCVAj3CTXBToDiFuFCnpfGh///fkOTVYWtlVhoz02GiD20mld7T27ti7LROFsixwOE1X33NOfjszojdD/aw2j+7e2VeqH31v0jpbm8ybRA91n3/RYNsNkfaxskuj8CDqPjLpaW0b8hurt/eZ2gZEfgQj+tvGXDxI32h8cTdJQG2Pk/3y4QAIPqrPWsXllJpCYcPW4oUUX4iCB5vmRAYiMN66WJsq0hNkcKaF6khFhCkQxCtN9hMLpqUdLcEBOkQRsrTlvNNVJcLV5oUVie6BSFIhTCX769mu6qDJUpJTVWEIBXC98NMV61MSRLVBak49UVa4iMEqRC+lDrK3heQKjWXK7JNPMoYPYPmI8qkeowRkCDELefvgtb9MJcqBBtX/IZ1hcmRaiZZRDntee0aIZiDABcouV05parjSljDw37bXw/1toPfTNeg2uysZ8bZUApGwOYgQEqjWPThivPzowwa+bGRSprYgIAxifJMaemLkQnsTriHx+BieTkp2nFDdFZSlQbuC5UFMSmofXQs2yCHi9qgMO6xDF5WhlJQAuvhaI8Y0qkv1E/1OZ7tFgJ3tTz8tL0cFTRXQI17MAailCAWpIthQfQOUqA+tOv3QY4EJ2zUApCRKEoZn77JvA5t7zFhCMH1DAF3GxGgCTTy0EgKJbcgsUI3T6WI0g1ozpGhSxfLFRNgmra//kwYMXD76ADz3VJ50WYFIAy5UkKPQtqrSi52+2niYjmGzhAm7W2I6X8xGmgQJv8xBqyvIBWH0Fd0ZdCGFpUkbmLt24nCOhgkm7OEnns0olowiaAM1Bb594FBFa/hIiVfSa9ERxGrxU8C82X3ZXBHC79Vk3JjDZyQ2YQwhFIwAuy7IGQ6OSx9JE2F5qid9qIlfvIHdBKZOYwQpEIQr6YHBLnIb4XmABGfIQSpEEQqGLjpb4WpbkQIUiGIUph7QyG9FcLzCRVdNUmb40CQDkFks8C0t8L016OAIB2CKMVYjfZWmGNRuLohyIGgajFPIoS3wrzHkhFBHgRRi9htjrS3QrmTiDna5hNkQpBpf98z0lth95MMOd7oCLIhMplPx/6JBf4AgKdhAU83Al8AAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 27x2 – 18x + 1 = 0
Therefore, required equation is 27x2 – 18x + 1 = 0
iii). Here, α = 2α + β and β = 2β + α
General form of equation whose roots are 2α + β and 2β + α
x2 – (2α + β + 2β + α)x + (2α + β)(2β + α) = 0
x2 – (3α + 3β) x + (4αβ + 2α2 + 2β2 + αβ) = 0
x2 – (3α + 3β) x + (2(α2 + β2 ) + 5αβ) = 0
α2 + β2 = (α + β)2 – 2αβ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
x2 – (3α + 3β) x + (2(α2 + β2 ) + 5αβ) = 0
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAS0AAAAoCAMAAACch1zbAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmaQZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLbbkNv/tmYAtmY6tpBmtrbbttv/tv//25A625Bm27Zm27aQ27a229u22////7Zm/9uQ/9u2//+2///b3mftjwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAExElEQVRoQ+1aa1fcIBAl22pNfbTarlvb2rjV2EdWq3nw/39aYSAwgYRMVtaT4zEfNK7kMnO584AsY6/XKwMvigH+88+M/KnOZmRMjym3n2dl38zMcbgpDmZFFuPZj3kZhK2p3t+0f9ZTQ7I8e9yBY+WbvztAjQOZG2lVR4Y3KnRxvAPHeDav1IDIsNJqloQI4LdALl8ni+/ypth/urpcTDZfcVl3rciGlbX5uAK2ioNHRXMEHXiYrFnOVFzWMJS/QnEINUHKUPMUQwcuJssjKJaaTKaMKxdtrqJIC1Ql2ToXT6kHIojLw2TWKqIvD6skeYY+rWjrT5USslYPW8wgEB3rGeauAKvSaaHYnF+zO7PuQUP4rxNUmIrTKY05z1rNU532ooZKc8AHD5Nn03tAWibh2adOWaqOaCIB65tlaxYxVfD8nZxNjG6to6bkzWB74mNuk7iKQTnyu8Mk2YPpdd7g9yJy1Sc0ktVSG8kb2hBQX8wk4hJmNatkcan+b+U5pJ4qlU8NBkrhY7I80KCK7iU58QLobjht5YtrVmcwv65JeXLJ6qWaY7AH8vcUZaKFaGjDQLRM5HrmzbJFrBbDSUiuTp25bBaBHA/lqJCethFuPwmIK8CWoQ0DCQ1dqJRY6999/LmejbBFgWQBtqBegrl5kuw/CmWK21KQVQYzEKxpZ920yxCcCMq46LNlrDJswWDzF8/2JF3NMrB0E9miQFoD/OUBt1XiyAVx4FOzlNEcYKteQ/HrdCZtTIA+DJSdMMBW12UbXJCVbC3o1ZZj5lgkEiADbKk02SzlT3k/nNytrZLMU1mcMFtGZ1DfeqCobGHBCpiH1UFoM9iVpZzYWeQqPU51BWrrwgikZUuEiL50JlNs6dJSpR880XtPwJz3qZQkapNsV626gS5UP4iRFNZWtz3nmcgOep2M6epGfYrZ6p2FX4kKpGqSujAkG8Xs6rnDVii/OWFQylJutYU6Ot07eamSlrec1rDjWm8kOvWr/zQPt+ejkIRIhD6xufhKPQuD+fE+z4SL2sH5UDS28k7cQSQGN7mjWR4oRrlvHDIQiaoVUy0xv7oJp1StZDkWIsCkGGiySuhnB6GC/VabbBAQBI3K8oHENdpv5dIqqy0CZCjCoHEC37kEhggLXlwmgVqnO7V1gQYeHsf3DhSlO8VAJpsGV9DdM3mzyOps+0mdoEfq7HB3KsuhwsjE/kCWuzG66rXYTKgzcdPLQ7aEB6U6RGrwoXy2zIqbZKXztLag/qZEVenfPYvo7YC9WeovwjKz2SdAivYnkI5qsceDfmCLyz9fmrJPnLyr9i2cerpC8pG4xydhdQY96QzCbonpB3CbQ30mr8ygPxjwzcHc5sSGyNxTzrdsv0Y9eBGFqHvyFkMGLuZO9EqkMzTMKmPK2xsU7KQ3RRRDcQKJolfKpBPHWElNcLte67MtXVImztk7HGNucxgYw4ZxDCspsrjy5K1hawLHQVsw5nzfkOGTVuJbH+H1ffvKI8YbH82iwTRd0fhSP/8IyxG/onytQb4da3cQ4okoBmPMOC/dopjVA4Kz6z/Ktxo2qekg6t+RrEKYs5aWfp8ayesYMLP+RhJjM/t62czMcQXw+k3KGCHxivGCGPgPe86daSmYc7MAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 3x2 –18x + 70 = 0
Therefore, the required equation is 3x2 –18x + 70 = 0
Question 12.Find a quadratic equation whose roots are the reciprocal of the roots of the equation
4x2 – 3x – 1 = 0
Answer:4x2 – 3x – 1 = 0 compare this with ax2 – bx + c = 0
∴ a = 4 , b = –3 and c = –1
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC4AAAAqCAMAAAD79pkTAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADo6OgAAOjqQOmaQOpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrb/kDoAkGY6kNv/tmYAttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bG+F7kAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAnklEQVRIS+2S2w6CQAxEy1VEAQVZWC7//5ss6JoQQjJjDPBAn/bhtDudjsghqrrnuI725ngljDeXlyJwM3dPvHBsuW9H9hSz8JgV89kBuVX/DMzmfoqwJ4M5sAgT1jajvnmcP/hJ/xDD/7pNhyKCLtIEDN5dEwYvIk3gOqwJvItzwfH+EcmIqww6grYBw/BpJi7mF5y7qtFPGA9ZsgoNResGUXcokvwAAAAASUVORK5CYII=)
Here ![](data:image/png;base64,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)
General form of quadratic equation whose roots are α2 and β2
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ x2 – 3x – 4 = 0
Therefore, required equation is x2 – 3x – 4 = 0
Question 13.If one root of the equation 3x2 + kx – 81 = 0 is the square of the other, find k.
Answer:Two roots of any quadratic equation are α and β.
Here, one root is square of the other i.e α = β2
3x2 – kx – 81 = 0 compare this with ax2 – bx + c = 0
∴ a = 3, b = –k and c = –81
![](data:image/png;base64,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)
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![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHgAAAArCAMAAACEoLy5AAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOjoAOjqQOmaQOma2OpDbZgAAZjoAZpDbZrb/kDoAkGY6kLaQkLbbkNu2kNv/tmYAtmY6tmZmtpBmtrZmttv/tv//25A627Zm27aQ27a229v/2////7Zm/9uQ/9u2//+2///bHjUuXwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABoUlEQVRYR+2X3VKDMBCFs1AtWAs1qFC1QlV+8v4v6IYqkgCWkBBmOs0VF5t8ObvJ2UDIdcybgRSceF7C0Or5FWwr8TzVKcDKFq/hcDCj669FwPmdHW4CvwPVouLMEldMao6buLEjWAI7ceGtrVeY8FSTHB7sk2toYt++sMROzCiAfcnqRPbqgbNTn6c9I3EPJIOt9jr/L8CofPCqAJlodtbBdSM1Ci5CWMW1ntboKk45WA7TkV94O5K5b9LN6gG776QM7825XYLOyagv1a4XnMLtZ1uj4P6q4k/JS9sm1r9eiorZ3jlvdvL0pjcJHz9Vy+WG0Z9qNDtjneWkOEE55w5XHcJ1N0Mr1fUFqULZtQdOdXeHqqX9i8drwl6eg+2HoLkHDPhQOnrmnItFeI3JHsSL0lfjR38RrxaqOyHBLPPxVTvh/0UXnDhPpKQTXgDaYG5Rgl+MTJsuuMZ0ru1IuGZYGW0OmktMmV4FABtzPUZpC0dPdkil6RrB+exPp4HNFQadb6R+RrnZLvAPwbh3lNRYVx2pF8PKyAOpIYyffAmR3wPvGq6zNWtgAAAAAElFTkSuQmCC)
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
αβ = –27
β2(β) = –27
β3 = –27
β3 = (–3)3
β = –3
now, we are going to apply in first equation
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFsAAAArCAMAAAAgygRqAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpDbZgAAZgA6ZjoAZpDbZrb/kDoAkGY6kLaQkLbbkNv/tmYAtmY6tpBmttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bq7BkzgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABn0lEQVRYR+2W0VqDMAyFG6YO5+acdm7oLNMB6/s/oUlB6GhL8WO5Gxdcnf5JT5O0Qty+2gEFyZ7Ni+LGdrwlTxTA3YHBdWJrOT8xoAWxiycWNLFzJrQoAOCeJ23Ku0znHG6L2m94Y4EbbtZrzu8NwGpyOLQ72WsJ8Gyhzi87kV95FujPFJLXOkj5GJ1htjy6yWx2EHmzA2VvxL/SlsfY5zXysEdJlzt2a9k7cFseQ5uSadjKPUmHbcmjaKGIbbIpEF300nTYnTyOFmr2JarN6oR8bNZ+zbvsP/kINLEVPPwEpD62Lc8oH/P5qpfy1h+Jp0e968LyJrs2GOC9jGzs0cDk8noSlvd2X7Prv/sF2K084okxKhvNHpT38wa82o5poCHdvAfljifvi3aeOKZ4PBmSe/0eU61GEzqYDqDzBb4gzMyLiy/ixuVZshOVNMcSF/+XTWNPmQa/OtukEqq70d4HhdV2yfFiw3g09JZMDxTEH9NAL073BO+Bi2v+GsSOUYYafVIYLWl88DysNLVNJXlem9U2xfcZX51McnXM4l+C3B6s/hA2dgAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 6 × 3 = k
⇒ k = 18
Question 14.If one root of the equation 2x2 – ax + 64 = 0 is twice the other, then find the value of a
Answer:Roots of any quadratic equation are α and β
Here one root is twice the other.
α = 2β
by comparing the given equation with general form of quadratic equation we get,
a = 2, b = –a and c = 64
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
α + β =![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADpmADqQAGa2OgA6Ojo6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkGY6kLbbkNv/tmY6tpBmtv//25A627Zm29u22////7Zm/9uQ/9u2//+2YkLolwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAY0lEQVQYV42OSQ6AMAwDY8q+lL2FQP//TVIqcShCwpdR4kQ2UaxzACoiV3c0p/vtsqctACEnExm1kuuhFmSv968Fgn7fP4dbC5TSpxnJSrbXkQca6SmyYvspgAWspb+P03HeBalyA2GH2v98AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
…(1)
αβ = 32
2β(β) = 32
2β2 = 32
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADwAAAAgCAMAAAB5NuugAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bb2f7vgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABFklEQVRIS9WUyRaCMAxFU3DAeR5wqKig8P8/aEKkAnpsw3FjFsAiN817JQH4kzj1lTfFXi8TpQbCnrPlBiJvC+mY3+K49Rgq3pIC99WM0/WwgrkICVWL4agq2VHIOZjTuU822wdsoV1IOt7CjeAE2YSKhP4BIsUKakLezDjxOelIYSCcjpDLFt0PQuxGJnRfDBdC7FCRoQnOjzdCBLB/hPtkEL+EuLOg/aNWnWudCMmSPL79hghDtvPId2uYgvyBLiGM99WOreiHBIb5WQrHtnNNYR1260OrbgznoDojbig1vH6uB1eilPemVlJDDJfHXgpXxl4Kk6om+8u4YRv7b7bV9pfE4QZj/yrfYOwNbPaXqNsfJz8AX9sRBWl2ldkAAAAASUVORK5CYII=)
⇒ β2 = 16
⇒ β = √16
⇒ β =± 4
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 12 × 2 = a
⇒ a = 24
Question 15.If α and β are roots of 5x2 – px + 1 = 0 and α – β = 1, then find P.
Answer:Roots of any quadratic equation are α and β
By comparing the given equation with ax2 + bx + c = 0
a = 5, b = –p and c = 1
![](data:image/png;base64,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)
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α – β = 1
α – β = (α + β)2 – 4αβ
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⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ p2 – 20 = 25
⇒ p2 = 25 + 20
⇒ p2 = 45
⇒ p = √45
⇒ p = 3√5
Find the sum and the product of the roots of the following equation.
x2 – 6x + 5 = 0
Answer:
x2 – 6x + 5 = 0
Compare this equation with ax2 + bx + c = 0
a = 1, b = –6 and c = 5
Question 2.
Find the sum and the product of the roots of the following equation.
kx2 + ax + pk = 0
Answer:
kx2 + ax + pk = 0
Compare this equation with ax2 + bx + c = 0
a = k, b = –a and c = pk
Question 3.
Find the sum and the product of the roots of the following equation.
3x2 – 5x = 0
Answer:
3x2 – 5x = 0
Compare this equation with ax2 + bx + c = 0
a = 3, b = –5 and c = 0
Question 4.
Find the sum and the product of the roots of the following equation.
8x2 – 25 = 0
Answer:
8x2 – 25 = 0
Compare this equation with ax2 + bx + c = 0
a = 8, b = 0 and c = –25
Question 5.
Form a quadratic equation whose roots are
(i) 3, 4 (ii) 3 + √7, 3 – √7 (iii)
Answer:
i: 3 and 4
Let
∴
∴
∴
∴
ii: 3 + √7 and 3 – √7
Let α=3 + √7 and β=3–√7
∴ α + β=3 + √7 + 3–√7=6 and αβ=(3 + √7)(3–√7)
=9–7=2
∴
∴
∴
iii:
⇒
⇒
Question 6.
If α and β are the roots of the equation 3x2 – 5x + 2= 0, then find the values of
(i) (ii) α – β (iii)
Answer:
3x2 – 5x + 2 = 0 compare this with ax2 – bx + c = 0
∴ a = 3 , b = –5 and c = 2
i).
⇒
⇒
⇒ =
⇒ =
⇒ =
ii). α – β
α – β = √(α + β)2 – 4α β
iii).
Question 7.
If α and β are the roots of the equation 3x2 – 6x + 4 = 0, find the value of α2 – β2.
Answer:
3x2 – 6x + 4 = 0 compare this with ax2 – bx + c = 0
∴ a = 3 , b = –6 and c = 4
Question 8.
If α, β are roots of 2x2 – 3x – 5 = 0, from an equation whose roots are α2 and β2.
Answer:
2x2 – 3x – 5 = 0 compare this with ax2 – bx + c = 0
∴ a = 2 , b = –3 and c = –5
Here α = α2 and β = β2
General form of quadratic equation whose roots are α2 and β2
⇒ x2 – (α2 + β2) x + α2β2 = 0
⇒ x2 – (α2 + β2) x + (αβ)2 = 0
α2 + β2 = (α + β)2 – 2(αβ)
x2 – (α2 + β2) x + (αβ)2 = 0
4x2 – 29x + 25 = 0
Therefore the required equation is 4x2 – 29x + 25 = 0
Question 9.
If α, β are roots of x2 – 3x + 2 = 0, form a quadratic equation whose roots are –α and –β
Answer:
x2 – 3x + 2 = 0 compare this with ax2 – bx + c = 0
∴ a = 1 , b = –3 and c = 2
Here α = –αand β = – β
General form of quadratic equation whose roots are α2 and β2
⇒ x2 – (–α – β) x + (–α) (–β) = 0
⇒ x2 + (α + β) x + (αβ) = 0
⇒ x2 + (3)x + (2) = 0
Therefore, the required quadratic equation is x2 – 3x + 2 = 0
Question 10.
If α and β are roots of x2 – 3x–1 = 0, then form a quadratic equation whose roots are
Answer:
x2 – 3x – 1 = 0 compare this with ax2 – bx + c = 0
∴ a = 1 , b = –3 and c = –1
Here
General form of quadratic equation whose roots are α2 and β2
⇒
⇒
⇒
= 32– 2 × (–1)
=9 + 2
= 11
⇒
⇒
⇒ x2 – 11x + 1 = 0
Therefore, the required equation is x2 + 11x + 1 = 0
Question 11.
If α and β are roots of 3x2 – 6x + 1 = 0, then form a quadratic equation whose roots are
(i) (ii) α2β, β2α (iii) 2α + β, 2β + α
Answer:
3x2 – 6x + 1 = 0 compare this with ax2 – bx + c = 0
∴ a = 3 , b = –6 and c = 1
i). Here
General form of quadratic equation whose roots are α2 and β2
⇒
⇒
⇒
⇒
⇒ x2 – 6x + 3 = 0
Therefore, required equation is x2 – 6x + 3 = 0
ii). Here, α = α2β and β = β2 α
General form of quadratic equation whose roots are α2β and β2 α
x2 – (α2β + β2 α) x + (α2β)(β2 α) = 0
⇒ x2 – αβ (α + β) x + (α3β3) = 0
⇒ x2 – αβ (α + β) x + (αβ)3 = 0
⇒
⇒
⇒
⇒ 27x2 – 18x + 1 = 0
Therefore, required equation is 27x2 – 18x + 1 = 0
iii). Here, α = 2α + β and β = 2β + α
General form of equation whose roots are 2α + β and 2β + α
x2 – (2α + β + 2β + α)x + (2α + β)(2β + α) = 0
x2 – (3α + 3β) x + (4αβ + 2α2 + 2β2 + αβ) = 0
x2 – (3α + 3β) x + (2(α2 + β2 ) + 5αβ) = 0
α2 + β2 = (α + β)2 – 2αβ
x2 – (3α + 3β) x + (2(α2 + β2 ) + 5αβ) = 0
⇒
⇒
⇒
⇒
⇒ 3x2 –18x + 70 = 0
Therefore, the required equation is 3x2 –18x + 70 = 0
Question 12.
Find a quadratic equation whose roots are the reciprocal of the roots of the equation
4x2 – 3x – 1 = 0
Answer:
4x2 – 3x – 1 = 0 compare this with ax2 – bx + c = 0
∴ a = 4 , b = –3 and c = –1
Here
General form of quadratic equation whose roots are α2 and β2
⇒
⇒
⇒
⇒
⇒ x2 – 3x – 4 = 0
Therefore, required equation is x2 – 3x – 4 = 0
Question 13.
If one root of the equation 3x2 + kx – 81 = 0 is the square of the other, find k.
Answer:
Two roots of any quadratic equation are α and β.
Here, one root is square of the other i.e α = β2
3x2 – kx – 81 = 0 compare this with ax2 – bx + c = 0
∴ a = 3, b = –k and c = –81
αβ = –27
β2(β) = –27
β3 = –27
β3 = (–3)3
β = –3
now, we are going to apply in first equation
⇒
⇒
⇒
⇒ 6 × 3 = k
⇒ k = 18
Question 14.
If one root of the equation 2x2 – ax + 64 = 0 is twice the other, then find the value of a
Answer:
Roots of any quadratic equation are α and β
Here one root is twice the other.
α = 2β
by comparing the given equation with general form of quadratic equation we get,
a = 2, b = –a and c = 64
α + β =
…(1)
αβ = 32
2β(β) = 32
2β2 = 32
⇒
⇒ β2 = 16
⇒ β = √16
⇒ β =± 4
⇒
⇒
⇒ 12 × 2 = a
⇒ a = 24
Question 15.
If α and β are roots of 5x2 – px + 1 = 0 and α – β = 1, then find P.
Answer:
Roots of any quadratic equation are α and β
By comparing the given equation with ax2 + bx + c = 0
a = 5, b = –p and c = 1
α – β = 1
α – β = (α + β)2 – 4αβ
⇒
⇒
⇒ p2 – 20 = 25
⇒ p2 = 25 + 20
⇒ p2 = 45
⇒ p = √45
⇒ p = 3√5
Exercise 3.19
Question 1.If the system 6x – 2y = 3, kx – y = 2 has a unique solution, then
A. k = 3
B. k ≠ 3
C. k = 4
D. k ≠ 4
Answer:Given: Two equations: 6x – 2y = 3 and kx – y = 2
Required: To find the value of k such that system of equations have unique solutions
To have unique solution to the System of equations, the required condition is ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADQAAAAjCAMAAADsddn6AAAAAXNSR0IArs4c6QAAAIFQTFRFAAAA///b//+22///tv///9u2/9uQ29u2ttv/kNv//7Zm27ZmZrb/ZrbbtpC225A6tpBmtpA6ZpDbOpDbtmY6tmYAZma2kGY6ZmaQOma2kDo6AGa2kDoAAGaQZjpmZjoAOjo6ADqQADpmZgBmZgA6ZgAAOgBmOgA6AABmAAA6AAAAFbCvRAAAAAF0Uk5TAEDm2GYAAADpSURBVHja7ZTZDoIwEEWni5Xiggq4L4Ab3P//QKMRImvRYOKD89bmnLSZTi/Rb5WDzQeW34XUWwCuUdp5SIbZkp9H5IXCJMUWObF83bLN0oZIXXW2HgRAKIgNLtIsPSk70eREko37xyYpFGwaZxTfIprcn4E3SgEQ3W9XoBqlOkqVJH9YlvKUDxRmRVV0pkwVB859b2iQVqIrAeTq9aBVHYT6Np00dXu9rg56zPAHv46t9T8jvpwRbA1E0iSl1DMj+NHiB9eUESmVZQTxvTRnRIFSS9Gi93nKngs1M49kjuJntHiudlR13QC/ux6jLGTR4gAAAABJRU5ErkJggg==)
∴ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACsAAAAXCAMAAAClO0YkAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAA///b//+22///tv///9u2/9uQttv/kNv//7Zm27aQ27ZmkLbbZrb/25A6OpDbtmYAkGY6Oma2OmaQkDo6AGa2kDoAOjqQOjoAADqQADpmZgA6ZgAAOgBmOgAAAABmAAA6AAAApuuiqAAAAAF0Uk5TAEDm2GYAAACxSURBVHja7ZHbEoIgFEVBvJEppIFlhe7//8kGhEns5msznQeGh8VhnX0I+bFqMJWbYf6NpdUJGMpNrJz2JNOOesnKenFXTrX2bAMMxQLNb+n68Zh6lmrFooFFTGaHrgi+vA+oRKiHWnIFOubZqmdxju3af2dmBwDniE0uz9NyiLlvbhT54OCmNSL4on7blmpHzZnZUy66cBWvza4h01aT29+oBkJM9FiuAzNAy8i/fN0BficNWR+kxI0AAAAASUVORK5CYII=)
∴ For every value of k except 3 the system equations have unique solutions.
∴ Correct option is – Option (B)
Question 2.A system of two linear equations in two variables is consistent, if their graphs
A. coincide
B. intersect only at a point
C. do not intersect at any point
D. cut the x-axis
Answer:Given: A system of two linear equations in two variables is consistent
We know that if a system of two linear equations in two variables is consistent then their graph do not intersect at any point
∴ Correct option is – Option (C)
Question 3.The system of equations x –4y = 8 , 3x –12y = 24
A. has infinitely many solutions
B. has no solution
C. has a unique solution
D. may or may not have a solution
Answer:Given: system of equations x –4y = 8 , 3x –12y = 24
Here,
a1 = 1, b1 = -4, c1 = 8 and a2 = 3, b2 = -12, c2 = 24
Now,
,
,![](data:image/png;base64,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)
Here, we can clearly see that
which is the condition for infinitely many solutions.
∴ The system of equations have infinitely many solutions.
∴ Correct option is – Option (A)
Question 4.If one zero of the polynomial p(x) = (k + 4)x2 + 13x + 3k is reciprocal of the other, then k is equal to
A. 2
B. 3
C. 4
D. 5
Answer:Given: A Quadratic equation p(x) = (k + 4)x2 + 13x + 3k
Required: To find the value of k
Let the roots of the given Quadratic equation be:
and ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAAgCAMAAAAYLsniAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6ADpmAGaQAGa2OgAAOgA6OgBmOmZmOpDbZgAAZjoAZmY6ZpDbZrb/kDoAkNv/tmYAtmY6tv//25A625Bm2////7Zm/9uQ/9u2//+2///b3hZfYwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXUlEQVQoU2NgwAJkhFnBomKcPBAGA4MoYYaMEIs4WCkjEHBhMxePGEgPCJCoDY9yKR5GJn4hVgYZAVZxSXYOEQZJdkEgB+gwOEOam4tBjJdVhoFBgo2JT5KNGdM4AJDaA6azg9qWAAAAAElFTkSuQmCC)
∴ Product of roots of the given Quadratic equation is ![](data:image/png;base64,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)
We know that, Product of roots of a given Quadratic equation is ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6ADpmADqQAGa2OgA6OgBmOjo6OpDbZgAAZjoAZrbbkDoAkGY6kNv/tmY6tmaQtpBmtv//25A627Zm2////7Zm/9uQ/9u2//+2///b4bzcqwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXklEQVQYV42OSw6AIAxEpwJ+UERFAeH+97Rq3LAwvM0knZdmgJI8E0mHbJSPnUNs7WN8mbTyefV8GEm8VRX0UuX+SifvG4CkJyySdzDhzqPn3R6hsdiFQzIkNlLlqwumzANbvrkdFAAAAABJRU5ErkJggg==)
∴ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 3k = k + 4
⇒ 2k = 4
⇒ k = 2
∴ The value of k is 2
∴ Correct option is – Option (A)
Question 5.The sum of two zeros of the polynomial f(x) = 2x2 + (p + 3)x + 5 is zero, then the value of p is
A. 3
B. 4
C. –3
D. –4
Answer:Given: A Quadratic equation f(x) = 2x2 + (p + 3)x + 5 and sum of roots is zero.
Required: to find the value of p
We know that, sum of roots = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAhCAMAAAAIybBlAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgA6Ojo6OpDbZgAAZrbbZrb/kDoAkGY6kNv/tmYAtmY6tpA6tpBmttv/tv//25A627Zm2////7Zm/9uQ/9u2//+2///b2RWMaQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAd0lEQVQoU71RSRKAIAxrEVdExQ1R8f/PdDtYtxm8mGMmaZsU4As0hle5cmGCEpmkTuU1oL0WrMANEtY5Q1QQ0QPjm6laXAdUjMip6UsaJ+1+H4GT63/RmCMmdK0VKZS+uVzSn5lu6e/E9KwAzWnFNkNe3//+mnAG5NcEKqQ/JaEAAAAASUVORK5CYII=)
∴
= 0
⇒ -(p + 3) = 0
⇒ P + 3 = 0
∴ p = –3
∴ The value of p is –3
∴ Correct option is – Option (C)
Question 6.The remainder when x2 – 2x + 7 is divided by x + 4 is
A. 28
B. 29
C. 30
D. 31
Answer:Given: x2 – 2x + 7
Required: Reminder of x2 – 2x + 7 at x + 4
By synthetic division we can find reminder of
x2 – 2x + 7 at x = -4
![](data:image/png;base64,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)
∴ Reminder of x2 – 2x + 7 at x + 4 = 31
∴ Correct option is –Option (D)
Question 7.The quotient when x3 – 5x2 + 7x – 4 is divided by x–1 is
A. x2 + 4x + 3
B. x2 – 4x + 3
C. x2 – 4x -3
D. x2 + 4x – 3
Answer:Given: x2 – 2x + 7
Required: Reminder of x3 – 5x2 + 7x – 4 at x – 1
By synthetic division we can find Quotient of x3 – 5x2 + 7x – 4 at x = 1
![](data:image/png;base64,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)
∴ Quotient of x2 – 2x + 7 when divided by x–1 = x2–4x + 3
∴ Correct option is –Option(B)
Question 8.The GCD of (x3 + 1) and x4 – 1 is
A. x3 – 1
B. x3 + 1
C. -(x + 1)
D. x–1
Answer:Given two polynomials: (x3 + 1) and x4 – 1
Required: To find GCD of the given two polynomials
Let f(x) = x3 + 1 and g(x) = X4 – 1
Here, degree of g(x) > f(x) ∴ Devisor = x3 + 1
![](data:image/png;base64,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)
Here reminder = –(x + 1) (Reminder ≠ zero)
![](data:image/png;base64,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)
∴ The GCD of given two polynomials is -(x + 1)
∴ Correct Option is - Option(C)
Question 9.The GCD of x2 – 2xy + y2 and x4 – y4 is
A. 1
B. x + y
C. x – y
D. x2 – y2
Answer:Given two polynomials: x2 – 2xy + y2 and x4 – y4
Required: To find GCD of the given two polynomials
Let f(x) = x2 – 2xy + y2 and g(x) = x4 – y4
f(x) = x2 – 2xy + y2
⇒ f(x) = (x–y)2
g(x) = x4 – y4
⇒ g(x) = (x2)2 – (y2)2
⇒ g(x) = (x2 – y2)(x2 + y2) (∵ a2–b2 = (a–b)(a + b))
⇒ g(x) = (x–y)(x + y)(x2 + y2) (∵ a2–b2 = (a–b)(a + b))
∴ The GCD of given two polynomials is (x–y)
∴ Correct Option is - Option (C)
Question 10.The LCM of x3 – a3 and (x - a)2 is
A. (x3 – a3) (x + a)
B. (x3 – a3) (x - a)2
C. (x - a)2 (x2 + ax + a2)
D. (x + a)2(x2 + ax + a2)
Answer:Given two polynomials: x3 – a3 and (x - a)2
Required: To find LCM of the given two polynomials
Let f(x) = x3 – a3 and g(x) = (x - a)2
here,
f(x) = x3 – a3
⇒f(x) = (x-a)(x2 + a2 + xa)
g(x) = (x - a)2
⇒g(x) = (x-a)(x-a)
∴ LCM = (x-a) (x2 + a2 + xa) (x-a) = (x-a)2(x2 + a2 + xa)
∴ Correct Option is - Option (C)
Question 11.The LCM of ak,ak + 3, ak + 5 where
is
A. a k + 9
B. ak
C. ak + 6
D. ak + 5
Answer:Given three polynomials: ak , ak + 3 and ak + 5
Required: To find LCM of the given two polynomials
Let f(x) = ak , g(x) = ak + 3 and h(x) = ak + 5
here,
f(x) = ak
⇒f(x) = ak
g(x) = ak + 3
⇒g(x) = ak ×a3
h(x) = ak + 5
⇒h(x) = ak ×a3 + 2 = ak×a3×a2
∴ LCM = ak×a3×a2 = ak + 5
∴ Correct Option is - Option (D)
Question 12.The lowest form of the rational expression
is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Given: Rational Expression: ![](data:image/png;base64,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)
Required: The lowest form of the given Rational Expression.
Let f(x) = ![](data:image/png;base64,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)
⇒ f(x) =
(∵ factorization)
⇒ f(x) = ![](data:image/png;base64,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)
⇒ f(x) = ![](data:image/png;base64,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)
⇒ f(x) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAAkCAMAAADM4ogkAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjqQOma2OpDbZgAAZgA6ZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkLb/kNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///b0mIaLAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABEklEQVQ4T9WT2xKCIBCGUTPK7JwdtEQt0vd/woRFWTQPzdRFe8EMy88u/PAR8otIN9fesmybyfV42SVjnmXvykW+EEoYURTBAWbF8UwSO6xqiYIJtUNOXdhRC8WEz4SQz28qfXez2FeVsfB52kPtsNofO+IwhOQrSwZ0jywHhOVp1H7mqcaN1ikVW7TwcSmCiTKpbp2vQ8JBWLaW7jA6LSKrdRlGwR55GRgGAqyO/SFdZfXYJxyq9+k6+Dscn9b9B3235xUwcIt32JjAgK6FjfxZBjC6oIGNcksDAwn4qRgbyCNgsJBIbBAzGBhDiLBRrTUwWIixKfMNYPRlDGxE2gRG2dOLDTa5DxvT5LFP+M3v+AIYwhsFNOv4fAAAAABJRU5ErkJggg==)
∴ The lowest form of the given rational expression is:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACwAAAAkCAMAAADFCSheAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjqQOma2OpDbZgAAZgA6ZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkLb/kNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///b0mIaLAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABHElEQVQ4T9WU2RaCIBCG0Rbad1usRCvS93/ChJlwEJdjpy7iJmL+GWbhk7Ffr2R1abxCrG/GHs3rpGLi+ZvcKKdvdbFDnyzYwS7bH1nsh/nOxFObmPuh5APwN2L1R46UWI6vxHQf3KIZ3kXFz8MW7lAuJk7UU8kxli48vSCTs9cDMWaGv2KCSZTSSLhys8WPUxb0sYEmjXQZMgliTENXKvgwO3tOgYJD60yBxaau27R1LJo1yLSJjqLLuNvifmqHGbSvT+P/o1/9XCiAUFkVhi6AoHUw1K/SAbAIbGGInbQBhEN47RRDOC8BSMVMY0gYLANoiQmGmIYNIBVTDPPzCgCLAi0M1bELILYOPzjVz6k8hCYM3SF0Gfe3n/ML5WMbBY9zRX4AAAAASUVORK5CYII=)
∴ Correct Option is - Option (B)
Question 13.If
and
are the two rational expressions, then their product is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGwAAAA7CAYAAACe7W+CAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAMvSURBVHja7VvLbRsxEGUBbsCA6ZsLUBgdc5UXhguwtNDN8MkgELgAem+6uAHfckwFubiC9KAO1ENiSl5LCrQKhxzOfvQOD/AKgmf0nuZDCk9VVaWA/gAkQDAAggF8gj2X5lYp9WcNM/sBUvPylxTM2WKkzezl82+lVnry+AQh8vGXFNDO598eFouz+vlxop/URfG2+xrAyx9/eUOwrPyxBvTfELTEvPzx9WM3PzeX169dqK7F4uGsLMbf/Uww5fNtH+KG8seWbGmMmzt33gWxigv1Vm9eUoKlxg3lj62UJb/JoTlJCpYSl8Ify6Dc7bv+uQvi9UUwKn8sye3DLKmtMceywiUYNTdK3Bj+Gvvxtbl83f4TvdJF6drcjqi51WSMS3tXFtr5RcA/+4MqZTGKFSw1brBg2+FpllPrrtYD8T1wzvYSSgolt5o4rYuf/r2bDW7zXsoVWqxgqXHpgu0c4Or7rlxnLLJgAbkdak1bwfWqsG4k1RJj4kbNMB/IluO7z7JuWTBKbk2zhDrbuGYY10xtvuf6IMP3Xjsx91yCHR60h9H04UJyayLoWLfgyC0mbrJgM6Nfdsu3Ky2Rktv/iJOuMGrcYMHqa/7dAdkVwSi5HW9NsjMsJm6wYIcI6IpglNyODn/CLwqsSwfDLxnN3+KPA5yz06uvWv+q11Jrp6N6pW6twgJy83Nu/fo7SfvrNe1gTxWMKy5phvlBviFGr8zE3q9vkpVa+md/IGxzhlFys2Vx80Wr3/WiEHN4jdlgOeJmu0sEZAESIBgAwQAIBsEACAZAsGEJFnpTDcgCFYaWCEAwII9g8IfJ8pcUDP6wSpy/pIDwh1Xi/PGXNwTLyh9rwK74w/poNwrlj68fd8Qf1le70cn6w/bmQU/sRhT+2JKEPyw+LvxhPRKMxR9GtfSk+sOos0LSbhQrWEhcFn9YG3Yj+mCXsxvFCjZouxFZMEG7EUdLHJzdKFY8CbsR1ww7CbtRU7wu2I2acjsJuxEF0nYjrgoblN0o+DagBbsRb0sciN2IcuaTthuxLh1DsRuRK0zQbkTFSdiNSIQI243iRIPdCIBgEAyAYMC/+AsUhHZZmvx9sAAAAABJRU5ErkJggg==)
Answer:Given: Two rational expressions:
and ![](data:image/png;base64,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)
Required: Product of the given two rational numbers
Let f(x) =
and g(x) = ![](data:image/png;base64,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)
Now, f(x)×g(x) = ![](data:image/png;base64,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)
⇒ f(x)×g(x) = ![](data:image/png;base64,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)
⇒ f(x)×g(x) = ![](data:image/png;base64,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)
∴ Product of the given rational expressions is:![](data:image/png;base64,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)
∴ Correct Option is - Option (A)
Question 14.On dividing
by
is equal to
A. (x – 5) (x – 3)
B. (x – 5) (x + 3)
C. (x + 5) (x – 3)
D. (x + 5) (x + 3)
Answer:Given: Two polynomials
and ![](data:image/png;base64,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)
Required: Divide
by ![](data:image/png;base64,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)
Let f(x) =
and g(x) = ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQcAAAAvCAMAAADZ/f96AAAAAXNSR0IArs4c6QAAALFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmZmOmaQOma2OpCQOpDbZgAAZgA6ZgBmZjoAZjpmZmaQZma2ZpDbZraQZrbbZrb/kDoAkDo6kGYAkGY6kGZmkLbbkLb/kNv/tmYAtmY6tmaQtpA6tpBmttv/tv//25A625Bm27Zm27aQ29u22/+22////7Zm/7aQ/9uQ/9u2//+2///bGm09RwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFVElEQVRoQ+1aa1fbOBC1QgNxl25hNzTb0mL6wtkWnNCNa1v//4ftjB62ZGsk2aS0OcWfOHhGGl3dmZGukyRPzxMCB40A/5qy+d1BL2EvwTcXN83yai9DHfogPPu9cOAfbx1btnnB2J5wqM4PghLrvx1hlvPbZnW5p/idM0wYm1+ziUUrwrNYuCJqVuzs/exmQrAOl31lWPMKNse1a8Eww57V8z0t1hdLedS2nk2K6Xb/WuXi5s0uuAjToPlnahMLeOZOOoyKzWdcMByfZ3oXy/nd9iYpuk2t/hwDRPXatv62YgyqT8EYC5C371mt2MzI+x9OB4lzS4jiGBZird3AJIh++WZXmUWrufiUbAGAIljRB57Ld0mVdm4irh/5yPLTLCUDSti4K0GHbTq7qVKYvfojzHRl/N8S3Pv1AXeSxoHyFBuTt2Pp+AZIwIT4hBGqPwxD0141sFbkBUyp8EbcZdksj3fFOWwCv44oUNrYEQ+CinlBNGfCE9Grs7YmlA9tCQ3wa01wimeLHX8v59ITdTgkxTNB8bhmoowlDg3yQjALeKXWb5LcQovwBOhedr2n6Ap5eOspC4Dc+UrUHtWWq1Ry0MBhcyr84nBQxoN5Ck2DrhT3bChPgPNCM5FnVHkoBnlIoLCFc2cEDsCNHg7fP/NsDt0zKi+0cT+IEmAor3AIig+UZ5LUHR2aJdk1uxriJQryvSAups3yfFdnJ5IsskBAnYQZMaU36YLniGBMnWyNe7HIBLnia2ib75xxUp5Aw2edh2arY4j8rxXDMhZ4mozNv6paODAtU3b2hclGlxsZaC19TN8MRTP1fUlfpfKTXb3Pi3dhVuQ2q3tnianrMPz0sbEec/D04QDcjcyNNgpfDBYO08/VQaSg6OB5BHIyaNoZtMGJslgvjYKHGIzFwReDB/IRAUeYitrvKXyuIXw4LCbkhSeGSBxydYAL3hdoRCCIb6tFuLQZI9hktYZen/bOiVER0jEIHNpF9v8I7DPpZ73Qg/Cs6+R+T+1B4WB4R0boi0G8i+RDBPGDJiYOQWNh4OED3g0mSB5kDJE4RLHOvzjByXHXRwMHnllXDTym2p09KkI6hkgc4vbPZ6Vr1JgCYQTXw2GaLuGJwUO9h6/dGEFdFcY1DHqTJuHgi+GxcKiVYtXXn7xg0+dqFCbuP4/cKV8M5rkahp0sUI4MKc7cwx64G0RcL0ABidQ3tRAjA5ssUA7WtTm1dMa4dfcVTfreHTteEqnz6nu3HHeyQNm1ZyUt8mupUMY9ndbRU/IersNE4qAycJvC0eTobrRACctsBU1csiXe+GqZ5WW49XB4sC6X5CdrNqPkYg7i5bGQYeVEzfIS/5ogUOIQStDs41B/cEsPkiKWPmnwwVY0SZ02jmdglR/dJhSrUKBUWoMsDwKH5wqHcQIlFhUpaFryJATADEHFEXarT5qqpkhOQ7YXeqHridarMS8oYnb/b3X7F1jV1LaMEighSENptPLi3lzQYDGmPmm5WYrmpHOCOVccDuo7TvNKJIkMZ5RAadhbeYFaKylVo6WlT7Y4DBXNyDpH5gkQnn8hVG9oyzv+L36jUN/14PYBF+rL8QIlskELmnZ9gO+l5ve5fpy2PtnhMFA0aUL0TtoEEjno1XOqb8FHRPFOf9YT/XJzrIuGHPJXECgxDPIg1On7JBviXrTbUIBQz9eoWHeq2bgPvXETTrJy/0oDtjL0CTl6tnYGFNjZGVbmX+tcLVbi/F0QZjR0uOil+gwP5HdBziVg2dhbXuwFzZ8zyBMOEnf8jr2Jvr/8nL16lFnLdPbW/J7xKJM+TfJbIfA//3+nV8jm2M4AAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJcAAAAmCAMAAAD+1/mbAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u22/+22////7Zm/7aQ/9uQ/9u2//+2///byvJu2wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACvklEQVRYR+1XYXPbIAyFrEncdluXtWuzOludNnbaxnPM//9zFRhsYcCA4+56t/Eph/XEk1DEEyH/19/MANtQOt+ZJ7r2O8vxyJD46uun+uabaena7yz9yCKh94Q83z41oOLuEMKotal/WPIFX137iNogspzv9hnJu6CrLzHEqlvdOqeUzjJC+vtmrH2L4pLO1mAm9/MFONa4II7ezJV3h4o7a1cOuYdl7Bue+hZs85vsISK5X0J89yJd+2SWVQmwrC7sN4NdS+PXFcC1+mp41cZ+C3YiearOsxbJ88VS7q1cHPIrSB7bwD34ljLu2/F7vBoEu5CEHH/9RIlveZH8TFxJw9K3pHFjJqIUmedRJ8N4F3JLz+y8iktea4G8pLGFPku1mzUs3MhnFFF3j38eWDqHbhF0j8q4dyrHevLlQNbfNSTU/ZKIui+SJdvSwLpvjfu8HqFNoNsw0+lCwvlNn8BL+wvG9AlfFZ76Pe/+Q3F91Xuw6vtHR3f3OBj9DvmIQc3ydlivhruHz83031kKNVuvoIonXdumM8lnb4xrIPZys4x5dAlpD+3/8BBw4vAH5YOl/G8uVxRyTBrCMRqvcNiw5UT32CVsKmKn+lF1H1dgp57qxQtavE9M/Yf0njxscJSC1i9g7X7erbGeFtdogW8cq3R6PB81GyDkaIHf9SgpBpVOD6bVyVA5GwjkPoFG92kXLfAFVA4E3I8mckGnD9JyIBGverUm5SwbIfDhYKzUMS9Np1sJ2pFoNhC8ziWvOIEPByqlrsl7out0e+LsSDQblJ+5SpTRRgl8OBApde0esU6383Ih1WxQX4tZsfEaJfCRvVZfPZ1up4U1fhsRng3gSQVBvo4X+DxbaiDQ696m03vs7EiGZgPRH4rF4aMJ/BwGM/bIJ6J3E/jBjQwb8tzRr/xh/aDv0Kio/jnQGystVKWcbu4EAAAAAElFTkSuQmCC)
⇒
(∵ a2–b2 = (a–b)(a + b))
⇒ ![](data:image/png;base64,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)
∴ Quotient when f(x) is divided g(x) we get is: (x-5)(x-3)
∴ Correct Option is - Option (A)
Question 15.If
is added with
, then the new expression is
A. a2 + ab + b2
B. a2 – ab + b2
C. a3 + b3
D. a3 – b3
Answer:Given: Two polynomials:
and ![](data:image/png;base64,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)
Required: Two add the given two polynomials
Let f(x) =
and g(x) = ![](data:image/png;base64,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)
Now, f(x) + g(x) =
+ ![](data:image/png;base64,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)
⇒ f(x) + g(x) = ![](data:image/png;base64,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)
⇒ f(x) + g(x) = ![](data:image/png;base64,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)
⇒ f(x) + g(x) = ![](data:image/png;base64,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)
⇒ f(x) + g(x) = ![](data:image/png;base64,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)
⇒ f(x) + g(x) =
(∵ a3–b3 = (a–b)(a2 + ab + b2))
∴ f(x) + g(x) = a2 + ab + b2
That is, the sum of given two polynomials is a2 + ab + b2
∴ Correct Option is - Option (A)
Question 16.The square root of 49 (x2 – 2x + y2)2 is
A. 7 |x – y|
B. 7(x + y) (x – y)
C. 7(x + y)2
D. 7(x – y)2
Answer:Given: A polynomial: 49 ![](data:image/png;base64,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)
Required: to find the square root of the given polynomial
Let f(x) = 49 ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
⇒
(∵ (x–y)2 = x2-2xy + y2)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
∴ Square root of the given polynomial is:![](data:image/png;base64,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)
∴ Correct Option is - Option (D)
Question 17.The square root of x2 + y2 + z2 – 2xy + 2yz – 2zx
A. |x + y – z|
B. |x – y + z|
C. |x + y + z|
D. |x – y – z|
Answer:Given: A polynomial: ![](data:image/png;base64,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)
Required: to find the square root of the given polynomial
Let f(x) = ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
⇒
(∵ (x–y–z)2 = x2 + y2 + z2-2xy + 2yz-2zx)
⇒ ![](data:image/png;base64,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)
∴ Square root of the given polynomial is:![](data:image/png;base64,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)
∴ Correct Option is - Option (D)
Question 18.The square root of 121 x4y8z6 (l – m)2 is
A. 11x2y4z4|l – m|
B. 11x4y4|z3(l – m)
C. 11x2y4z6|(l – m)|
D. 11x2y4|z3(l – m)|
Answer:Given: A polynomial: 121 ![](data:image/png;base64,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)
Required: to find the square root of the given polynomial
Let f(x) = 121 ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
∴ Square root of the given polynomial is:![](data:image/png;base64,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)
∴ Correct Option is - Option (D)
Question 19.If ax2 + bx + c = 0 has equal roots, then c is equal
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACwAAAA5CAYAAABNn8kpAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAIQSURBVGje7ZkxbsIwFIZ9AC6AxGPjANQwdg1WxQEAKxtiQpYqDmDYsnABtm7tCbpwgt6BG+QOLS+pE6OmjUND7CAPlrBJxGfzv/f83iO73Y60abQK1gPfPfCW0ykh5DMZdPHiNLAUbAh0sc8+ExJDsN44CyzC8HEVRR01XwewIT121Nec1nAijzYB4wk3LokoWnU4Gz+jHinfTo31LMMu7U8OZadbKzDCsh45KquvAswplaGUXSteIjGeCsD4fJXNWQVGQ9N1i/Oy96wBq+cuBz2VSeNmwGMuZpyBRAPEOQYJE6OyJgkA9jYXcpB6DpBVwq91SeQeBGIm5NB54Gu8h3VgdTMziWZOAbfmhNP1lmg4MzqD21jjwIKPZ4kLO8NdurXyoGAt4xCcPT0A+VARrK6g4ZNQD+yBPfA9AxdnAMXjvxcY6ydsutFrhtewB/bAHtgD15Ey1ZNt3BxY9S5aAYxF6jxNagHwgsIeGJdpicpx4KTue04885raT2D8bkL7h/zeADFusHFglMIIRq8I+Buwvo6lAFxTFc7GC9ooBVXdKQXWCiumtbfaXZj+g39JQn8Giy9Z4bspYL0VawqsQPE9EdBlo8BmGUsOjtLRi4NWJFH0dxedcBZQtBaC08BFcO044e91KeaDEcC7atwIMR8qd+cEcGJwZyNLoSGmgVgm/WZCTjjHlpm/Xnpgw/EFAJR03YDvN5wAAAAASUVORK5CYII=)
Answer:Given: The quadratic equation has equal roots
Here,
b2–4ac = 0 (∵ The quadratic equation has equal roots)
⇒ b2–4ac = 0
⇒ b2 = 4ac
⇒ 4ac = b2
⇒ c = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAjCAMAAAB8ZqA1AAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6Oma2OpDbZgAAZgBmZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkNv/tmYAtmY6tpA6tpBmttv/tv//25A627Zm29uQ2////7Zm/9uQ/9u2//+2///brSd/0QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAsElEQVQoU6WR6w7CIAyF21103uZ1Y7rNTRQQ3v8BLVuQGWNCYv/AR0+hnAIEhqkQk5sX622vdxsAjnN3qPc23zhWB2EzjuVRqJPlWYtRCTpHRKqHJu6Bx/5mq1eL+v3SF6fCXKf6FTXm5YHfCZZRl9MIrvtDqLJyWm2q5Qfzgll+nkf/5NpY1nkBbSqAhmBYcRmsl8R86J42d/KJFopBL6Ma+DjQR0Z2aoZJ52f5q/0XHlcL0iBCXFoAAAAASUVORK5CYII=)
∴ The value of c = ![](data:image/png;base64,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)
∴ Correct Option is - Option (B)
Question 20.If x2 + 5kx + 16 = 0 has no real roots, then
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Given: The quadratic equation
and has no real roots.
Required: To find the value of k
Here,
b2–4ac<0 (∵ Quadratic equation has no real roots)
⇒ (5k)2–4(1)(16)<0
⇒ 25k2<64
⇒ k2<![](data:image/png;base64,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)
Applying Square root on both sides
⇒ √k2 < ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
∴ The value of k is ![](data:image/png;base64,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)
∴ Correct Option is - Option (C)
Question 21.A quadratic equation whose one root is 3 is
A. x2 – 6x – 5 = 0
B. x2 + 6x – 5 = 0
C. x2 – 5x – 6 = 0
D. x2 – 5x + 6 = 0
Answer:Given: One of the root of the Quadratic Equation that is 3
Required: To find the Quadratic equation, which satisfies the given root.
Now,
We substitute the zero in every equation given in the options
∴ Case (i): x2–6x-5 = 0
⇒ (3)2–6(3)–5 = 0
⇒ 9–18–5 = –14≠0
Case (ii): x2 + 6x–5 = 0
⇒ (3)2 + 6(3)–5 = 0
⇒ 9 + 18–5 = 22≠0
Case(iii): x2–5x–6 = 0
⇒ (3)2–5(3)–6 = 0
⇒ 9–15–6 = –12≠0
Case(iv): x2–5x + 6 = 0
⇒ (3)2–5(3) + 6 = 0
⇒ 9–15 + 6 = 0
∴ The Quadratic equation with one root as 3 is x2–5x–6 = 0
∴ Correct Option is - Option (D)
Question 22.The common root of the equation x2 – bx + c = 0 and x2 + bx – a = 0 is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Given: Two Quadratic Equations
and ![](data:image/png;base64,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)
Required: to find the common root of the given Quadratic Equations
Let the common root be α
Α is the root of x2–bx + c = 0
∴ x2–bx + c = 0
⇒ α2–bα + c = 0 -eq(1)
Also, α is the root of x2 + bx–a = 0
∴ α2 + bα–a = 0 -eq(2)
Here, eq(1) = eq(2)
∴ α2 + bα–a = α2–bα + c
⇒ 2bα = c + a
⇒ α = ![](data:image/png;base64,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)
∴ The common root is α = ![](data:image/png;base64,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)
∴ Correct Option is - Option(A)
Question 23.If α, β are the roots of ax2 + bx + c = 0 a ≠ 0, then the wrong statement is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Given:
are the roots of ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Required: To find the wrong statement in the given options
We know that sum of roots is given by :![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAhCAMAAAAIybBlAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgA6Ojo6OpDbZgAAZrbbZrb/kDoAkGY6kNv/tmYAtmY6tpA6tpBmttv/tv//25A627Zm2////7Zm/9uQ/9u2//+2///b2RWMaQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAd0lEQVQoU71RSRKAIAxrEVdExQ1R8f/PdDtYtxm8mGMmaZsU4As0hle5cmGCEpmkTuU1oL0WrMANEtY5Q1QQ0QPjm6laXAdUjMip6UsaJ+1+H4GT63/RmCMmdK0VKZS+uVzSn5lu6e/E9KwAzWnFNkNe3//+mnAG5NcEKqQ/JaEAAAAASUVORK5CYII=)
∴ ![](data:image/png;base64,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)
We, also know that product of roots is given by: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6ADpmADqQAGa2OgA6OgBmOjo6OpDbZgAAZjoAZrbbkDoAkGY6kNv/tmY6tmaQtpBmtv//25A627Zm2////7Zm/9uQ/9u2//+2///b4bzcqwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXklEQVQYV42OSw6AIAxEpwJ+UERFAeH+97Rq3LAwvM0knZdmgJI8E0mHbJSPnUNs7WN8mbTyefV8GEm8VRX0UuX+SifvG4CkJyySdzDhzqPn3R6hsdiFQzIkNlLlqwumzANbvrkdFAAAAABJRU5ErkJggg==)
∴ ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKwAAAAjCAMAAAA+CRHnAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6Oma2OpDbZgAAZgA6ZgBmZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNu2kNv/tmYAtmY6tmZmtpA6tpBmtrZmttv/tv//25A627Zm27aQ27a229u229v/2////7Zm/9uQ/9u2//+2///byPRH5wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACZ0lEQVRYR+2Y61LbMBCFpSRASompWy4pN6cFKkNrO9L7P1x3pdhRpDXadjwZZ0C/mOGs99PRaiVFiI/xnh0w36Wc/ToQB/SXZ50vRg6r5ElLqL+O3tmihW2+VSM3VogWtr6omsux0xbHKzm5FjqTUo69ZkUxfRZqSlXr71zKs/14zU2FZdB8eoih9PmdKCfEP4bnZ6fqhUUmchYMWPM4l5P0DvBlrFTFUWV+kmUAUCpdxeYFqmUWLADWVpneAr6MkQq6wec4V+tdGZWsWV4HxhbySqyz3dnqDCZpll0HdyFRrC+LUzEW0JeoeHsRsIik5M4caiz1NKwnI1KxYJscVhUnXQNrHRgZw9pv1ha2i1QIa33zRxS7lVGpOLDN/FKU06cJ8ELn3XWMWEr3yQLLoIu0rXCdnwUnYgzbyshUHFjYcLCCp0G9tZG0s80cjd1GAqySx3+CdBQsIXOzR6PseKt5usULanBjX98HzBJjvEh01qzgXOwGmTyWJdzsANwfm5Q12EsOytnNTnKwNtKdiOj02zVLyjjLv9G4lLYGqUHBFicWyot0sOEhTtZsLPuHMrA26byvUghYhQbWC9eqXKTd5tGM6W7QbwzDYkhkftxni1fS2xjWHpOmgPXYRsKNvhIv82TrImUMxk5ibvDwXMmw8TgBdYLZYscjq4tU09vT+G5AlAElS8MO+16kr5wRBVMWxQ37XmRSMGWU1QO+F5kUTBkBO+R7kUnBlHmw6xu7Ow7ivaizK7E6qg7jvYge9x6u6S6yZ0UJL4W+m8CeUZLp8LauDuUXOb2Us8ft713JyX0I/tOBv6YvPoeiSOY5AAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
∴ We can clearly see in the option that
is wrong.
∴ Correct Option is - Option (C)
Question 24.If α and β are the roots of ax2 + bx + c = 0, then one of the quadratic equations whose roots are
is
A. ax2 + bx + c = 0
B. bx2 + ax + c = 0
C. cx2 + bx + a = 0
D. cx2 + ax + b = 0
Answer:Given:
and
are roots of ![](data:image/png;base64,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)
Required:- Quadratic equation with roots
and ![](data:image/png;base64,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)
Sum of roots of given quadratic equation = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAhCAMAAAAIybBlAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgA6Ojo6OpDbZgAAZrbbZrb/kDoAkGY6kNv/tmYAtmY6tpA6tpBmttv/tv//25A627Zm2////7Zm/9uQ/9u2//+2///b2RWMaQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAd0lEQVQoU71RSRKAIAxrEVdExQ1R8f/PdDtYtxm8mGMmaZsU4As0hle5cmGCEpmkTuU1oL0WrMANEtY5Q1QQ0QPjm6laXAdUjMip6UsaJ+1+H4GT63/RmCMmdK0VKZS+uVzSn5lu6e/E9KwAzWnFNkNe3//+mnAG5NcEKqQ/JaEAAAAASUVORK5CYII=)
∴
=
-eq(1)
Product of roots of given quadratic equation = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6ADpmADqQAGa2OgA6OgBmOjo6OpDbZgAAZjoAZrbbkDoAkGY6kNv/tmY6tmaQtpBmtv//25A627Zm2////7Zm/9uQ/9u2//+2///b4bzcqwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXklEQVQYV42OSw6AIAxEpwJ+UERFAeH+97Rq3LAwvM0knZdmgJI8E0mHbJSPnUNs7WN8mbTyefV8GEm8VRX0UuX+SifvG4CkJyySdzDhzqPn3R6hsdiFQzIkNlLlqwumzANbvrkdFAAAAABJRU5ErkJggg==)
∴
=
-eq(2)
Sum of roots of required quadratic equation = ![](data:image/png;base64,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)
Product of roots of required quadratic equation = ![](data:image/png;base64,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)
Here,
Dividing eq(1) by eq(2) we get,
![](data:image/png;base64,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)
∴ Sum of roots of the required quadratic equation = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAhCAMAAAAIybBlAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgA6OgBmOjo6ZgAAZjoAZrb/kDoAkGY6kNv/tmYAtmY6tmaQtpA6ttv/tv//25A62////9uQ//+2///bvFci7QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAcElEQVQoU72QSRaAIAxDWxFRERUnpvvf02lhHRawMcu8/L40ACkyKJ5xHeMUA2aKkprNYNgCXuIhBfsdV/Yk9OFwG6aNuqRrxJxCKd9EZc9+RFHU/6HQIXI6TWiFdRV17vPuFd+Ol8KG0dL+rkmaeAXrRwRREggKCgAAAABJRU5ErkJggg==)
Again by making the reciprocal of eq(2), we get
![](data:image/png;base64,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)
∴ Product of roots of the required quadratic equation = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6ADpmADqQAGa2OgA6OgBmOjo6OpDbZgAAZjoAZrbbkDoAkGY6kNv/tmY6tmaQtpBmtv//25A627Zm2////7Zm/9uQ/9u2//+2///b4bzcqwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAX0lEQVQYV42O2wqAIBBEZ1O7WGZlqen//2emBFEEnZcDO7MwwJN9JOqAIAdM3ObUnd5aomRXaazMIChiC4nX+9eBCr/7n8WY9nGDqIT1jYGvda5eDlLYOKe9vidWojsHgKwDW3yeGrMAAAAASUVORK5CYII=)
We know that, when roots of the quadratic equation are known, we can calculate the quadratic equation as:
x2-(sum of roots)x + (product of roots) = 0
∴ Required quadratic equation: x2 –(
) + (
) = 0
⇒
= 0
⇒ cx2 + bx + a = 0
∴ Required quadratic equation is: cx2 + bx + a = 0
∴ Correct option is -Option(C)
Question 25.Let b = a + c. Then the equation ax2 + bx + c = 0 has equal roots, if
A. a = c
B. a = - c
C. a = 2 c
D. a = -2c
Answer:Given: b = a + c and the quadratic equation has equal roots
Here,
b2–4ac = 0 (∵ The quadratic equation has equal roots)
⇒ b2–4ac = 0
⇒ (a + c)2–4ac = 0 (∵ b = a + c)
⇒ a2 + 2ac + c2–4ac = 0
⇒ a2–2ac + c2 = 0
⇒ (a–c)2 = 0
Applying sq.rt on both sides
⇒ √(a–c)2 = √0
⇒ a–c = 0
⇒ a = c
∴ a = c
∴ Correct option is -Option (A)
If the system 6x – 2y = 3, kx – y = 2 has a unique solution, then
A. k = 3
B. k ≠ 3
C. k = 4
D. k ≠ 4
Answer:
Given: Two equations: 6x – 2y = 3 and kx – y = 2
Required: To find the value of k such that system of equations have unique solutions
To have unique solution to the System of equations, the required condition is
∴
⇒
⇒
⇒
∴ For every value of k except 3 the system equations have unique solutions.
∴ Correct option is – Option (B)
Question 2.
A system of two linear equations in two variables is consistent, if their graphs
A. coincide
B. intersect only at a point
C. do not intersect at any point
D. cut the x-axis
Answer:
Given: A system of two linear equations in two variables is consistent
We know that if a system of two linear equations in two variables is consistent then their graph do not intersect at any point
∴ Correct option is – Option (C)
Question 3.
The system of equations x –4y = 8 , 3x –12y = 24
A. has infinitely many solutions
B. has no solution
C. has a unique solution
D. may or may not have a solution
Answer:
Given: system of equations x –4y = 8 , 3x –12y = 24
Here,
a1 = 1, b1 = -4, c1 = 8 and a2 = 3, b2 = -12, c2 = 24
Now,
,
,
Here, we can clearly see that which is the condition for infinitely many solutions.
∴ The system of equations have infinitely many solutions.
∴ Correct option is – Option (A)
Question 4.
If one zero of the polynomial p(x) = (k + 4)x2 + 13x + 3k is reciprocal of the other, then k is equal to
A. 2
B. 3
C. 4
D. 5
Answer:
Given: A Quadratic equation p(x) = (k + 4)x2 + 13x + 3k
Required: To find the value of k
Let the roots of the given Quadratic equation be: and
∴ Product of roots of the given Quadratic equation is
We know that, Product of roots of a given Quadratic equation is
∴
⇒
⇒ 3k = k + 4
⇒ 2k = 4
⇒ k = 2
∴ The value of k is 2
∴ Correct option is – Option (A)
Question 5.
The sum of two zeros of the polynomial f(x) = 2x2 + (p + 3)x + 5 is zero, then the value of p is
A. 3
B. 4
C. –3
D. –4
Answer:
Given: A Quadratic equation f(x) = 2x2 + (p + 3)x + 5 and sum of roots is zero.
Required: to find the value of p
We know that, sum of roots =
∴ = 0
⇒ -(p + 3) = 0
⇒ P + 3 = 0
∴ p = –3
∴ The value of p is –3
∴ Correct option is – Option (C)
Question 6.
The remainder when x2 – 2x + 7 is divided by x + 4 is
A. 28
B. 29
C. 30
D. 31
Answer:
Given: x2 – 2x + 7
Required: Reminder of x2 – 2x + 7 at x + 4
By synthetic division we can find reminder of
x2 – 2x + 7 at x = -4
∴ Reminder of x2 – 2x + 7 at x + 4 = 31
∴ Correct option is –Option (D)
Question 7.
The quotient when x3 – 5x2 + 7x – 4 is divided by x–1 is
A. x2 + 4x + 3
B. x2 – 4x + 3
C. x2 – 4x -3
D. x2 + 4x – 3
Answer:
Given: x2 – 2x + 7
Required: Reminder of x3 – 5x2 + 7x – 4 at x – 1
By synthetic division we can find Quotient of x3 – 5x2 + 7x – 4 at x = 1
∴ Quotient of x2 – 2x + 7 when divided by x–1 = x2–4x + 3
∴ Correct option is –Option(B)
Question 8.
The GCD of (x3 + 1) and x4 – 1 is
A. x3 – 1
B. x3 + 1
C. -(x + 1)
D. x–1
Answer:
Given two polynomials: (x3 + 1) and x4 – 1
Required: To find GCD of the given two polynomials
Let f(x) = x3 + 1 and g(x) = X4 – 1
Here, degree of g(x) > f(x) ∴ Devisor = x3 + 1
Here reminder = –(x + 1) (Reminder ≠ zero)
∴ The GCD of given two polynomials is -(x + 1)
∴ Correct Option is - Option(C)
Question 9.
The GCD of x2 – 2xy + y2 and x4 – y4 is
A. 1
B. x + y
C. x – y
D. x2 – y2
Answer:
Given two polynomials: x2 – 2xy + y2 and x4 – y4
Required: To find GCD of the given two polynomials
Let f(x) = x2 – 2xy + y2 and g(x) = x4 – y4
f(x) = x2 – 2xy + y2
⇒ f(x) = (x–y)2
g(x) = x4 – y4
⇒ g(x) = (x2)2 – (y2)2
⇒ g(x) = (x2 – y2)(x2 + y2) (∵ a2–b2 = (a–b)(a + b))
⇒ g(x) = (x–y)(x + y)(x2 + y2) (∵ a2–b2 = (a–b)(a + b))
∴ The GCD of given two polynomials is (x–y)
∴ Correct Option is - Option (C)
Question 10.
The LCM of x3 – a3 and (x - a)2 is
A. (x3 – a3) (x + a)
B. (x3 – a3) (x - a)2
C. (x - a)2 (x2 + ax + a2)
D. (x + a)2(x2 + ax + a2)
Answer:
Given two polynomials: x3 – a3 and (x - a)2
Required: To find LCM of the given two polynomials
Let f(x) = x3 – a3 and g(x) = (x - a)2
here,
f(x) = x3 – a3
⇒f(x) = (x-a)(x2 + a2 + xa)
g(x) = (x - a)2
⇒g(x) = (x-a)(x-a)
∴ LCM = (x-a) (x2 + a2 + xa) (x-a) = (x-a)2(x2 + a2 + xa)
∴ Correct Option is - Option (C)
Question 11.
The LCM of ak,ak + 3, ak + 5 where is
A. a k + 9
B. ak
C. ak + 6
D. ak + 5
Answer:
Given three polynomials: ak , ak + 3 and ak + 5
Required: To find LCM of the given two polynomials
Let f(x) = ak , g(x) = ak + 3 and h(x) = ak + 5
here,
f(x) = ak
⇒f(x) = ak
g(x) = ak + 3
⇒g(x) = ak ×a3
h(x) = ak + 5
⇒h(x) = ak ×a3 + 2 = ak×a3×a2
∴ LCM = ak×a3×a2 = ak + 5
∴ Correct Option is - Option (D)
Question 12.
The lowest form of the rational expression is
A.
B.
C.
D.
Answer:
Given: Rational Expression:
Required: The lowest form of the given Rational Expression.
Let f(x) =
⇒ f(x) = (∵ factorization)
⇒ f(x) =
⇒ f(x) =
⇒ f(x) =
∴ The lowest form of the given rational expression is:
∴ Correct Option is - Option (B)
Question 13.
If and
are the two rational expressions, then their product is
A.
B.
C.
D.
Answer:
Given: Two rational expressions: and
Required: Product of the given two rational numbers
Let f(x) = and g(x) =
Now, f(x)×g(x) =
⇒ f(x)×g(x) =
⇒ f(x)×g(x) =
∴ Product of the given rational expressions is:
∴ Correct Option is - Option (A)
Question 14.
On dividing by
is equal to
A. (x – 5) (x – 3)
B. (x – 5) (x + 3)
C. (x + 5) (x – 3)
D. (x + 5) (x + 3)
Answer:
Given: Two polynomials and
Required: Divide by
Let f(x) = and g(x) =
Now,
⇒
⇒
⇒ (∵ a2–b2 = (a–b)(a + b))
⇒
∴ Quotient when f(x) is divided g(x) we get is: (x-5)(x-3)
∴ Correct Option is - Option (A)
Question 15.
If is added with
, then the new expression is
A. a2 + ab + b2
B. a2 – ab + b2
C. a3 + b3
D. a3 – b3
Answer:
Given: Two polynomials: and
Required: Two add the given two polynomials
Let f(x) = and g(x) =
Now, f(x) + g(x) = +
⇒ f(x) + g(x) =
⇒ f(x) + g(x) =
⇒ f(x) + g(x) =
⇒ f(x) + g(x) =
⇒ f(x) + g(x) = (∵ a3–b3 = (a–b)(a2 + ab + b2))
∴ f(x) + g(x) = a2 + ab + b2
That is, the sum of given two polynomials is a2 + ab + b2
∴ Correct Option is - Option (A)
Question 16.
The square root of 49 (x2 – 2x + y2)2 is
A. 7 |x – y|
B. 7(x + y) (x – y)
C. 7(x + y)2
D. 7(x – y)2
Answer:
Given: A polynomial: 49
Required: to find the square root of the given polynomial
Let f(x) = 49
Now,
⇒ (∵ (x–y)2 = x2-2xy + y2)
⇒
⇒
∴ Square root of the given polynomial is:
∴ Correct Option is - Option (D)
Question 17.
The square root of x2 + y2 + z2 – 2xy + 2yz – 2zx
A. |x + y – z|
B. |x – y + z|
C. |x + y + z|
D. |x – y – z|
Answer:
Given: A polynomial:
Required: to find the square root of the given polynomial
Let f(x) =
Now,
⇒ (∵ (x–y–z)2 = x2 + y2 + z2-2xy + 2yz-2zx)
⇒
∴ Square root of the given polynomial is:
∴ Correct Option is - Option (D)
Question 18.
The square root of 121 x4y8z6 (l – m)2 is
A. 11x2y4z4|l – m|
B. 11x4y4|z3(l – m)
C. 11x2y4z6|(l – m)|
D. 11x2y4|z3(l – m)|
Answer:
Given: A polynomial: 121
Required: to find the square root of the given polynomial
Let f(x) = 121
Now,
⇒
⇒
∴ Square root of the given polynomial is:
∴ Correct Option is - Option (D)
Question 19.
If ax2 + bx + c = 0 has equal roots, then c is equal
A.
B.
C.
D.
Answer:
Given: The quadratic equation has equal roots
Here,
b2–4ac = 0 (∵ The quadratic equation has equal roots)
⇒ b2–4ac = 0
⇒ b2 = 4ac
⇒ 4ac = b2
⇒ c =
∴ The value of c =
∴ Correct Option is - Option (B)
Question 20.
If x2 + 5kx + 16 = 0 has no real roots, then
A.
B.
C.
D.
Answer:
Given: The quadratic equation and has no real roots.
Required: To find the value of k
Here,
b2–4ac<0 (∵ Quadratic equation has no real roots)
⇒ (5k)2–4(1)(16)<0
⇒ 25k2<64
⇒ k2<
Applying Square root on both sides
⇒ √k2 <
⇒
∴ The value of k is
∴ Correct Option is - Option (C)
Question 21.
A quadratic equation whose one root is 3 is
A. x2 – 6x – 5 = 0
B. x2 + 6x – 5 = 0
C. x2 – 5x – 6 = 0
D. x2 – 5x + 6 = 0
Answer:
Given: One of the root of the Quadratic Equation that is 3
Required: To find the Quadratic equation, which satisfies the given root.
Now,
We substitute the zero in every equation given in the options
∴ Case (i): x2–6x-5 = 0
⇒ (3)2–6(3)–5 = 0
⇒ 9–18–5 = –14≠0
Case (ii): x2 + 6x–5 = 0
⇒ (3)2 + 6(3)–5 = 0
⇒ 9 + 18–5 = 22≠0
Case(iii): x2–5x–6 = 0
⇒ (3)2–5(3)–6 = 0
⇒ 9–15–6 = –12≠0
Case(iv): x2–5x + 6 = 0
⇒ (3)2–5(3) + 6 = 0
⇒ 9–15 + 6 = 0
∴ The Quadratic equation with one root as 3 is x2–5x–6 = 0
∴ Correct Option is - Option (D)
Question 22.
The common root of the equation x2 – bx + c = 0 and x2 + bx – a = 0 is
A.
B.
C.
D.
Answer:
Given: Two Quadratic Equations and
Required: to find the common root of the given Quadratic Equations
Let the common root be α
Α is the root of x2–bx + c = 0
∴ x2–bx + c = 0
⇒ α2–bα + c = 0 -eq(1)
Also, α is the root of x2 + bx–a = 0
∴ α2 + bα–a = 0 -eq(2)
Here, eq(1) = eq(2)
∴ α2 + bα–a = α2–bα + c
⇒ 2bα = c + a
⇒ α =
∴ The common root is α =
∴ Correct Option is - Option(A)
Question 23.
If α, β are the roots of ax2 + bx + c = 0 a ≠ 0, then the wrong statement is
A.
B.
C.
D.
Answer:
Given: are the roots of
Required: To find the wrong statement in the given options
We know that sum of roots is given by :
∴
We, also know that product of roots is given by:
∴
Now,
⇒
⇒
⇒
⇒
Now,
∴ We can clearly see in the option that is wrong.
∴ Correct Option is - Option (C)
Question 24.
If α and β are the roots of ax2 + bx + c = 0, then one of the quadratic equations whose roots are is
A. ax2 + bx + c = 0
B. bx2 + ax + c = 0
C. cx2 + bx + a = 0
D. cx2 + ax + b = 0
Answer:
Given: and
are roots of
Required:- Quadratic equation with roots and
Sum of roots of given quadratic equation =
∴ =
-eq(1)
Product of roots of given quadratic equation =
∴ =
-eq(2)
Sum of roots of required quadratic equation =
Product of roots of required quadratic equation =
Here,
Dividing eq(1) by eq(2) we get,
∴ Sum of roots of the required quadratic equation =
Again by making the reciprocal of eq(2), we get
∴ Product of roots of the required quadratic equation =
We know that, when roots of the quadratic equation are known, we can calculate the quadratic equation as:
x2-(sum of roots)x + (product of roots) = 0
∴ Required quadratic equation: x2 –() + (
) = 0
⇒ = 0
⇒ cx2 + bx + a = 0
∴ Required quadratic equation is: cx2 + bx + a = 0
∴ Correct option is -Option(C)
Question 25.
Let b = a + c. Then the equation ax2 + bx + c = 0 has equal roots, if
A. a = c
B. a = - c
C. a = 2 c
D. a = -2c
Answer:
Given: b = a + c and the quadratic equation has equal roots
Here,
b2–4ac = 0 (∵ The quadratic equation has equal roots)
⇒ b2–4ac = 0
⇒ (a + c)2–4ac = 0 (∵ b = a + c)
⇒ a2 + 2ac + c2–4ac = 0
⇒ a2–2ac + c2 = 0
⇒ (a–c)2 = 0
Applying sq.rt on both sides
⇒ √(a–c)2 = √0
⇒ a–c = 0
⇒ a = c
∴ a = c
∴ Correct option is -Option (A)
Exercise 3.2
Question 1.Solve the following systems of equation using cross multiplication method.
3x + 4y = 24, 20x – 11y = 47
Answer:The given system of equations is
3x + 4y – 24 = 0 and
20x – 11y – 47 = 0
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADAAAAAgCAMAAABjCgsuAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6ADo6ADqQAGaQAGa2OgAAOgA6OjoAOmZmOmaQOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bvlAauQAAAAF0Uk5TAEDm2GYAAAC6SURBVHja7VHLEoIwDExQWx9QK0UEAUvb//9Hp2DB4KkXHWfYy84k2WweAD+Cu7Oo+vYoWaRF8/8CV+90VH9ExAxWBGAk1otFoj1gcnnjCUYinjS4ksbd9QZdUk08J4oMjMigYbrfV9QkBEjCCgVQM09eS7zLnHBwSLURyp4rrwMrxpf7LrgZCkeeE36HVAcBwYMrwuNIMgdXfI7kO/RcTTwvxgcB1NvF0i1/nZUvztrx4axWYpJ/4b1PVP8OGmkm214AAAAASUVORK5CYII=)
⇒
=
= ![](data:image/png;base64,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)
⇒ x =
= 4
⇒ y =
= 3
∴ (4, 3) is the solution to the given system.
Question 2.Solve the following systems of equation using cross multiplication method.
0.5x + 0.8y = 0.44, 0.8x + 0.6y = 0.5
Answer:The given equations are
0.5x + 0.8y – 0.44 = 0
0.8x +0.6y – 0.5 = 0
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGMAAAAjCAMAAAC3vUTLAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OgBmOjo6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kNv/tmYAtmY6ttuQttv/tv/btv//25A625Bm27Zm27aQ29u229vb2////7Zm/9uQ/9u2//+2///bxdlVdAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABiklEQVRIS+1Va1ODMBA8sIKvIkjVQm1Rm1ZFIP//33l3ARIQ28qIjkPzBWbvsXubEABGumTiDD355iocnANAHDkO3shf8EquTtOD9fRLFBYuv1/tsWpEDtA5GXiNyM1/POrL7dMn9Zu7H72nROeFlF3uIflamXyw7Hulmi48ewnczIBhO5tS+wa1juehZVGcwkZVCXMz4aTZ2VJxROpBY2hYJtdqgOx8rR2s4zL2IQ98pcyoKmHqVgQRUJrmkDFSGXD9U5ILJYWWjtMbrBxWZlRVMCqD4gYLMaXyykNG4qhhGU9nF+wVByoOI+6leRC1qjCZYaxaNDjID7ednbne22tMIrBNEah7JzKkkfFe2uYABVNVwyvu47e84m5iQtndXoVzrHLaDhcKZg5YTXjP30/W1APnUHtew884NCc397yOP7qaw2iWKZi9giK07DkwR4JHF1/5/Gi4Grp5dnV86/LZZWUahRKmPe9awutC93yDjZNt1Hd/0AC97pJeyjpn3AH2UvZdkj/M/wDPQTYoDK046QAAAABJRU5ErkJggg==)
⇒
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,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)
⇒ x =
= 0.4
⇒ y =
= 0.3
∴ (0.4, 0.3) is the solution to the given system.
Question 3.Solve the following systems of equation using cross multiplication method.
![](data:image/png;base64,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)
Answer:The given equations are
–
= – 2 … (1)
+
=
… (2)
By taking LCM,
(1) becomes 9x – 10y + 12 = 0
(2) becomes 2x + 3y – 13 = 0
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACcAAAAgCAMAAACmeJG/AAAAAXNSR0IArs4c6QAAAGNQTFRFAAAAAAAAAAA6ADqQAGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kDqQkGY6kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bhSeGTQAAAAF0Uk5TAEDm2GYAAACwSURBVHja5ZLJDsIwDETHhUCXQMrSEkpC8v9fWWVreuCQA0gIfLGSGT3bsoGPh72xEtt9z1kZUH63z45bVUIjImrwq0Flgb+LByeqAdu78dkL4ULVATDdFVM1wJ4V7Gnwh9ULZEEy9dz5f6Ssa2RfFEwrYPtwZzKkiFv5ZAPTDcDo+5kCBtq9TBsWJaKQfTLaEm7heWGp65rSIuFWviiMGz9HqCNWuDRvFAyn6vj2bcyZYAvLhSLozAAAAABJRU5ErkJggg==)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAgCAMAAAD68tKbAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAA///b//+22///tv///9uQ29uQkNv//7ZmZrb/Zrbb25Bm25A6OpDbtmY6kGaQtmYAOma2kDqQOmZmkDo6AGa2kDoAAGaQZjpmZjoAOjoAADqQOgAAAABmAAAAzHOQRAAAAAF0Uk5TAEDm2GYAAABySURBVHjavZDJCoAwDEQTW7XWfWnd8/+fKakK9SKC6DsEhhkIMwCPwML4Mm1Hcw1k7zRWvfBdIqIcvoCuwA+ESwKAlt9xa+wm1o0A7GIuq22yB1XtDh7a2cEg0eqS11HmnIZmcdgut+fVOWm0rNKzb9kA95MHNB7ntrMAAAAASUVORK5CYII=)
⇒ x =
= 2
⇒ y =
= 3
∴ (2, 3) is the solution to the given system.
Question 4.Solve the following systems of equation using cross multiplication method.
![](data:image/png;base64,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)
Answer:The given equations are
–
= – 2 … (1)
+
= 13 … (2)
Let a =
and b =
.
⇒ 5a – 4b + 2 = 0 … (3)
⇒ 2a + 3b – 13 = 0 … (4)
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAgCAMAAADdXFNzAAAAAXNSR0IArs4c6QAAAFFQTFRFAAAA///b//+22////9u2/9uQkNv/27ZmZrb/Zrbb25Bm25A6OpDbtmY6tmYAOma2OmaQOmZmkDo6AGa2kDoAAGaQZjoAZgAAOgAAAAA6AAAAiFJZxQAAAAF0Uk5TAEDm2GYAAACRSURBVHja1ZLLDoMgFEQHX73WB7ZawPn/D21EA9gFmzZpnAULTuZOLgzwU6lhzuH7w8z5Ad1/uRpfZc5Nkj2uJOaF6+vo3PY3a5PcV4acYuc6CYZF/NmjsrETn7ywAowJJ6lPfl2mfgC1ExR2f1jx+frEt8g437RQS+icejbeH3jtdn50Tg3k2ib5uDm/33d6A1PKC8ItJE55AAAAAElFTkSuQmCC)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAgCAMAAAD68tKbAAAAAXNSR0IArs4c6QAAAGlQTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bsvReGQAAAAF0Uk5TAEDm2GYAAAB6SURBVHjatY/LDoIwFETPoFSt8qgKtqhg+/8f6YKY0JUmhrOZTObmZgZ+It3KpQ2HuswP/H8+9dvnMpWkI2ugHFbnUUsWCHsVDcTqylB0pPOsANMuVz9Pf11aAAYLQK9NC+Dt5/XdOBgtjI5YdUzGEU+S5CAYFc3Xdm/lbAWNmSZcSQAAAABJRU5ErkJggg==)
⇒ a =
= 2
⇒ b =
= 3
When a = 2,
= 2. Thus, x = 1/2.
When b = 3,
= 3. Thus, y =
.
∴ (
,
) is the solution to the given system.
Question 5.Formulate the following problems as a pair of equations, and hence find their solutions:
One number is greater than thrice the other number by 2. If 4 times the smaller number exceeds the greater by 5, find the numbers.
Answer:Let x be the greater number and y be the smaller number.
First condition is x = 3y + 2
Equation is x – 3y – 2 = 0 … (1)
Second condition is x = 4y – 5
Equation is x – 4y + 5 = 0 … (2)
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAgBAMAAAAGZY7BAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAAGaQAGa2OmZmOpDbZgAAZjoAZrb/tmYAtmY625A625Bm2/////+2///bWBw9lgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAOUlEQVQoU2NgwA74F8PEjxTDVWwmjsU/6QJU4UZBURzmExAWhAIB8rQj6yLCJ8+AtjUwYPEdYZ8AAFnQECxSqz28AAAAAElFTkSuQmCC)
⇒ x =
= 23
⇒ y =
= 7
∴ The numbers are 23 and 7.
Question 6.Formulate the following problems as a pair of equations, and hence find their solutions:
The ratio of income of two persons is 9: 7 and the ratio of their expenditure is 4: 3. If each of them manages to save ₹ 2000 per month, find their monthly income.
Answer:Let the incomes be x and expenditure be y.
We know that savings = income – expenditure
First condition is
9x – 4y = 2000
Second condition is
7x – 3y = 2000
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgBAMAAAAs1X9uAAAAAXNSR0IArs4c6QAAACdQTFRFAAAAAAAAAGaQAGa2OmZmOpDbZjoAtmYAtmY625A625Bm2//////bcdiRUwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAALElEQVQYV2NgQIATBgwMXG5AgoETC3FcGyguKBiApAGNKSgoKIBbFiJDqh0AeeMH1iO4ctMAAAAASUVORK5CYII=)
⇒ x = 2000
⇒ y = 4000
∴ Income of first person = 9x = 9 × 2000 = Rs. 18, 000
Income of second person = 7x = 7 × 2000 = Rs. 14, 000
Question 7.Formulate the following problems as a pair of equations, and hence find their solutions:
A two digit number is seven times the sum of its digits. The number formed by reversing the digits is 18 less than the given number. Find the given number.
Answer:Let x denote the digit in the tenth place and y denote the digit in the unit place. So the number may be written as 10x + y.
When digits are reversed, the number becomes 10y + x.
First condition is
10x + y = 7 (x + y)
⇒ 10x + y – 7x – 7y = 0
⇒ 3x – 6y = 0
⇒ x – 2y = 0
Second condition is
10y + x = (10x + y) – 18
⇒ 10y + x – 10x – y + 18 = 0
⇒ – 9x + 9y + 18 = 0
⇒ – x + y + 2 = 0
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAA///b//+22////9uQ29u2kNv//7Zm27ZmkLbbZrb/25Bm25A6OpDbtmY6tmYAOma2OmZmAGa2AGaQZjoAZgA6ZgAAOgA6AAA6AAAALw4FZgAAAAF0Uk5TAEDm2GYAAABpSURBVHjazZHJDoAgDEQHXFAErYhL//9HPXAw0XIwxoR37LRN2gd8QI1BrA9zDJkR93+gpqUS+5mZPcqAM6BU7q6byEyCa7326I5WFlTvmcD5S6ne0issgI5k145k14YAY5+u0077/uwToBgHyupUFSAAAAAASUVORK5CYII=)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAgBAMAAAAGZY7BAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAAGaQAGa2OmZmOpDbZgAAZjoAZrb/tmYAtmY625A625Bm2/////+2///bWBw9lgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAOUlEQVQoU2NgwA74F8PEjxTDVWwmjsU/6QJU4UZBURzmExAWhAIB8rQj6yLCJ8+AtjUwYPEdYZ8AAFnQECxSqz28AAAAAElFTkSuQmCC)
⇒ x =
= 4
⇒ y =
= 2
∴ The number is 10x + y = 10 (4) + 2 = 40 + 2 = 42.
Question 8.Formulate the following problems as a pair of equations, and hence find their solutions:
Three chairs and two tables cost ₹ 700 and five chairs and three tables cost ₹1100. What is the total cost of 2 chairs and 3 tables?
Answer:Let chairs be x and tables be y.
Then the equations are
3x + 2y = 700 i.e. 3x + 2y – 700 = 0
5x + 3y = 1100 i.e. 5x + 3y – 1100 = 0
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAgBAMAAAAGZY7BAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAAGaQAGa2OmZmOpDbZgAAZjoAZrb/tmYAtmY625A625Bm2/////+2///bWBw9lgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAOUlEQVQoU2NgwA74F8PEjxTDVWwmjsU/6QJU4UZBURzmExAWhAIB8rQj6yLCJ8+AtjUwYPEdYZ8AAFnQECxSqz28AAAAAElFTkSuQmCC)
⇒ x =
= 100 (Cost of one chair)
⇒ y =
= 200 (Cost of one table)
Now, total cost of two chairs and three tables,
2x + 3y = 2 (100) + 3 (200) = 200 + 600 = Rs. 800
Question 9.Formulate the following problems as a pair of equations, and hence find their solutions:
In a rectangle, if the length is increased and the breadth is reduced each by 2 cm then the area is reduced by 28 cm2. If the length is reduced by 1 cm and the breadth increased by 2 cm , then the area increases by 33 cm2. Find the area of the rectangle.
Answer:Let length be l and breadth be b.
Then the first condition is
(l + 2) (b – 2) = lb – 28
⇒ lb – 2l + 2b – 4 – lb + 28 = 0
⇒ – 2l + 2b + 24 = 0
⇒ – l + b + 12 = 0
The second condition is
(l – 1) (b + 2) = lb + 33
⇒ lb + 2l – b – 2 – lb – 33 = 0
⇒ 2l – b – 35 = 0
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAdElEQVQ4T7VRSRKAIAwr7oobLgj//6iteKr14IA5Mc0kaQlABPxciOql0jIBYP4n/JTv0lpGIZqIc1NKaRcJKTOSevGuN61UjQm8a9cNsGYjZT+7PsoXwoReLoVrw8f0+F4pQrAy95x3bXFuUci7Dp7k+BEnQ4sDtY8XdUcAAAAASUVORK5CYII=)
⇒
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAgBAMAAAAGZY7BAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAAGaQAGa2OmZmOpDbZgAAZjoAZrb/tmYAtmY625A625Bm2/////+2///bWBw9lgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAOUlEQVQoU2NgwA74F8PEjxTDVWwmjsU/6QJU4UZBURzmExAWhAIB8rQj6yLCJ8+AtjUwYPEdYZ8AAFnQECxSqz28AAAAAElFTkSuQmCC)
⇒ l =
= 23
⇒ b =
= 11
∴ Area of rectangle= l × b = 23 × 11 = 253 cm2
Question 10.Formulate the following problems as a pair of equations, and hence find their solutions:
A train travelled a certain distance at a uniform speed. If the train had been 6 km/hr faster, it would have taken 4 hours less than the scheduled time. If the train were slower by 6 km/hr, then it would have taken 6 hours more than the scheduled time. Find the distance covered by the train.
Answer:Let the speed be x km/hr and distance travelled be y km.
We know that distance = speed × time
Scheduled time to cover distance = y/x hr
Then the first condition is
⇒
=
– 4
⇒
= ![](data:image/png;base64,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)
⇒ xy = (x + 6) (y – 4x)
⇒ xy = xy – 4x2 + 6y – 24x
⇒ 4x2 – 6y + 24x = 0
⇒ 2x2 – 3y + 12x = 0 i.e. 12x – 3y + 2x2 = 0
Second condition is
⇒
=
+ 6
⇒
= ![](data:image/png;base64,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)
⇒ xy = (x – 6) (y + 6x)
⇒ xy = xy + 6x2 – 6y – 36x
⇒ 6x2 – 6y – 36x = 0
⇒ x2 – y – 6x = 0 i.e. – 6x – y + x2 = 0
For cross multiplication method, we write the coefficients as
![](data:image/png;base64,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)
Hence, we get
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,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)
⇒
=
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
⇒ – 30x = – x2
⇒ 30km/hr = x
Now,
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
⇒ y = 24 × 30 = 720 km
∴ The distance covered by train = 720 km
Solve the following systems of equation using cross multiplication method.
3x + 4y = 24, 20x – 11y = 47
Answer:
The given system of equations is
3x + 4y – 24 = 0 and
20x – 11y – 47 = 0
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ x = = 4
⇒ y = = 3
∴ (4, 3) is the solution to the given system.
Question 2.
Solve the following systems of equation using cross multiplication method.
0.5x + 0.8y = 0.44, 0.8x + 0.6y = 0.5
Answer:
The given equations are
0.5x + 0.8y – 0.44 = 0
0.8x +0.6y – 0.5 = 0
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ x = = 0.4
⇒ y = = 0.3
∴ (0.4, 0.3) is the solution to the given system.
Question 3.
Solve the following systems of equation using cross multiplication method.
Answer:
The given equations are
–
= – 2 … (1)
+
=
… (2)
By taking LCM,
(1) becomes 9x – 10y + 12 = 0
(2) becomes 2x + 3y – 13 = 0
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ x = = 2
⇒ y = = 3
∴ (2, 3) is the solution to the given system.
Question 4.
Solve the following systems of equation using cross multiplication method.
Answer:
The given equations are
–
= – 2 … (1)
+
= 13 … (2)
Let a = and b =
.
⇒ 5a – 4b + 2 = 0 … (3)
⇒ 2a + 3b – 13 = 0 … (4)
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ a = = 2
⇒ b = = 3
When a = 2, = 2. Thus, x = 1/2.
When b = 3, = 3. Thus, y =
.
∴ (,
) is the solution to the given system.
Question 5.
Formulate the following problems as a pair of equations, and hence find their solutions:
One number is greater than thrice the other number by 2. If 4 times the smaller number exceeds the greater by 5, find the numbers.
Answer:
Let x be the greater number and y be the smaller number.
First condition is x = 3y + 2
Equation is x – 3y – 2 = 0 … (1)
Second condition is x = 4y – 5
Equation is x – 4y + 5 = 0 … (2)
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ x = = 23
⇒ y = = 7
∴ The numbers are 23 and 7.
Question 6.
Formulate the following problems as a pair of equations, and hence find their solutions:
The ratio of income of two persons is 9: 7 and the ratio of their expenditure is 4: 3. If each of them manages to save ₹ 2000 per month, find their monthly income.
Answer:
Let the incomes be x and expenditure be y.
We know that savings = income – expenditure
First condition is
9x – 4y = 2000
Second condition is
7x – 3y = 2000
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ x = 2000
⇒ y = 4000
∴ Income of first person = 9x = 9 × 2000 = Rs. 18, 000
Income of second person = 7x = 7 × 2000 = Rs. 14, 000
Question 7.
Formulate the following problems as a pair of equations, and hence find their solutions:
A two digit number is seven times the sum of its digits. The number formed by reversing the digits is 18 less than the given number. Find the given number.
Answer:
Let x denote the digit in the tenth place and y denote the digit in the unit place. So the number may be written as 10x + y.
When digits are reversed, the number becomes 10y + x.
First condition is
10x + y = 7 (x + y)
⇒ 10x + y – 7x – 7y = 0
⇒ 3x – 6y = 0
⇒ x – 2y = 0
Second condition is
10y + x = (10x + y) – 18
⇒ 10y + x – 10x – y + 18 = 0
⇒ – 9x + 9y + 18 = 0
⇒ – x + y + 2 = 0
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ x = = 4
⇒ y = = 2
∴ The number is 10x + y = 10 (4) + 2 = 40 + 2 = 42.
Question 8.
Formulate the following problems as a pair of equations, and hence find their solutions:
Three chairs and two tables cost ₹ 700 and five chairs and three tables cost ₹1100. What is the total cost of 2 chairs and 3 tables?
Answer:
Let chairs be x and tables be y.
Then the equations are
3x + 2y = 700 i.e. 3x + 2y – 700 = 0
5x + 3y = 1100 i.e. 5x + 3y – 1100 = 0
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ x = = 100 (Cost of one chair)
⇒ y = = 200 (Cost of one table)
Now, total cost of two chairs and three tables,
2x + 3y = 2 (100) + 3 (200) = 200 + 600 = Rs. 800
Question 9.
Formulate the following problems as a pair of equations, and hence find their solutions:
In a rectangle, if the length is increased and the breadth is reduced each by 2 cm then the area is reduced by 28 cm2. If the length is reduced by 1 cm and the breadth increased by 2 cm , then the area increases by 33 cm2. Find the area of the rectangle.
Answer:
Let length be l and breadth be b.
Then the first condition is
(l + 2) (b – 2) = lb – 28
⇒ lb – 2l + 2b – 4 – lb + 28 = 0
⇒ – 2l + 2b + 24 = 0
⇒ – l + b + 12 = 0
The second condition is
(l – 1) (b + 2) = lb + 33
⇒ lb + 2l – b – 2 – lb – 33 = 0
⇒ 2l – b – 35 = 0
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ l = = 23
⇒ b = = 11
∴ Area of rectangle= l × b = 23 × 11 = 253 cm2
Question 10.
Formulate the following problems as a pair of equations, and hence find their solutions:
A train travelled a certain distance at a uniform speed. If the train had been 6 km/hr faster, it would have taken 4 hours less than the scheduled time. If the train were slower by 6 km/hr, then it would have taken 6 hours more than the scheduled time. Find the distance covered by the train.
Answer:
Let the speed be x km/hr and distance travelled be y km.
We know that distance = speed × time
Scheduled time to cover distance = y/x hr
Then the first condition is
⇒ =
– 4
⇒ =
⇒ xy = (x + 6) (y – 4x)
⇒ xy = xy – 4x2 + 6y – 24x
⇒ 4x2 – 6y + 24x = 0
⇒ 2x2 – 3y + 12x = 0 i.e. 12x – 3y + 2x2 = 0
Second condition is
⇒ =
+ 6
⇒ =
⇒ xy = (x – 6) (y + 6x)
⇒ xy = xy + 6x2 – 6y – 36x
⇒ 6x2 – 6y – 36x = 0
⇒ x2 – y – 6x = 0 i.e. – 6x – y + x2 = 0
For cross multiplication method, we write the coefficients as
Hence, we get =
=
⇒ =
=
⇒ =
=
⇒ =
⇒ – 30x = – x2
⇒ 30km/hr = x
Now, =
⇒ =
⇒ y = 24 × 30 = 720 km
∴ The distance covered by train = 720 km
Exercise 3.3
Question 1.Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
x2 – 2x – 8
Answer:Let f(x) = x2 – 2x – 8
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ x2 – 2x – 8 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 2 and product is 8.
∴ x2 – (4 – 2)x – 8 = 0
⇒ x2 – 4x + 2x – 8 = 0
⇒ x(x – 4) + 2(x – 4) = 0
⇒ (x + 2)(x – 4) = 0
∴ x = – 2 and x = 4.
⇒ Our zeros are α = – 2 and β = 4.
⇒ sum of zeros = α + β = – 2 + 4 = 2.
⇒ Product of zeros = αβ = ( – 2) × 4 = – 8.
⇒ Comparing f(x) = x2 – 2x – 8 with standard equation ax2 + bx + c = 0.
We get, a = 1, b = – 2 and c = – 8
We can verify,
⇒ Sum of zeros = ![](data:image/png;base64,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)
i.e. α + β = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAgCAMAAACrZuH4AAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAdklEQVQ4T2NgoC0Q4WZkZMdnhQQXL4MwEx8BV4izElIhyIFphAQnIxjwgKSE8ToDqECQkAIxoAIxsFk4AMQ+fCpoG9hkmM4PCSEgAMcBnAdjkGHkkNMiKcCC381CbNwEVABTz+BQIcnPLIrXM4KgmMWSV8iMNQClogLRDEyV+gAAAABJRU5ErkJggg==)
∴ α + β = 2
⇒ Product of zeros = ![](data:image/png;base64,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)
αβ = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOmaQOma2OpDbZgAAZjoAZrbbZrb/kDoAkNv/tmYAtmY625A625Bm27Zm2////9uQ/9u2//+2///bcIk3dwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAdklEQVQ4T8WS2wqAIBBEx8pKLbOLqf//oynUkxsREs2bHHdmGRYol1OMdTb3CVrASZEDL0fANNREb12CmVJGT2R4NSBowmrnNwAbp9ctL+KNg2GXqinOnY83Dv/+DTNRelxpbRUNgOV7EExNHEmMTv0Sl/jY4QESsQOzBxoF9QAAAABJRU5ErkJggg==)
αβ = – 8.
Hence, relationship between zeros and coefficient is verified.
Question 2.Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
4x2 – 4x + 1
Answer:Let f(x) = 4x2 – 4x + 1
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ 4x2 – 4x + 1 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 4 and product is 4.
∴ 4x2 – (2 + 2)x + 1 = 0
⇒ 4x2 – 2x – 2x + 1 = 0
⇒ 2x(2x – 1) – 1(2x – 1) = 0
⇒ (2x – 1)(2x – 1) = 0
∴ 2x – 1 = 0
∴ x =
.
Again, 2x – 1 = 0
∴ x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAWUlEQVQYV2NgwACSAiwgMSE2bjDNwCCIn5bkZxYFqWIEAg5M43CIgFQDAdHq4QpFuBkZ2RkYJLh4GYSZ+MDC4qwQWhBivTBQGsSDUGJASoyHQYITZB0Pun0AG8sCbODE2DAAAAAASUVORK5CYII=)
⇒ Our zeros are α =
and β =
.
⇒ sum of zeros = α + β =
+
= 1.
⇒ Product of zeros = αβ =
.
Now, Comparing f(x) = 4x2 – 4x + 1 with standard equation ax2 + bx + c = 0.
We get, a = 4, b = – 4 and c = 1
We can verify,
⇒ Sum of zeros = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAAAhCAMAAAA27N8rAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OmaQOma2OpDbZgAAZgBmZjoAZjpmZmY6ZmZmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkLb/kNv/tmYAtmaQtpA6tpBmtrZmttv/tv//25A625Bm27Zm29vb2////7Zm/7aQ/9uQ/9u2//+2///bpzWZjAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB5ElEQVRIS+2W63KDIBCFwVaxV216115imkaTalHf/+G6LOCFRDGZ6a+EmQAJLIc9aPgIOZXWgbVPIyKqzMvVr21v2CgMO6QUbrqZYzUqmLH+4hhxiB5BGXtKiSFojyCkfqfUS0k5o/QGRKo3Sm9JQSmNVAXSOOebQk8Pe0vficgXTEPHVTRGELLxnTn3m6PoZ1zHLOeXqWjKkJE6DkgZRp0MYdNqDvbUsPNBElhRZaijtSeFl2e3+ugNh/kFeo5NdpZyHzZJg76gmgOC3WFhnxLU0c0hZOcvQyfZE4QVpf6gYGd4SxA2oM9wfTVgKJxZyPJ6kYumjJn8upz3BdUcWK07jIJe/gPb09Fa8Pezjt3VQI58Rl3IChoqbIfWeRUPDZMV9+G4cI7uwTA4y+ATiFH0TkVjBIHXl9WJfJpO5eTA3g4k4rXG4oj3rfn2T52993cKOGYHarhW3XSyAxportv/wwl3eZdnqrtVNQumCjZA8zAqaACNyTPV/eQMDwMawwP+tOu2GAeaR3HFyCmFFWgMnimec75969uApuUZcbVagEZlKHmmCpFUzGIDGkExLVRYgEZbOsIzEp00Qe0AGkNwHGgm8IwVaDTFLBCPLECjBEd5RsLKINC0PNOgzTDQHCfP/AFei11E98V7+wAAAABJRU5ErkJggg==)
i.e. α + β = ![](data:image/png;base64,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)
∴ α + β = 1
⇒ Product of zeros = ![](data:image/png;base64,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)
αβ = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAABmADqQAGaQAGa2OgAAOjoAOmZmOma2OpDbZjoAZjpmZrbbZrb/kDoAkDo6kGaQtmYAtmY6tv//25A625Bm29uQ2////7Zm/9uQ//+2///bTz76BgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAWklEQVQYV2NgwAAywqwgMTEObjDNwCCKn5YRYpEAqWIEAi5M43CIgFQDAdHqkRRKsfGBeDIC7GBalJcfREtyyoBoaR4RGX5eQYhzGJnBCsHyDAzibEwi6PYBAGYKA1Xe9vSfAAAAAElFTkSuQmCC)
Hence, relationship between zeros and coefficient is verified.
Question 3.Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
6x2 – 3 – 7x
Answer:Let f(x) = 6x2 – 3 – 7x
Arranging equation in proper form.
Now, f(x) = 6x2 – 7x – 3
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ 6x2 – 7x – 3 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 7 and product is – 18.
∴ 6x2 – (9 – 2)x – 3 = 0
⇒6x2 – 9x + 2x – 3 = 0
⇒ 3x(2x – 3) + 1(2x – 3) = 0
⇒ (3x + 1)(2x – 3) = 0
∴ 3x + 1 = 0
∴ x =
.
Again, 2x – 3 = 0
∴ x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAA///b//+2/9u2/9uQ29u2kNv//7Zm27aQ27ZmkLbbZrb/Zrbb25Bm25A6ZpDbOpDbtmYAkGY6Oma2kDo6ZjoAOjoAADo6ZgA6ZgAAOgA6OgAAAAA6AAAAWdJFPQAAAAF0Uk5TAEDm2GYAAABeSURBVHjalY45EoAgEARHRBABLzxQ2P9/0wIyTLSTrp3aC6ixB8UJaLYBKooc8Tu7XU3SQqcprX3QYF6ABw3YkOe+QgX8ptuJHMD86z+Mc5ZypSqSDpAa7ErndL3tAYu4BPqa/gHrAAAAAElFTkSuQmCC)
⇒ Our zeros are α =
and β =
.
⇒ sum of zeros = α + β =
+ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAA///b//+2/9u2/9uQ29u2kNv//7Zm27aQ27ZmkLbbZrb/Zrbb25Bm25A6ZpDbOpDbtmYAkGY6Oma2kDo6ZjoAOjoAADo6ZgA6ZgAAOgA6OgAAAAA6AAAAWdJFPQAAAAF0Uk5TAEDm2GYAAABeSURBVHjalY45EoAgEARHRBABLzxQ2P9/0wIyTLSTrp3aC6ixB8UJaLYBKooc8Tu7XU3SQqcprX3QYF6ABw3YkOe+QgX8ptuJHMD86z+Mc5ZypSqSDpAa7ErndL3tAYu4BPqa/gHrAAAAAElFTkSuQmCC)
⇒ sum of zeros = α + β = ![](data:image/png;base64,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)
⇒ Product of zeros = αβ =
.
Now, Comparing f(x) = 6x2 – 7x – 3 with standard equation ax2 + bx + c.
We get, a = 6, b = – 7 and c = – 3.
We can verify,
⇒ Sum of zeros = ![](data:image/png;base64,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)
i.e. α + β = ![](data:image/png;base64,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)
∴ α + β = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFFQTFRFAAAA///b//+22////9u2/9uQ29vb29u2kNv//7Zm27Zm25A6OpC2tmY6Oma2OmaQkDqQkDo6AGa2kDoAZjo6ZjoAOjo6ADqQZgA6OgAAAAAAMLm0dgAAAAF0Uk5TAEDm2GYAAABbSURBVHjanY1JDoAwDAMdwlIotNCULf9/KILCpQeEmNvYkg1kkKiqBtDIoMlcWetS9Wh4049oAj/oomeAhp5Ps/e9+Lh4RrW7upGAYjOAXUHimCQA5azqON86ANvNBCIH4kyeAAAAAElFTkSuQmCC)
⇒ Product of zeros = ![](data:image/png;base64,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)
αβ = ![](data:image/png;base64,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)
Hence, relationship between zeros and coefficient is verified.
Question 4.Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
4x2 + 8x
Answer:Let f(x) = 4x2 + 8x
Arranging equation in proper form.
Now, f(x) = 4x2 + 8x + 0
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ f(x) = 4x2 + 8x + 0 = 0
∴4x2 + 8x = 0
⇒4x(x + 2) = 0
Now, 4x = 0
∴ x = 0
When, (x + 2) = 0
Then, x = – 2
⇒ Our zeros are α = 0 and β = – 2.
⇒ sum of zeros = α + β = 0 + ( – 2)
⇒ sum of zeros = α + β = – 2.
⇒ Product of zeros = αβ = 0 × ( – 2) = 0.
Now, Comparing f(x) = 4x2 + 8x + 0 with standard equation ax2 + bx + c.
We get, a = 4, b = 8 and c = 0.
We can verify,
⇒ Sum of zeros = ![](data:image/png;base64,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)
i.e. α + β = ![](data:image/png;base64,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)
∴ α + β = – 2
⇒ Product of zeros = ![](data:image/png;base64,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)
αβ = ![](data:image/png;base64,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)
Hence, relationship between zeros and coefficient is verified.
Question 5.Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
x2 – 15
Answer:Let f(x) = x2 – 15
Arranging equation in proper form.
Now, f(x) = x2 + 0x – 15
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ x2 – 15 = 0
∴ x2 –
= 0
So, (x +
)(x –
) = 0
When, (x +
) = 0
Then, x = –
.
When. (x –
) = 0
Then, x = ![](data:image/png;base64,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)
⇒ Our zeros are α = –
and β =
.
⇒ sum of zeros = α + β = –
+ ![](data:image/png;base64,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)
⇒ sum of zeros = α + β = 0
⇒ Product of zeros = αβ = –
= – 15
Now, Comparing f(x) = x2 + 0x – 15 with standard equation ax2 + bx + c.
We get, a = 1, b = 0 and c = – 15.
We can verify,
⇒ Sum of zeros = ![](data:image/png;base64,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)
i.e. α + β = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6ADqQAGaQAGa2OgA6OmZmOpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkGY6kNv/tmYAtmY625A625Bm2////9uQ//+2///b+5PvmwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbElEQVQ4T2NgoBxICjAy8WAzRoRVTJxNEFNGgpOPQZKfA4sEF1C1ECspEriMYhBixm45gwQ3IxMv5b6m0AQhRhhgAoUQlEOhoXTULimMJZqA9ouyc2OXYGAQob2EpBCLGNaUCApfLAmOYIABACYFA8+LGlC1AAAAAElFTkSuQmCC)
∴ α + β = 0
⇒ Product of zeros = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGAAAAAgCAMAAADaHo1mAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmaQOma2OpDbZgAAZgBmZjoAZjpmZmZmZmaQZma2ZpC2ZpDbZrbbZrb/kDoAkDo6kGY6kJBmkLbbkLb/kNv/tmYAtmY6tmaQtpA6tpBmtrZmttv/tv//25A625Bm27Zm29u229vb2////7Zm/7aQ/9uQ/9u2//+2///bEgdS2gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACM0lEQVRIS+2W2XabMBCGZ2ihkG62626OaevYadyYuBUR8P5v1n9GCAuCfZqLXmTRObLwrJoB9EH0FEaR3q3Sy8Z09+7JZiSBl43pxhI035mTHVVz5veGSk62WbQkusmYF1cMnalgMe00Tkbk1vob88Tpkr36rq44uggSNXlq7OudLNUM+y2iFW0SY8+WVExJdlnPzukSIVuNymTI2uRTqmYwjdb2ncEvbWSGdduztVjrUrzYUSGhEtPk0Vds05uWTqozTGBRp5Qncvh7q6MJgjD0e45OabD9W23KaAK3vVMJ6llqmp9GliqXFrlQ2LKUj2b9KVF0EXe1qUxbhFW9t9D3Krh8JfX4YeccYxtYeGIINaeY01vUjqA2ixZ1zvFWpapRmXZVVrhF5+olAk5LNEzmvZ/XZ4cH2gF5R/7neKBted72v3WgASVw0BwbYM1SpjvBDofl6eDi4S3qD9f1/Oh5VMa7/Q/M9ckEQyKLlzu63ag/Hq3An/unNzwk8mE36mc/S/V9HJPnK/MnvH9LgNY4k19yFdAXJPJkPrAcHmBTBnZmwMkXYxcAag/HHV9bejlyqoleefoqsz04Dyx3NwygKibAl4J0gOOQr1owfloTuQroG5C5x3J9Hl4qjVyfQt53wTr2DhIE9L2TwNeMoDdvumcP3yM9HId89RW0Jvg71LZk7rEc8W8vmjy+7kro4bjlK4gqZE2FuSuRxbht7qqlb5/MjuXqIZ9rabPR77JHMf4CjOFddYmkn3cAAAAASUVORK5CYII=)
αβ = ![](data:image/png;base64,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)
Hence, relationship between zeros and coefficient is verified.
Question 6.Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
3x2 – 5x + 2
Answer:Let f(x) = 3x2 – 5x + 2.
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ 3x2 – 5x + 2 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 5 and product is 6.
∴ 3x2 – (3 + 2)x + 2 = 0
⇒3x2 – 3x – 2x + 2 = 0
⇒ 3x(x – 1) – 2(x – 1) = 0
⇒ (3x – 2)(x – 1) = 0
When, 3x – 2 = 0
Then, x =
.
Again when, x – 1 = 0
∴ then, x = 1
⇒ Our zeros are α =
and β = 1.
⇒ sum of zeros = α + β =
+ 1
⇒ sum of zeros = α + β = ![](data:image/png;base64,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)
⇒ Product of zeros = αβ =
.
Now, Comparing f(x) = 3x2 – 5x + 2 with standard equation ax2 + bx + c.
We get, a = 3, b = – 5 and c = 2.
We can verify,
⇒ Sum of zeros = ![](data:image/png;base64,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)
i.e. α + β = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACwAAAAhCAMAAACVxLntAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OgBmOjoAOjqQOma2OpDbZgAAZgA6ZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ2////9uQ/9u2//+2///bY3Wj3wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAz0lEQVQ4T+VTyxLCIAwMtb7fVq1KtVj+/x8loe0IZJzYgxf3Ukg3SzYEgF/isb4mx1Wbmi1BL97DWimVnQHMjGNHUb1vU0MNPtiTzeSWFGIPnZT/hWXMcWELV0wET26WjuNAiSbHT6xCApEyhdCygIyHt8pMGRC6thfXuR0Vwxjkg+g0aH/nVJP5GPylAHxz3ewMDAmWvv3tvEC/SxdD1P80p5qqbCv0bosT3PGtSmHGcvLzSKMsQqlGcrKbVnqrAjSr7vkJyFDl8tZJ9D5wXmx2Cpc6KP0vAAAAAElFTkSuQmCC)
∴ α + β = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgA6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ/9uQ/9u2//+2///bXfyARAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZElEQVQYV42O2xKAIAhEwexelqXd/P//DKF6sGmmfWFYWDgAqRwiKgvgOpk8lfxSrE3LLPQVBGO5DzPF2te1T4OuRf0P3Ju+QNXQbzPCEjkjT851H/j/hNnFsRLXUQsfeM25RCeHOgM8qAgctAAAAABJRU5ErkJggg==)
⇒ Product of zeros = ![](data:image/png;base64,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)
αβ = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGa2OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bBE+F5wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbUlEQVQoU6WPSRKAIAwEE3fBHQUV9f/PFAWDFw6Wc+qahC4CEMpSIRZ2uPMBVCRoU2eep5Jq5dZNMXlcDa6t9TA0sfwnl8Xks2JnCb15813KHKPajo8ueO/WN49ixJgYYE7djVyAdgwyJWfgsyfRqAP1pExd2gAAAABJRU5ErkJggg==)
Hence, relationship between zeros and coefficient is verified.
Question 7.Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
2x2 – 2√2 x + 1
Answer:Let f(x) = 2x2 – 2
x + 1
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ 2x2 – 2
x + 1 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 2
and product is 2.
∴ 2x2 – (
)x + 1 = 0
⇒2x2 –
x –
x + 1 = 0
⇒
x(
x – 1) – 1(
x – 1) = 0
⇒ (
x – 1)(
x – 1) = 0
⇒ (
x – 1)2 = 0
∴ x =
,![](data:image/png;base64,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)
⇒ Our zeros are α =
and β =
.
⇒ sum of zeros = α + β =
+ ![](data:image/png;base64,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)
⇒ sum of zeros = α + β = ![](data:image/png;base64,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)
⇒ Product of zeros = αβ =
.
Now, Comparing f(x) = 2x2 – 2
x + 1 with standard equation ax2 + bx + c.
We get, a = 2, b = – 2
and c = 1.
We can verify,
⇒ Sum of zeros = ![](data:image/png;base64,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)
i.e. α + β = ![](data:image/png;base64,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)
∴ α + β = ![](data:image/png;base64,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)
⇒ Product of zeros = ![](data:image/png;base64,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)
αβ = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAY0lEQVQoU5WOSRKAIAwEgzviFlGE/3/UIEi8UIVz6qo0wwBk47bmve2dSgyA/9it9RmLUFCG/JeFF99CKbRZs5KXf/kxDiVEH1w7zqCrJT28Wmbk+Trq5CGjITRT6JF+ZuBMbtcPArbUXiyhAAAAAElFTkSuQmCC)
Hence, relationship between zeros and coefficient is verified.
Question 8.Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
x2 + 2x – 143
Answer:Let f(x) = x2 + 2x – 143
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ x2 + 2x – 143 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is 2 and product is – 143.
∴ x2 + (13 – 11)x – 143 = 0
⇒x2 + 13x – 11x – 143 = 0
⇒ x (x + 13) – 11(x + 13) = 0
⇒ (x – 11) (x + 13) = 0
∴ When, (x – 11) = 0
∴ Then, x = 11.
Again, When, (x + 13) = 0
∴ Then, x = – 13
⇒ Our zeros are α = 11 and β = – 13.
⇒ sum of zeros = α + β = 11 + ( – 13)
⇒ sum of zeros = α + β = – 2
⇒ Product of zeros = αβ = 11 × ( – 13) = – 143
Now, Comparing f(x) = x2 + 2x – 143 with standard equation ax2 + bx + c.
We get, a = 1, b = 2 and c = – 143.
We can verify,
⇒ Sum of zeros = ![](data:image/png;base64,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)
i.e. α + β = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbUlEQVQ4T2NgoByIcDMysmMxRoKLl0GYiQ+7BeKsOCQEObBrEMZmBVCpIA5xMaC4GA+mWRKcjECARYLygCDFBH6QK8AAHEJQNikmDKxaSQEWrA4QYuPGLgGMKdpLSPIzi2JzliAofHEkLLzhCABVhgKUS4EiGgAAAABJRU5ErkJggg==)
∴ α + β = – 2
⇒ Product of zeros = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGAAAAAgCAMAAADaHo1mAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmaQOma2OpDbZgAAZgBmZjoAZjpmZmZmZmaQZma2ZpC2ZpDbZrbbZrb/kDoAkDo6kGY6kJBmkLbbkLb/kNv/tmYAtmY6tmaQtpA6tpBmtrZmttv/tv//25A625Bm27Zm29u229vb2////7Zm/7aQ/9uQ/9u2//+2///bEgdS2gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACM0lEQVRIS+2W2XabMBCGZ2ihkG62626OaevYadyYuBUR8P5v1n9GCAuCfZqLXmTRObLwrJoB9EH0FEaR3q3Sy8Z09+7JZiSBl43pxhI035mTHVVz5veGSk62WbQkusmYF1cMnalgMe00Tkbk1vob88Tpkr36rq44uggSNXlq7OudLNUM+y2iFW0SY8+WVExJdlnPzukSIVuNymTI2uRTqmYwjdb2ncEvbWSGdduztVjrUrzYUSGhEtPk0Vds05uWTqozTGBRp5Qncvh7q6MJgjD0e45OabD9W23KaAK3vVMJ6llqmp9GliqXFrlQ2LKUj2b9KVF0EXe1qUxbhFW9t9D3Krh8JfX4YeccYxtYeGIINaeY01vUjqA2ixZ1zvFWpapRmXZVVrhF5+olAk5LNEzmvZ/XZ4cH2gF5R/7neKBted72v3WgASVw0BwbYM1SpjvBDofl6eDi4S3qD9f1/Oh5VMa7/Q/M9ckEQyKLlzu63ag/Hq3An/unNzwk8mE36mc/S/V9HJPnK/MnvH9LgNY4k19yFdAXJPJkPrAcHmBTBnZmwMkXYxcAag/HHV9bejlyqoleefoqsz04Dyx3NwygKibAl4J0gOOQr1owfloTuQroG5C5x3J9Hl4qjVyfQt53wTr2DhIE9L2TwNeMoDdvumcP3yM9HId89RW0Jvg71LZk7rEc8W8vmjy+7kro4bjlK4gqZE2FuSuRxbht7qqlb5/MjuXqIZ9rabPR77JHMf4CjOFddYmkn3cAAAAASUVORK5CYII=)
αβ =
– 143
Hence, relationship between zeros and coefficient is verified.
Question 9.Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
3, 1
Answer:Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = 3 and product of roots = 1
∴ Quadratic equation is,
⇒ x2 – 3x + 1 = 0
Hence, Quadratic equation is x2 – 3x + 1 = 0.
Question 10.Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
2, 4
Answer:Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = 2 and product of roots = 4
∴ Quadratic equation is,
⇒ x2 – 2x + 4 = 0
Hence, Quadratic equation is x2 – 2x + 4 = 0.
Question 11.Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
0, 4
Answer:Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = 0 and product of roots = 4
∴ Quadratic equation is,
⇒ x2 – 0x + 4 = 0
⇒ x2 + 4 = 0
Hence, Quadratic equation is x2 + 4 = 0.
Question 12.Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
![](data:image/png;base64,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)
Answer:Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots =
and product of roots = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAEJQTFRFAAAAAAAAAAA6AGaQAGa2OgAAOmZmOma2OpDbZjoAZrbbZrb/kDoAkDo6kNv/tmYAtmY625A625Bm2////9uQ///b7M5aaQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAU0lEQVQYV2NgwACiAiwgMSE2TjDNwCCInxblZxYGqWIEAg5M43CIgFQDAdHq4QpB9jDxAe3jhroORgPF2SFCIqwQMVEeDgZRXj4wX1QAqI0LwzoABl4CTEIhdkMAAAAASUVORK5CYII=)
∴ Quadratic equation is,
⇒ x2 –
x +
= 0
Hence, Quadratic equation is x2 –
x +
= 0.
Question 13.Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
![](data:image/png;base64,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)
Answer:Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots =
and product of roots = 1
∴ Quadratic equation is,
⇒ x2 –
x + 1 = 0
⇒ ![](data:image/png;base64,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)
Hence, Quadratic equation is ![](data:image/png;base64,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)
Question 14.Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
![](data:image/png;base64,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)
Answer:formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots =
and product of roots = – 4
∴ Quadratic equation is,
⇒ x2 –
x – 4 = 0
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHcAAAAeCAMAAAAhHHEqAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmY6OmaQOma2OpDbZgAAZgA6ZjoAZjpmZmYAZmZmZpDbZrb/kDoAkDo6kGY6kLbbkLb/kNv/tmYAttv/tv//25A625Bm27Zm29u22////7Zm/7aQ/9uQ/9u2//+2///bwbLKNQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABcklEQVRIS+2VbVPDIAzHk6pbdbp1PuJDUVaLHXz/72do67muUnJw9YUnLyh3DfmRfwIB+B8/KVDlWdnki/rX1dGLWq3nwb5fI669AanT+3mCNdtnqLLS53x3MaPKzbmP+/FixdnrPBEDqI3H8y5fWolTESu/VAc+7RPianT8yp/eUKQNFXzIBsCKRb0XJ29DSxWPNcUdh6udkcYHAImknkK31ITV9I0ZctO6DA3pQjXFkuwk2VtBOFMgjTiuXtUcrpPZgdzs1t5yCp2//2+2JRxxScd+HOjQcbsZmvxylNfRrt5Hyxn9tIKuAXFVSKwBF3g3YCJ0/RUaj9vlF8zN7XFhM+UdmHF0hr6u3CthH8v+ADG07z2cugLV3hs3Welyg753in8WVrZcKXdJFphRvJgMphxzLvCeet7VPO2ULxHHcro5czxE2QSac5RP7iZ/c+Z6iLNLfU3jqJDQnCOJ7baE5pyCTWnOCdyU5pyA/TNbPwHvvxpMmLwGmgAAAABJRU5ErkJggg==)
Hence, Quadratic equation is
.
Question 15.Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
![](data:image/png;base64,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)
Answer:Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots =
and product of roots = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bjLFCZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbklEQVQ4T72Qxw6AMAxDXfZeLS2z//+ZFMQNSxwYPkV6ieMEeEG2D6mLSQsOAP09sCqYWCwtnLIXzn5kofYUhzzpjM76keWfwyYRXkkW2qbDcFxENMccLG1F+5XwOQDGqL6OrLnEzABMxOPevmwDtI0De6zhlboAAAAASUVORK5CYII=)
∴ Quadratic equation is,
⇒ x2 –
x –
= 0
⇒
= 0
Hence, Quadratic equation is
= 0.
Question 16.Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADAAAAAfCAYAAACh+E5kAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAKoSURBVFjD7ZjNattAEMdX9jX0ntDRraTHRGO/QXC3NDn0YBxZ+JKCDsIIGgd0CHTlmyFtz6WXfhxzbymkvYfmERI/QJo8glp315ZVZSvrO2CBBYOxEKP5z878dlZkOBySKlulgy8sgF/1LLaMAiZZbJkFeGlsWQV4Ve6BSWWb2G/MSgv4rzH5VQtRp1Zwde/44qbch4Cg/h2rvYkA34LGBu2ibTmbOfwqwpcG5CLsa8c6xigRpdS/4/Q2uPfLMHGU6cu1cc9xNrL4PbZ2EIhyIxHsDyFwizrbK0WAXP9dhJOmYXfm/5mtP/IFTdBwd9P6HY3MNQrKGVD9lTkarQW+gJxPE/KQ/pjfLyogcWNyDdwVmcOe+yytX3dAt5q0/0K+z1hvnSfkSqwCtdl2WQK8hFLgZYznPcbWy6BNF8mnUgUkEWRf01z5ZYUEaOQzIXglJ6RU/p/ywAfm88dTgvgUWoS/LCYgoRFlDNQ+LNzEcfXPl/mDT43fs+fgVuCvqACm4x7P/mVUOcaOyHnqX5DENpodIOQXmZHju0yOrNlvAHylA3crcSPzA/dCU2Y97/wjiMJFcJ7jOG8j8wAUjug3cSSTg/PmvwtGhUzzj4HkY14Bonf6LRgAtQ5zzUIhEfUs/A8bD+CIAD3LWkLT4PWmHrUnEHJaS93EcsCL6r8dMXA5ZvuB2FXl5f83pE2UxcFjF7T9t/KRVBAO1SfvwwlJEnBnFaKyH+ySHJutjiV237rJgzcQXgN2T+RnOWJ/Cj/Q6h/JiJ0HH3cslVGaZjkDEVECBHVaDfVdePhS1cYXarx8GimWzzWzYQ9u5I1O0Cv+WKpeyzTKfHgv5RDCM20gsjLGjLQv9Mo6/4rgmU23oxr0Xk9kZQQvys2g9CCqvFZf5lYCVgLy2V+lKB5nHz/UVQAAAABJRU5ErkJggg==)
Answer:Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots =
and product of roots = 2
∴ Quadratic equation is,
⇒ x2 –
x + 2 = 0
Hence, Quadratic equation is x2 –
x + 2 = 0.
Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
x2 – 2x – 8
Answer:
Let f(x) = x2 – 2x – 8
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ x2 – 2x – 8 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 2 and product is 8.
∴ x2 – (4 – 2)x – 8 = 0
⇒ x2 – 4x + 2x – 8 = 0
⇒ x(x – 4) + 2(x – 4) = 0
⇒ (x + 2)(x – 4) = 0
∴ x = – 2 and x = 4.
⇒ Our zeros are α = – 2 and β = 4.
⇒ sum of zeros = α + β = – 2 + 4 = 2.
⇒ Product of zeros = αβ = ( – 2) × 4 = – 8.
⇒ Comparing f(x) = x2 – 2x – 8 with standard equation ax2 + bx + c = 0.
We get, a = 1, b = – 2 and c = – 8
We can verify,
⇒ Sum of zeros =
i.e. α + β =
∴ α + β = 2
⇒ Product of zeros =
αβ =
αβ = – 8.
Hence, relationship between zeros and coefficient is verified.
Question 2.
Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
4x2 – 4x + 1
Answer:
Let f(x) = 4x2 – 4x + 1
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ 4x2 – 4x + 1 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 4 and product is 4.
∴ 4x2 – (2 + 2)x + 1 = 0
⇒ 4x2 – 2x – 2x + 1 = 0
⇒ 2x(2x – 1) – 1(2x – 1) = 0
⇒ (2x – 1)(2x – 1) = 0
∴ 2x – 1 = 0
∴ x = .
Again, 2x – 1 = 0
∴ x =
⇒ Our zeros are α = and β =
.
⇒ sum of zeros = α + β = +
= 1.
⇒ Product of zeros = αβ = .
Now, Comparing f(x) = 4x2 – 4x + 1 with standard equation ax2 + bx + c = 0.
We get, a = 4, b = – 4 and c = 1
We can verify,
⇒ Sum of zeros =
i.e. α + β =
∴ α + β = 1
⇒ Product of zeros =
αβ =
Hence, relationship between zeros and coefficient is verified.
Question 3.
Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
6x2 – 3 – 7x
Answer:
Let f(x) = 6x2 – 3 – 7x
Arranging equation in proper form.
Now, f(x) = 6x2 – 7x – 3
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ 6x2 – 7x – 3 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 7 and product is – 18.
∴ 6x2 – (9 – 2)x – 3 = 0
⇒6x2 – 9x + 2x – 3 = 0
⇒ 3x(2x – 3) + 1(2x – 3) = 0
⇒ (3x + 1)(2x – 3) = 0
∴ 3x + 1 = 0
∴ x = .
Again, 2x – 3 = 0
∴ x =
⇒ Our zeros are α = and β =
.
⇒ sum of zeros = α + β = +
⇒ sum of zeros = α + β =
⇒ Product of zeros = αβ = .
Now, Comparing f(x) = 6x2 – 7x – 3 with standard equation ax2 + bx + c.
We get, a = 6, b = – 7 and c = – 3.
We can verify,
⇒ Sum of zeros =
i.e. α + β =
∴ α + β =
⇒ Product of zeros =
αβ =
Hence, relationship between zeros and coefficient is verified.
Question 4.
Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
4x2 + 8x
Answer:
Let f(x) = 4x2 + 8x
Arranging equation in proper form.
Now, f(x) = 4x2 + 8x + 0
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ f(x) = 4x2 + 8x + 0 = 0
∴4x2 + 8x = 0
⇒4x(x + 2) = 0
Now, 4x = 0
∴ x = 0
When, (x + 2) = 0
Then, x = – 2
⇒ Our zeros are α = 0 and β = – 2.
⇒ sum of zeros = α + β = 0 + ( – 2)
⇒ sum of zeros = α + β = – 2.
⇒ Product of zeros = αβ = 0 × ( – 2) = 0.
Now, Comparing f(x) = 4x2 + 8x + 0 with standard equation ax2 + bx + c.
We get, a = 4, b = 8 and c = 0.
We can verify,
⇒ Sum of zeros =
i.e. α + β =
∴ α + β = – 2
⇒ Product of zeros =
αβ =
Hence, relationship between zeros and coefficient is verified.
Question 5.
Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
x2 – 15
Answer:
Let f(x) = x2 – 15
Arranging equation in proper form.
Now, f(x) = x2 + 0x – 15
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ x2 – 15 = 0
∴ x2 – = 0
So, (x + )(x –
) = 0
When, (x + ) = 0
Then, x = – .
When. (x – ) = 0
Then, x =
⇒ Our zeros are α = – and β =
.
⇒ sum of zeros = α + β = – +
⇒ sum of zeros = α + β = 0
⇒ Product of zeros = αβ = – = – 15
Now, Comparing f(x) = x2 + 0x – 15 with standard equation ax2 + bx + c.
We get, a = 1, b = 0 and c = – 15.
We can verify,
⇒ Sum of zeros =
i.e. α + β =
∴ α + β = 0
⇒ Product of zeros =
αβ =
Hence, relationship between zeros and coefficient is verified.
Question 6.
Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
3x2 – 5x + 2
Answer:
Let f(x) = 3x2 – 5x + 2.
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ 3x2 – 5x + 2 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 5 and product is 6.
∴ 3x2 – (3 + 2)x + 2 = 0
⇒3x2 – 3x – 2x + 2 = 0
⇒ 3x(x – 1) – 2(x – 1) = 0
⇒ (3x – 2)(x – 1) = 0
When, 3x – 2 = 0
Then, x = .
Again when, x – 1 = 0
∴ then, x = 1
⇒ Our zeros are α = and β = 1.
⇒ sum of zeros = α + β = + 1
⇒ sum of zeros = α + β =
⇒ Product of zeros = αβ = .
Now, Comparing f(x) = 3x2 – 5x + 2 with standard equation ax2 + bx + c.
We get, a = 3, b = – 5 and c = 2.
We can verify,
⇒ Sum of zeros =
i.e. α + β =
∴ α + β =
⇒ Product of zeros =
αβ =
Hence, relationship between zeros and coefficient is verified.
Question 7.
Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
2x2 – 2√2 x + 1
Answer:
Let f(x) = 2x2 – 2x + 1
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ 2x2 – 2x + 1 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is – 2 and product is 2.
∴ 2x2 – ()x + 1 = 0
⇒2x2 – x –
x + 1 = 0
⇒ x(
x – 1) – 1(
x – 1) = 0
⇒ (x – 1)(
x – 1) = 0
⇒ (x – 1)2 = 0
∴ x = ,
⇒ Our zeros are α = and β =
.
⇒ sum of zeros = α + β = +
⇒ sum of zeros = α + β =
⇒ Product of zeros = αβ = .
Now, Comparing f(x) = 2x2 – 2x + 1 with standard equation ax2 + bx + c.
We get, a = 2, b = – 2 and c = 1.
We can verify,
⇒ Sum of zeros =
i.e. α + β =
∴ α + β =
⇒ Product of zeros =
αβ =
Hence, relationship between zeros and coefficient is verified.
Question 8.
Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.
x2 + 2x – 143
Answer:
Let f(x) = x2 + 2x – 143
To find out zeros of the given polynomial.
We put f(x) = 0
⇒ x2 + 2x – 143 = 0
To find out roots of this polynomial we use splitting of middle term method.
According to this method we need to find two numbers whose sum is 2 and product is – 143.
∴ x2 + (13 – 11)x – 143 = 0
⇒x2 + 13x – 11x – 143 = 0
⇒ x (x + 13) – 11(x + 13) = 0
⇒ (x – 11) (x + 13) = 0
∴ When, (x – 11) = 0
∴ Then, x = 11.
Again, When, (x + 13) = 0
∴ Then, x = – 13
⇒ Our zeros are α = 11 and β = – 13.
⇒ sum of zeros = α + β = 11 + ( – 13)
⇒ sum of zeros = α + β = – 2
⇒ Product of zeros = αβ = 11 × ( – 13) = – 143
Now, Comparing f(x) = x2 + 2x – 143 with standard equation ax2 + bx + c.
We get, a = 1, b = 2 and c = – 143.
We can verify,
⇒ Sum of zeros =
i.e. α + β =
∴ α + β = – 2
⇒ Product of zeros =
αβ = – 143
Hence, relationship between zeros and coefficient is verified.
Question 9.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
3, 1
Answer:
Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = 3 and product of roots = 1
∴ Quadratic equation is,
⇒ x2 – 3x + 1 = 0
Hence, Quadratic equation is x2 – 3x + 1 = 0.
Question 10.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
2, 4
Answer:
Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = 2 and product of roots = 4
∴ Quadratic equation is,
⇒ x2 – 2x + 4 = 0
Hence, Quadratic equation is x2 – 2x + 4 = 0.
Question 11.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
0, 4
Answer:
Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = 0 and product of roots = 4
∴ Quadratic equation is,
⇒ x2 – 0x + 4 = 0
⇒ x2 + 4 = 0
Hence, Quadratic equation is x2 + 4 = 0.
Question 12.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
Answer:
Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = and product of roots =
∴ Quadratic equation is,
⇒ x2 – x +
= 0
Hence, Quadratic equation is x2 – x +
= 0.
Question 13.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
Answer:
Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = and product of roots = 1
∴ Quadratic equation is,
⇒ x2 – x + 1 = 0
⇒
Hence, Quadratic equation is
Question 14.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
Answer:
formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = and product of roots = – 4
∴ Quadratic equation is,
⇒ x2 – x – 4 = 0
⇒
Hence, Quadratic equation is .
Question 15.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
Answer:
Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = and product of roots =
∴ Quadratic equation is,
⇒ x2 – x –
= 0
⇒ = 0
Hence, Quadratic equation is = 0.
Question 16.
Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
Answer:
Formula for quadratic equation is,
x2 – (sum of roots) x + Product of roots = 0
Given, sum of roots = and product of roots = 2
∴ Quadratic equation is,
⇒ x2 – x + 2 = 0
Hence, Quadratic equation is x2 – x + 2 = 0.
Exercise 3.4
Question 1.Find the quotient and remainder using synthetic division.
x3 + x2 – 3x + 5) ÷ ( x – 1)
Answer:Let p(x) = x3 + x2 – 3x + 5 be the dividend. Arranging p(x) according to the descending powers of x.
p(x) = x3 + x2 – 3x + 5
Divisor, q(x) = x – 1
⇒ To find out Zero of the divisor –
q(x) = 0
x – 1 = 0
x = 1
So, zero of divisor is 1.
⇒ p(x) = x3 + x2 – 3x + 5
Put zero for the first entry in the second row.
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)
∴ Quotient = x2 + 2x – 1
Hence, when p(x) is divided by (x – 1) the quotient is x2 + 2x – 1 and remainder is 4.
Question 2.Find the quotient and remainder using synthetic division.
(3x3 – 2x2 + 7x – 5) ÷ ( x + 3)
Answer:Let p(x) = 3x3 – 2x2 + 7x – 5 be the dividend and arranging p(x) according to the descending powers of x.
Divisor, q(x) = x + 3
⇒ To find out Zero of the divisor –
q(x) = 0
x + 3 = 0
x = – 3
So, zero of divisor is – 3.
And, p(x) = 3x3 – 2x2 + 7x – 5
Put zero for the first entry in the second row.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZ0AAACaCAIAAAD9zwL5AAAAAXNSR0IArs4c6QAAGbNJREFUeF7tXb1vF0e3frlBSqRQRAoFIlGEHf8BKSIKZFBEAf4DXNhpkiKyhJUiRWQ3SC+R3OAgJTQgpYkr4gIpRRrbBU7AouAvSOFgZCUIRUR6C5AoiHKfy7l3dO7s13zt/mZ3ny3QD+/MmXOec+aZj509e+Sff/75Fy8iQASIwIAQ+K8B2UJTiAARIAL/gwB5jXFABIjA0BAgrw3No7SHCBAB8hpjgAgQgaEhQF4bmkdpDxEgAh68duXKFeJFBIgAEcgfgSPu5zyOHPEonL/l1JAIEIGhIuAxXxsqBLSLCBCBgSFAXhuYQ2kOESACPL/GGCACRGBwCHC+NjiX0iAiMHoEyGujDwECQAQGhwB5bXAupUFEYPQIkNdGHwIEgAgMDgHy2uBcSoOIwOgRIK+NPgQIABEYHALktcG5lAYRgdEjQF4bfQgQACIwOATKeW1ubg5vg5rr4cOHjoZvbm7qiuvr644VcyumrQAauanXqj57e3vafPk9MzPTaqP5C7c6BTBBtOev9jg1LOE1sNj+/j7eh5fr6tWr77//viM6Gxsb9+7dk4q//fbb6upqH32Pjn3p0iWDANBYXl52RGAAxWZnZ43t8uPixYtLS0sDMC3SBB0VgGVhYSFSIKu3hEAJr01PT6Mnm/bOnDmD345Ttq2tLfQKqQs5IMTDw8OWVG9PLEy4ceOGkY8uvbOz015zmUvGyLS9vb2yspK5nlSPCBgEmvfX1tbWMFyDpHxRQ3/AlG1+ft63IstnhcDly5cxZ89KJSpDBBoQsFYc+r9S84cffpA/4ndNYX0L03UUBhs6ls+5GKgZtqBj56xke7rB++5+b0+NHCQjnnVfQmDkoBV1KEXg/7GeobAiSXnxmlTHRpumxTwdILRlLrM5aLTFUnoYBB2GP8wfIafrkCjtFDKBJbWFBVUHtZxS4OLRD7y7uLgo7OZ+YbsdG3PYdHOvklVJPATE1V/9I8HETkKA0yMb7Ut16RR8dJCnv5r31xyfGJSaF1N34njJ4YbRkhrwB6lxZ23icUgFQhAozgkxCunn2ViFYTHiuA7VezGyNVNc2XUwC41sQlbQI1x/adyE0SKRHEx1hITejpAd5MFYNzxDyn2jCVJIzZHXxN8121W9QLB0klK6z9ILcwKUHPnTksataNMpArBllQ4QcNpfE57i96hC5sOsQwSIQOcINO+vda4SGyQCRIAIRCFAXouCj5WJABHIEAHyWoZOoUpEgAhEIUBei4KPlYkAEcgQAfJahk6hSkSACEQhQF6Lgo+ViQARyBAB8lqGTqFKRIAIRCFAXouCj5WJABHIEAHyWoZOoUpEgAhEIUBei4KPlYkAEcgQAfJahk6hSkSACEQhQF6Lgo+ViQARyBABD1778MMPMzSAKhEBIkAELAQ88nlMTU198sknX331VSoQjx07dvz48VTS0so5evTou+++m1ZmQmmnTp1KKC2tqBMnTrz++utpZaaShnhD1KWSllZOzt3hrbfe+uCDD9La26o0P147ODhIqM2zZ8+ePn2aUGBCUS9evHjy5ElCgWlFPXr0KK3AhNKAG9BLKDChqL/++gtRl1BgQlE5dweQ2jfffJPQ2LZFTZLX2raN8okAERgnAh77awAo5ynMOP1Hq4kAESgi4Mdr2a4v6FoiQASIgEHAj9cIHBEgAkQgfwTIa/n7iBoSASLgh4AHr2X7gNzPYpYmAkRg6Ah48Fq2Z82G7iPaRwSIgB8CHrzmJ5iliQARIAITQsCP17I909g2euvr6/h86ubmZkBDqIW6kBBQV1dJJcdS4+HDh1Bvbm4uUj2pHgNUTN0kyicXAmBnZmbCxMbUDWtxULXcv7380Ucf7e7u1peX74TLdfHiRXfhCUtqHZKoIQLv3btXr6QOC+t74KiLu5ATYyYkOH5z/tKlS1Xg41P2pd8qdxdeY0I8UDAQmsSg5FgXEBU9YjyY5HPuNV4wSsIdOmws/0INFHC0iMU0Ah4x5MJrcJLxBLrWRKgNCuhoQHAgwmK8juqNEhCRukzRdhchNUrCIhcwUaZqUJGejKu000L/+M7sYmMjUDCh7c5s2ESPNDpOYEgkGkLxjSMZWjFlZPDTw6f8JSZ0R1vXA7VGXrP6RpJJSrxjrGGzyDj1/Os4B7H0LE6LItHwmk8FzNegv9XEpIBqe8om8i13WAxiOb2KcWqm8C6TtWJsF73MKVsYA/jtr9WvwO/evat3E2ZnZ1H+8ePH9bVavYvNo5s3b3766aemla2tLfw220nyQ/5Yej148ACxJba4X6urq0tLS7o8JEAOpLkLMSX39vbwe2FhIaCuexV0RXhw4kCJmWJy8gtiFxcXrZ0KtHL//n14xzQ3PT2N//7+++/yF/gOFHb27FnRCv/iN/5SExU7Ozs66lwMkR1Yy8uIojt37rhUZxmNgAevhZ1fM8HRMe6yyy6rCStWDLU1khp0Pjw8vHDhgqPyiHg0igs9Z2VlxaoFOZDmKEoXA4ZmgRlQ3bHKuXPnMAzowpMCCsa2ETawDnwE14C2XDDRzjLUBvZpJDUIRyunT592aQVlEIeIGeRokVmzvs6cObO/v+8oh8UMAh681q/za+AymcEiYpaXly2XmwlazUxNqhQzAhnyEgrTXIDol0ZBYaVPGMPyCxXZsEaHmOAudqFJARU2ANTYDjdhkMMky5HUiqKE2jATr5+poaI1PIgoLGUkYIoPxwEyYgbjCm4VJ6rF2WWMi0dS14PXwhCZeHZG7KdgKWopb61DvUwz5CUUVtpPNjY2tre3S+Pbq62qwi46JGmo70BZIGCqJcyCH7gFpiuOeabKe++9p6vL8hM7p2ZB6oUwxgyzVVScy0MURmJMVG/duuUlloVLEfDjtfrpBgYcPeDLyHPy5MnJQl8c+c3y09prK9UzLDNt1TIqTJrVwdrD0zpsNUGgkpuM4UfvQJvnBjdu3ACeWO7paREGJPxXD8lmTw2UpPfaSn0RPCUsXXLqjb/2XD8wyX68Vm88BhxEgzmAura2hvEn2MfBQKM3msk8AhQLB31KyNpTE2qrH7SxB9yoDCToU7vYnMY2vGU75IR1V3QwzP4adYgsgIcGWuEJAgVjO57my1MdEwbXrl3TD4usBwVmr61mPu7yjAjV9UCCjoPu8+WXX2o/4oFG8MneyHjod3X3x6h4vvP999/Xl8/hXK48yDdX5GEox3MeMv6bq3jerZtzHtbjBX3kzZxfEyWL57PwR8dzv6UxkAqots95iPKl7jDuizy8BvmO5zwsfxXPu/GchztB6ZIe59dceC1MicxruRw3bTQhUojjudxGNaoKdHYut1HDDs7lNuoQX8DxXG59QzyXG+wI8lozdI4zkRpBkZM1kRw5n6q3M4nweKC6maw1uzxFCccpW01TnKwF+8GD1z7//PPGdWiwHplXlB26sGWa9NXI5TDwSSXHglrIyOUlLRcfxQAVU9dFt+7LlK73HdWIqevYxICLeXyP6sqVK3ic53uKut+7j9SeCBCBHiLg9zw028999hB5qkwEiEBbCPjx2mjzr7UFP+USASLQAgJ+vNaCAhRJBIgAEUiMAHmtDlD9GmaqjLKJHdiCOP0mI1450i3g5Kp5ybH0ZcYW1MlaZDGLsuQfNpd15FvfHU9Ede9C8lol5uaUuTlmMYZAhNXgNfOkDIcV9Hl3vGlrbuH5LN6UbO8d2O47g3uLhp6KVZCKSj/7BmI6Bbw+uoG6Y4god1RTlnR/1vvvV5d7+b6XxNEH67UB4N6YDbzvVlv61xwoiz+tNgCsEBL1p3/0KbY8E68OwAtFEzzma2H511Jy8KRlYbDF+3qT1qLT9i9fvmy9gGWat16i7FStfjaWYeLVfgLZrLUHryH/2t9//90sMr8SVrYyvf1Ro+z58+exiNDrrAFkwpIUhsXL+lyWWWch4ZJkvNCXbMDhMGMfUx46IpAkihF4Vrrmotg2MmgmUb7XQjx4DXb21AdWtjI9azXO011dMnMgI418psDcGkDGGElhWLysjGAmqw+SshTzSUgqMUBkpdXsRU9wRMDYIl//k8s3tQb2HzHbbTuBey9g71hJP17rWLkum9Nd3QQierv5OyZruJCoq0utJt4WJmuwuvRrAwAHRH/79u2JK9mqAjoGvOan4EGQWnG2a2nbcUamVrHKR/goeC1sHWo5CR044AMu+XhaNPFdhdV/dqePC3NfBAI8KKv4IqnlmXg1wMAeVHF/GoKX3vFyqHv5IZWUV7LH8DC0+PVV80o8zC8mdBuSlwNsQVRYz0PlCXLVQ1LcmvgHdgPM7F0Vj3weI+Q1k/YvPtFgXyLDmoJZeT6sHca+GJVcz+JE1RBZ6eNjMyLmkHg1ORoZCvTI54FHY7/88gvYrQezUKpIBIjAiBHw2F/j+bURxwlNJwJ9QsCD13B+7cmTJ30yjroSASIwSgQ8eA34vHjxYpQo0WgiQAT6hIAfr/XJMupKBIjAWBEgr43V87SbCAwXAfLacH1Ly4jAWBEgr43V87SbCAwXAfLacH1Ly4jAWBHw4LUTJ06Y8/djhYt2EwEi0AMEPN43+Pbbb69fv+5rE07z4uBbZ7XefvvtztoKsOuNN97A8OCrYVgt31ZYnggMBgEPXvvPq8vX8i5r4TOAAV84ff78eUCtsLa6rBXGhl3WOnr06DvvvOMbUagVkNsn/1q+OLB8DQIevEYc+4UADlEHvB+Sfy0kbQ7Ib/ry5cvMa4UxL2ISiYsDIjOsVl9eDyevBYQEqxCB9AiEMS/0ePToUYA2YbWQqSygre6rkNe6x5wtEgEi0C4CHs9D21WE0okAESACiRAgryUCkmK6RQBf+eu2QbbWJwS4Du2Tt6irQWBqaurg4ICAEIFSBDhfY2D0FYGAU0d9NZV6eyJAXvMEjMWzQYC8lo0rslOEvJadS6iQCwI4P+xSjGXGiQB5bZx+773VAa+j9d5mGuCMAHnNGSoWzAwBrkMzc0hG6pDXMnIGVfFCgLzmBdeoCpPXPNy9t7d35MiR5eVljzr/VzSmbkBzg6+Ctynzt3EiTp+bm0OUPnz4MH98WtQw8lvNWrOAj6LjO9hGAj6UHalMfXWtqvUZc8d2YeDVq1frC2uLrFbkW9/m09+OjbZRTCtpYPFyH3DQeJqvnUNbS3hLbsWLiru7u5HgFPsVfBQpU1d3CRgpDzyL+Fufjrd003etoMKtsAhPaPtkRf0rpnlEs45aQFmKJoqV9hnp56ZLoExLfUA6mxZe1VaVqpBQc8tgKF3aBFkREIRvngEHrUopG0BpzjKW4u+mm1lWA2ddBYA0DgYBQQhe+/HHHwMq6io6/BxFuYSBiHIsaeip2Ef0X2QgMUrq/1r4S5lMRlBHVJMXi+I1S5vSMafGwVYnRxxozyU3VQv0VRV1XcZesINFzVZ45TNl02jUIF/Fa5Z3agiiiEkSz4LXkDMnUlSrvOYSMJqn6ufLVuRYfimCPPIpW8r9tdXV1aWlJfc18507d6anp03506dP43ee+wLQCoE1Pz/vbp2URPzdv3/f1IK9CEH9F1+BbZS/fPmyta70amV9fR3lFxYWirWwwbS9vf3xxx97CRxA4eCAcbFdhOvkmufPn7c6DjAH8i7SBlkmAa/J5iguYL2yshIJ0+PHjyMlNFZHBPhSMLQCQ2kWLm0F4XXz5k0dYTLM6gtlwlJfNdoVVmBzczPYcbJFDXNk4aMvkB1uYVaFW7Ozs2G61dQKSJlbKm1xcVGid2ZmJqGSjgHj3iKeViECa5Dc39/X0qRknrMEd6uDSybgNSAo0+kLFy4g0EUVACrhgguhg55j/hv2PNHdQsOz0mLRtdATkyZDwS6qFlOt6lpoBY1CQ8iUFa4xFr+LmmcVbcXJGhxk9IfjTM8vgrm1tQW/nzt3ziBgjAUUuLWzs4NbMqFLe7322mvu5zyqQkIvY8FrVdTmEiGWdaW5eQ2q+IHhxB0QAIjJF8B0ryIlO5gl+KrUUfnIHQpdXfYvi0+UqjZQrU0BmdqkfSBVtA5EU7NzX6Wq4x6w1VzpblpWjw4a9zQd99dqNtGqtjIjA+/fr65IIS7RazXhGAmOxVz212SLQPeLYlyVBhVq5fDwPaGP3EUlmK8ZAvbNH29tCjx48ACiGtd6MXwvyw3MMnyFhK16bt++Xbp2aNVGL9MwF4vZWTNtWYsgrUNWi24vcGIKhwVMsUXMnbFnAiLTMYPfiCvd3aytaiPn5MmTMVb0uK47BRZLFp/olx7UyOScB5zUeOCgXlWvuaTwRXHArDpREeOIsLrW0YFSIaXzNeBQPIJgwCkeASk9KRKms6mFfczI+RrM1/FQP5E37TpOxHwXH6WzWqhU9ZC08ZyHLJ4iQe5v9SjLHU9g1oRCZ+dyS2clxf5Wo6rjY3vz6eiaI3s5rA6k4zUSfdU61PpCtmb8miO7CfsJDnl88cUXMQIdozdsHYpajgFjHb41YVN84gR/6S2UmnO5aH3k5zyieC0mqnpX13Ggrrcrq8213rlAKwxek4et2V5JAibYujFvrgG0lPtrPV6NO6gu57NiHu3J+RI8f3RojUV6j0B8wARDgF05zOzaOF4TrFLXFYMHhBFWlJVL2MteMXVHCHWjyT/99FPm8zWYMBGnyxaB115wI9q9K8DvtnQ9kLC9JAj8/PPP169fxyuiSaRRyMAQ4Dp0YA4dkTnu53JHBApNfYUAeY2BQASIwNAQIK8NzaMjsefYsWMjsZRmBiBAXgsAjVUmj8Dx48efPXs2eT2oQWoEcOQgPgfBBHhNv/0bb0BqVP9XHhSrOtKBd/vbfnW/JaMixUr2Di1EvxBuUh5EtuJe/enTp+6FOygpaEgGBHPh/faa1931rao0DZGa6ywGydOW1OsmOV0i9Q+r3jWvwc36nAToo/v+UI+UhFfpaW9oi1vjTGuF7lE0XB+pB6q5uTKsSwTUknwhpblb7t69a14vwQ+8kGsRn/XSSxvvDuu3FLp0k+R0CcAzvkrXvIbDijdu3DB649X3mlem480LkCBHdUpjFKrilvUKUUATvauCgRdJciT5h56JACWT7gnnjUF8naVgyuq7yJKqq3QsRLSb87H4YeUZ7T4S1tbWrPFJZlVymexJMvfUk0381umeNDtXrcCkutgoq0stUNuuJePsur5VtSaQZZOZjVqB1zWvWY70ze/YfRywRcQi3FQcfjAT0dsI0ns7S/iF7yJzfy0gOA8PD3Ut0M13330nY7nk2tOEhbFKbuH9P9wCJ5r/4lC0kaNfhcYfq3ZpIB8xIxIwOdC5Gs+ePWtmtfr9YrAVhJtblnDkPkDuPxFozXOjeM3Krahpu37cTptit9S71raC0W2cW2NVHaC4vyN/McGNHwho99WEb66qgJ5pqiTZX2tEIEZDqy5IpJhN3j1hL0LXWsMG6IYhShMH/mvSVYIasEd069YtI3ZjY0N+nzlzBv+alRb+qyeneszDlwCq8l+CoYwE0KKpde3aNbRb+tYXbkFbcws8Cy4z6qFWafZ5FIjiNQBR9YJF/TZBaYrdACfVVAGCpboZZK0N3bSt90ValfskktCLMJCWLq/6YmCjnvUIFKvLwxO5vMZIgAkSwVpedw3dek3CXpmnnDp1KuyVTyw8jc6Y+5itA2FJnd7ZymJfj56Zu+j5jbWKrJFg4go8CNNKS+IWBBrlEY0o5rLXEcVrjUHTWAADQpebMlofML2OqkZVR1vAxD1mFgABQVbz8n+qfIouaE/k08iS+lwuvVNcr7CMECC1qvkFqmMygq5e2mmBORhH9/DSaWaVDua5AQpgLWkVs94k9c26ivmBTmaVJE2p1rCYSsvl0UoUrwWvQ43e7S1buA51oYb6VZiZVks3NknDZcDH1oZegMjg32WCVnBo/BZbB+tQ9PxGUqt3luxwNSZcaPQ4KAzTCDPNlNmf5KkOvrBlBt6s4etGyZioViVVxqdIkAq4UUKxQBSvBaxDgan+YgWmAPCWCwH72ta4DvUVOMjyvqswa8KLfmLmbpgIIL7bcGUN8vFbbDEIuIQEAl6+W1Ts+YBOz3yx5VQDIMYSLNYi99fgHaxDMfUz7aL3QT0zSYR8r8U1EIBWenhzX4ca9GC4Xv9qCfK1QI2S64nXKr+29PewJKUtKVMq1gpWK1W0vts4fnapdgdtFT/yorfeuv+OPXrUwcFBB4Y3NmFFNYLEPMIrPTAkArvpC8XEueJH4y9r5SiaW5+GsT7JZP1Xn3zSCc11wFiJzq1Y0jpYJa3tXdPp6r+3zTxFLoMuy+SIwNTU1O7ubtWWc44aU6euEIhah3alJNshAiUIJNlfI7KDRIC8Nki3jsIoPA+N318bBVLjM5K8Nj6f02IiMHQEyGtD9zDtIwLjQ4C8Nj6fD8ViviI6FE+mt4O8lh5TSuwGAaT04P5aN1D3rhXyWu9cRoWJABFoQIC8xhAhAkRgaAiQ14bm0fHYw08cjMfXvpaS13wRY/lcEHjzzTe5v5aLMzLTg7yWmUOoDhEgAtEIkNeiIaQAIkAEMkOAvJaZQ6iOMwL4NPLz58+di7PgiBBgPo8ROXtgpiLZMvJ2/fnnn1V24cX4qpy6OPuGY71VFfFEApt3VXdrMoiAalG3qiJarPqMFvTsMtXwwCKhaA55bfAuHqyBSIL466+/fvbZZ8ZCpF9++fJlqcF4wlCVXBd/r3r+AGk1KZ3/+OOPbpp769U1QfoGX8/Pz/cokshrPXIWVR01AhNkbUxCv/766x6hT17rkbOoKhEgAk4I8LmBE0wsRASIQI8QIK/1yFlUlQgQAScE/hu1r33foy4V/AAAAABJRU5ErkJggg==)
∴ Quotient = 3x2 – 11x + 40
Hence, when p(x) is divided by (x – 1) the quotient is 3x2 – 11x + 40 and remainder is – 125.
Question 3.Find the quotient and remainder using synthetic division.
(3x3 + 4x2 – 10x + 6) ÷ ( 3x – 2)
Answer:Let p(x) = 3x3 + 4x2 – 10x + 6 be the dividend and arranging p(x) according to the descending powers of x.
Divisor, q(x) = 3x – 2
⇒ To find out Zero of the divisor –
q(x) = 0
3x – 2= 0
x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAY0lEQVQYV42OyQ6AIAxEp4j7vqCC+P+/aYHEA8TEuby0M20GiHW2RCVgmxm7WLxr8sCt8tjZdlOAZugBtibWkLz7Wrg063f+DaqCRAfcY9LvmnqXWinzBA7JvZoFhgkl/V2kB6PpA2t1TzLqAAAAAElFTkSuQmCC)
So, zero of divisor is
.
And, p(x) = 3x3 + 4x2 – 10x + 6
Put zero for the first entry in the second row.
![](data:image/png;base64,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)
∵ p(x) = (Quotient)×q(x) + remainder.
So, 3x3 + 4x2 – 10x + 6 = (x –
)(3x2 + 6x – 6) + 2
= (3x – 2)
(3x2 + 6x – 6) + 2
Thus, the Quotient =
(3x2 + 6x – 6)= x2 + 2x – 2 and remainder is 2.
Hence, when p(x) is divided by (3x – 2) the quotient is x2 + 2x – 2 and remainder is 2.
Question 4.Find the quotient and remainder using synthetic division.
(3x3 – 4x2 – 5) ÷ (3x + 1)
Answer:Let p(x) = 3x3 – 4x2 – 5 be the dividend. Arranging p(x) according to the descending powers of x and insert zero for missing term.
p(x) = 3x3 – 4x2 + 0x – 5
Divisor, q(x) = 3x + 1
⇒ To find out Zero of the divisor –
q(x) = 0
3x + 1 = 0
x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bjLFCZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbklEQVQ4T72Qxw6AMAxDXfZeLS2z//+ZFMQNSxwYPkV6ieMEeEG2D6mLSQsOAP09sCqYWCwtnLIXzn5kofYUhzzpjM76keWfwyYRXkkW2qbDcFxENMccLG1F+5XwOQDGqL6OrLnEzABMxOPevmwDtI0De6zhlboAAAAASUVORK5CYII=)
zero of divisor is
.
And, p(x) = 3x3 – 4x2 + 0x – 5
Put zero for the first entry in the 2nd row.
![](data:image/png;base64,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)
∵ p(x) = (Quotient)×q(x) + remainder.
So, 3x3 – 4x2 – 5 = (x +
)(3x2 – 5x +
) + (
)
= (3x + 1)
(3x2 – 5x +
) ![](data:image/png;base64,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)
Thus, the Quotient =
(3x2 – 5x +
)= (x2 –
x +
) and remainder is
.
Hence, when p(x) is divided by (3x + 1) the quotient is (x2 –
x +
) and remainder is
.
Question 5.Find the quotient and remainder using synthetic division.
(8x4 – 2x2 + 6x + 5) ÷ (4x + 1)
Answer:Let p(x) = 8x4 – 2x2 + 6x – 5 be the dividend. Arranging p(x) according to the descending powers of x and insert zero for missing term.
p(x) = 8x4 + 0x3 – 2x2 + 6x – 5
Divisor, q(x) = 4x + 1
⇒ To find out Zero of the divisor –
q(x) = 0
4x + 1 = 0
x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOjoAOmZmOma2OpDbZjoAZjpmZrbbZrb/kDoAkDo6kGaQkNv/tmYAtmY6tv//25A625Bm29uQ2////7Zm/9uQ//+2///bL3fpFQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbElEQVQ4T2NgoAKQE2XDaooEJw92CQYGcdpLyImwSmFzljgjEHBTwdsUGSECcgUYMAkDDYKyKTKSvppl2PmxWignyIFdQpxPAKuENJccVglZXjE5AT4hzDgERyAjC9bIxW4U0KGS7MxiZAQQABeaBAesSn63AAAAAElFTkSuQmCC)
zero of divisor is
.
And, p(x) = 8x4 + 0x3 – 2x2 + 6x – 5
Put zero for the first entry in the 2nd row.
![](data:image/png;base64,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)
∵ p(x) = (Quotient)×q(x) + remainder.
So, 8x4 – 2x2 + 6x – 5 = (x +
)( 8x3 – 2x2 –
x +
) + (
)
= (4x + 1)
(8x3 – 2x2 –
x +
) ![](data:image/png;base64,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)
Thus, the Quotient =
(8x3 – 2x2 –
x +
)= (2x3 –
x2 –
x +
) and remainder is
.
Hence, when p(x) is divided by (4x + 1) the quotient is (2x3 –
x2 –
x +
) and remainder is
.
Question 6.Find the quotient and remainder using synthetic division.
(2x4 – 7x3 – 13x2 + 63x – 48) ÷ (2x – 1)
Answer:Let p(x) = 2x4 – 7x3 – 13x2 + 63x – 48 be the dividend. Arranging p(x) according to the according descending powers of x.
p(x) = 2x4 – 7x3 – 13x2 + 63x – 48
Divisor, q(x) = 2x – 1
⇒ To find out Zero of the divisor –
q(x) = 0
2x – 1 = 0
x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAWUlEQVQYV2NgwACSAiwgMSE2bjDNwCCIn5bkZxYFqWIEAg5M43CIgFQDAdHq4QpFuBkZ2RkYJLh4GYSZ+MDC4qwQWhBivTBQGsSDUGJASoyHQYITZB0Pun0AG8sCbODE2DAAAAAASUVORK5CYII=)
zero of divisor is
.
And, p(x) = 2x4 – 7x3 – 13x2 + 63x – 48
Put zero for the first entry in the 2nd row.
![](data:image/png;base64,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)
∴ Quotient = 2x3 – 6x2 – 16x + 55
∵ p(x) = (Quotient)×q(x) + remainder.
So, 2x4 – 7x3 – 13x2 + 63x – 48
= (x –
)( 8x3 – 2x2 –
x +
) + (
)
= (2x – 1)
(2x3 – 6x2 – 16x + 55) ![](data:image/png;base64,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)
Thus, the Quotient =
(2x3 – 6x2 – 16x + 55) = (x3 – 3x2 – 8x +
) and remainder is
.
Hence, when p(x) is divided by (2x – 1) the quotient is (x3 – 3x2 – 8x +
) and remainder is
.
Question 7.If the quotient on dividing x4 + 10x3 + 35x2 + 50x + 29 by x + 4 is x3 – ax2 + bx + 6, then find a, b and also the remainder.
Answer:Let p(x) = x4 + 10x3 + 35x2 + 50x + 29 be the dividend. Arranging p(x) according to the descending powers of x.
p(x) = x4 + 10x3 + 35x2 + 50x + 29
Divisor, q(x) = x + 4
⇒ To find out Zero of the divisor –
q(x) = 0
x + 4 = 0
x = – 4
zero of divisor is – 4.
And, p(x) = x4 + 10x3 + 35x2 + 50x + 29
Put zero for the first entry in the 2nd row.
![](data:image/png;base64,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)
∴ Quotient = x3 + 6x2 + 11x + 6
Hence, when p(x) is divided by (x + 4) the quotient is x3 + 6x2 + 11x + 6 and remainder is 5.
Comparing x3 + 6x2 + 11x + 6 with x3 – ax2 + bx + 6 we get,
a = – 6 and b = 11.
Question 8.If the quotient on dividing, 8x4 – 2x2 + 6x – 7 by 2x + 1 is 4x3 + px2 – qx + 3,then find p, q and also the remainder.
Answer:Let p(x) = 8x4 – 2x2 + 6x – 7 be the dividend. Arranging p(x) according to the descending powers of x and write zero in place of missing term.
p(x) = 8x4 + 0x3 – 2x2 + 6x – 7
Divisor, q(x) = 2x + 1
⇒ To find out Zero of the divisor –
q(x) = 0
2x + 1 = 0
x =
.
zero of divisor is
.
And, p(x) = 8x4 + 0x3 – 2x2 + 6x – 7
Put zero for the first entry in the 2nd row.
![](data:image/png;base64,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)
∴ Quotient = 8x3 – 4x2 + 0x + 6
Hence, when p(x) is divided by (2x + 1) the quotient is 8x3 – 4x2 + 0x + 6
6 and remainder is – 10.
Comparing 8x3 – 4x2 + 0x + 6 with 4x3 + px2 – qx + 3 we get,
p = – 4 and q = 0.
Find the quotient and remainder using synthetic division.
x3 + x2 – 3x + 5) ÷ ( x – 1)
Answer:
Let p(x) = x3 + x2 – 3x + 5 be the dividend. Arranging p(x) according to the descending powers of x.
p(x) = x3 + x2 – 3x + 5
Divisor, q(x) = x – 1
⇒ To find out Zero of the divisor –
q(x) = 0
x – 1 = 0
x = 1
So, zero of divisor is 1.
⇒ p(x) = x3 + x2 – 3x + 5
Put zero for the first entry in the second row.
∴ Quotient = x2 + 2x – 1
Hence, when p(x) is divided by (x – 1) the quotient is x2 + 2x – 1 and remainder is 4.
Question 2.
Find the quotient and remainder using synthetic division.
(3x3 – 2x2 + 7x – 5) ÷ ( x + 3)
Answer:
Let p(x) = 3x3 – 2x2 + 7x – 5 be the dividend and arranging p(x) according to the descending powers of x.
Divisor, q(x) = x + 3
⇒ To find out Zero of the divisor –
q(x) = 0
x + 3 = 0
x = – 3
So, zero of divisor is – 3.
And, p(x) = 3x3 – 2x2 + 7x – 5
Put zero for the first entry in the second row.
∴ Quotient = 3x2 – 11x + 40
Hence, when p(x) is divided by (x – 1) the quotient is 3x2 – 11x + 40 and remainder is – 125.
Question 3.
Find the quotient and remainder using synthetic division.
(3x3 + 4x2 – 10x + 6) ÷ ( 3x – 2)
Answer:
Let p(x) = 3x3 + 4x2 – 10x + 6 be the dividend and arranging p(x) according to the descending powers of x.
Divisor, q(x) = 3x – 2
⇒ To find out Zero of the divisor –
q(x) = 0
3x – 2= 0
x =
So, zero of divisor is .
And, p(x) = 3x3 + 4x2 – 10x + 6
Put zero for the first entry in the second row.
∵ p(x) = (Quotient)×q(x) + remainder.
So, 3x3 + 4x2 – 10x + 6 = (x – )(3x2 + 6x – 6) + 2
= (3x – 2)(3x2 + 6x – 6) + 2
Thus, the Quotient = (3x2 + 6x – 6)= x2 + 2x – 2 and remainder is 2.
Hence, when p(x) is divided by (3x – 2) the quotient is x2 + 2x – 2 and remainder is 2.
Question 4.
Find the quotient and remainder using synthetic division.
(3x3 – 4x2 – 5) ÷ (3x + 1)
Answer:
Let p(x) = 3x3 – 4x2 – 5 be the dividend. Arranging p(x) according to the descending powers of x and insert zero for missing term.
p(x) = 3x3 – 4x2 + 0x – 5
Divisor, q(x) = 3x + 1
⇒ To find out Zero of the divisor –
q(x) = 0
3x + 1 = 0
x =
zero of divisor is .
And, p(x) = 3x3 – 4x2 + 0x – 5
Put zero for the first entry in the 2nd row.
∵ p(x) = (Quotient)×q(x) + remainder.
So, 3x3 – 4x2 – 5 = (x + )(3x2 – 5x +
) + (
)
= (3x + 1)(3x2 – 5x +
)
Thus, the Quotient = (3x2 – 5x +
)= (x2 –
x +
) and remainder is
.
Hence, when p(x) is divided by (3x + 1) the quotient is (x2 – x +
) and remainder is
.
Question 5.
Find the quotient and remainder using synthetic division.
(8x4 – 2x2 + 6x + 5) ÷ (4x + 1)
Answer:
Let p(x) = 8x4 – 2x2 + 6x – 5 be the dividend. Arranging p(x) according to the descending powers of x and insert zero for missing term.
p(x) = 8x4 + 0x3 – 2x2 + 6x – 5
Divisor, q(x) = 4x + 1
⇒ To find out Zero of the divisor –
q(x) = 0
4x + 1 = 0
x =
zero of divisor is .
And, p(x) = 8x4 + 0x3 – 2x2 + 6x – 5
Put zero for the first entry in the 2nd row.
∵ p(x) = (Quotient)×q(x) + remainder.
So, 8x4 – 2x2 + 6x – 5 = (x + )( 8x3 – 2x2 –
x +
) + (
)
= (4x + 1)(8x3 – 2x2 –
x +
)
Thus, the Quotient = (8x3 – 2x2 –
x +
)= (2x3 –
x2 –
x +
) and remainder is
.
Hence, when p(x) is divided by (4x + 1) the quotient is (2x3 – x2 –
x +
) and remainder is
.
Question 6.
Find the quotient and remainder using synthetic division.
(2x4 – 7x3 – 13x2 + 63x – 48) ÷ (2x – 1)
Answer:
Let p(x) = 2x4 – 7x3 – 13x2 + 63x – 48 be the dividend. Arranging p(x) according to the according descending powers of x.
p(x) = 2x4 – 7x3 – 13x2 + 63x – 48
Divisor, q(x) = 2x – 1
⇒ To find out Zero of the divisor –
q(x) = 0
2x – 1 = 0
x =
zero of divisor is .
And, p(x) = 2x4 – 7x3 – 13x2 + 63x – 48
Put zero for the first entry in the 2nd row.
∴ Quotient = 2x3 – 6x2 – 16x + 55
∵ p(x) = (Quotient)×q(x) + remainder.
So, 2x4 – 7x3 – 13x2 + 63x – 48
= (x – )( 8x3 – 2x2 –
x +
) + (
)
= (2x – 1)(2x3 – 6x2 – 16x + 55)
Thus, the Quotient = (2x3 – 6x2 – 16x + 55) = (x3 – 3x2 – 8x +
) and remainder is
.
Hence, when p(x) is divided by (2x – 1) the quotient is (x3 – 3x2 – 8x + ) and remainder is
.
Question 7.
If the quotient on dividing x4 + 10x3 + 35x2 + 50x + 29 by x + 4 is x3 – ax2 + bx + 6, then find a, b and also the remainder.
Answer:
Let p(x) = x4 + 10x3 + 35x2 + 50x + 29 be the dividend. Arranging p(x) according to the descending powers of x.
p(x) = x4 + 10x3 + 35x2 + 50x + 29
Divisor, q(x) = x + 4
⇒ To find out Zero of the divisor –
q(x) = 0
x + 4 = 0
x = – 4
zero of divisor is – 4.
And, p(x) = x4 + 10x3 + 35x2 + 50x + 29
Put zero for the first entry in the 2nd row.
∴ Quotient = x3 + 6x2 + 11x + 6
Hence, when p(x) is divided by (x + 4) the quotient is x3 + 6x2 + 11x + 6 and remainder is 5.
Comparing x3 + 6x2 + 11x + 6 with x3 – ax2 + bx + 6 we get,
a = – 6 and b = 11.
Question 8.
If the quotient on dividing, 8x4 – 2x2 + 6x – 7 by 2x + 1 is 4x3 + px2 – qx + 3,then find p, q and also the remainder.
Answer:
Let p(x) = 8x4 – 2x2 + 6x – 7 be the dividend. Arranging p(x) according to the descending powers of x and write zero in place of missing term.
p(x) = 8x4 + 0x3 – 2x2 + 6x – 7
Divisor, q(x) = 2x + 1
⇒ To find out Zero of the divisor –
q(x) = 0
2x + 1 = 0
x = .
zero of divisor is .
And, p(x) = 8x4 + 0x3 – 2x2 + 6x – 7
Put zero for the first entry in the 2nd row.
∴ Quotient = 8x3 – 4x2 + 0x + 6
Hence, when p(x) is divided by (2x + 1) the quotient is 8x3 – 4x2 + 0x + 6
6 and remainder is – 10.
Comparing 8x3 – 4x2 + 0x + 6 with 4x3 + px2 – qx + 3 we get,
p = – 4 and q = 0.
Exercise 3.5
Question 1.Factorize each of the following polynomials.
x3 – 2x2 – 5x + 6
Answer:Given,
x3 – 2x2 – 5x + 6, put x = 1
then, 1 – 2 – 5 + 6 = 0, since,
this equation is divisible by (x – 1)
according to the question,
x3 – 2x2 – 5x + 6 = (x – 1)(x2 – x – 6)
= (x – 1)[x2 – 3x + 2x – 6]
= (x – 1)[x(x – 3) + 2(x – 3)]
= (x – 1)(x + 2)(x – 3)
Question 2.Factorize each of the following polynomials.
4x3 – 7x + 3
Answer:Given,
4x3 – 7x + 3,put x = 1
Then, 4 × 1 – 7 + 3 = 0.
since, this equation is divisible by (x – 1).
according to the question,
4x3 – 7x + 3 = (x – 1)(4x2 + 4x – 3)
= (x – 1)[4x2 + 6x – 2x – 3]
= (x – 1)[2x(2x + 3) – 1(2x + 3)]
= (x – 1)(2x + 3)(2x – 1)
Question 3.Factorize each of the following polynomials.
x3 – 23x2 + 142x – 120
Answer:Given,
x3 – 23x2 + 142x – 120, put x = 1
then, 1 – 23 × 1 + 142 – 120 = – 143 + 143 = 0
since, this equation is divisible by (x – 1).
according to the question,
x3 – 23x2 + 142x – 120 = (x – 1)[x2 – 22x + 120]
= (x – 1)[x2 – 12x – 10x + 120]
= (x – 1)[x(x – 12) – 10(x – 12)]
= (x – 1)(x – 12)(x – 10).
Question 4.Factorize each of the following polynomials.
4x3 – 5x2 + 7x – 6
Answer:Given,
4x3 – 5x2 + 7x – 6,put x = 1
Then, 4 × 1 – 5 + 7 – 6 = 0
since, this equation is divisible by (x – 1).
according to the question,
4x3 – 5x2 + 7x – 6 = (x – 1)[4x2 – x + 6]
Question 5.Factorize each of the following polynomials.
x3 – 7x + 6
Answer:Given,
x3 – 7x + 6,
put x = 1
then, 1 – 7 + 6 = 0
since, this equation is divisible by (x – 1).
according to the question,
x3 – 7x + 6 = (x – 1)(x2 + x – 6)
= (x – 1)[x2 + 3x – 2x – 6]
= (x – 1)[x(x + 3) – 2(x + 3)]
= (x – 1)(x – 2)(x + 3)
Question 6.Factorize each of the following polynomials.
x3 + 13x2 + 32x + 20
Answer:Given,
x3 + 13x2 + 32x + 20,
put x = – 1
then, – 1 + 13 – 32 + 20 = 0
since, this equation is divisible by (x + 1).
according to the question,
x3 + 13x2 + 32x + 20 = (x + 1)(x2 + 12x + 20)
= (x + 1)[x2 + 10x + 2x + 20]
= (x + 1)[x(x + 10) + 2(x + 10)]
= (x + 1)(x + 2)(x + 10)
Question 7.Factorize each of the following polynomials.
2x3 – 9x2 + 7x + 6
Answer:Given,
2x3 – 9x2 + 7x + 6,
put x = 2
Then, 16 – 36 + 14 + 6 = 0
since, this equation is divisible by (x – 2).
according to the question,
2x3 – 9x2 + 7x + 6 = (x – 2)(2x2 – 5x – 3)
= (x – 2)[2x2 – 6x + x – 3]
= (x – 2)[2x(x – 3) + 1(x – 3)]
= (x – 2)(x – 3)(2x + 1)
Question 8.Factorize each of the following polynomials.
x3 – 5x + 4
Answer:Given,
x3 – 5x + 4,put x = 1
then,1 – 5 + 4 = 0
since, this equation is divisible by (x – 1).
according to the question,
x3 – 5x + 4 = (x – 1)(x2 + x – 4).
Question 9.Factorize each of the following polynomials.
x3 – 10x2 – x + 10
Answer:Given,
x3 – 10x2 – x + 10,put x = 1
then, 1 – 10 – 1 + 10 = 0
since, this equation is divisible by (x – 1).
according to the question,
x3 – 10x2 – x + 10 = (x – 1)(x2 – 9x – 10)
= (x – 1)[x2 – 10x + x – 10]
= (x – 1)[x(x – 10) + 1(x – 10)]
= (x – 1)(x + 1)(x – 10)
Question 10.Factorize each of the following polynomials.
2x3 + 11x2 – 7x – 6
Answer:Given,
2x3 + 11x2 – 7x – 6,
put x = 1
Then, 2 + 11 – 7 – 6 = 0
since, this equation is divisible by (x – 1).
according to the question,
2x3 + 11x2 – 7x – 6 = (x – 1)(2x2 + 13x + 6)
= (x – 1)[2x2 + 12x + x + 6]
= (x – 1)[2x(x + 6) + 1(x + 6)]
= (x – 1)(2x + 1)(x + 6)
Question 11.Factorize each of the following polynomials.
x3 + x2 + x – 14
Answer:Given,
x3 + x2 + x – 14,
put x = 2
then, 8 + 4 + 2 – 14 = 0
since, this equation is divisible by (x – 2).
according to the question,
x3 + x2 + x – 14 = (x – 2)(x2 + 3x + 7).
Question 12.Factorize each of the following polynomials.
x3 – 5x2 – 2x + 24
Answer:Given,
x3 – 5x2 – 2x + 24,put x = – 2
then, – 8 – 20 + 4 + 24 = 0
since, this equation is divisible by (x + 2).
according to the question,
x3 – 5x2 – 2x + 24 = (x + 2)(x2 – 7x + 12)
= (x + 2)[x2 – 4x – 3x + 12]
= (x + 2)[x(x – 4) – 3(x – 4)]
= (x + 2)(x – 4)(x – 3)
Factorize each of the following polynomials.
x3 – 2x2 – 5x + 6
Answer:
Given,
x3 – 2x2 – 5x + 6, put x = 1
then, 1 – 2 – 5 + 6 = 0, since,
this equation is divisible by (x – 1)
according to the question,
x3 – 2x2 – 5x + 6 = (x – 1)(x2 – x – 6)
= (x – 1)[x2 – 3x + 2x – 6]
= (x – 1)[x(x – 3) + 2(x – 3)]
= (x – 1)(x + 2)(x – 3)
Question 2.
Factorize each of the following polynomials.
4x3 – 7x + 3
Answer:
Given,
4x3 – 7x + 3,put x = 1
Then, 4 × 1 – 7 + 3 = 0.
since, this equation is divisible by (x – 1).
according to the question,
4x3 – 7x + 3 = (x – 1)(4x2 + 4x – 3)
= (x – 1)[4x2 + 6x – 2x – 3]
= (x – 1)[2x(2x + 3) – 1(2x + 3)]
= (x – 1)(2x + 3)(2x – 1)
Question 3.
Factorize each of the following polynomials.
x3 – 23x2 + 142x – 120
Answer:
Given,
x3 – 23x2 + 142x – 120, put x = 1
then, 1 – 23 × 1 + 142 – 120 = – 143 + 143 = 0
since, this equation is divisible by (x – 1).
according to the question,
x3 – 23x2 + 142x – 120 = (x – 1)[x2 – 22x + 120]
= (x – 1)[x2 – 12x – 10x + 120]
= (x – 1)[x(x – 12) – 10(x – 12)]
= (x – 1)(x – 12)(x – 10).
Question 4.
Factorize each of the following polynomials.
4x3 – 5x2 + 7x – 6
Answer:
Given,
4x3 – 5x2 + 7x – 6,put x = 1
Then, 4 × 1 – 5 + 7 – 6 = 0
since, this equation is divisible by (x – 1).
according to the question,
4x3 – 5x2 + 7x – 6 = (x – 1)[4x2 – x + 6]
Question 5.
Factorize each of the following polynomials.
x3 – 7x + 6
Answer:
Given,
x3 – 7x + 6,
put x = 1
then, 1 – 7 + 6 = 0
since, this equation is divisible by (x – 1).
according to the question,
x3 – 7x + 6 = (x – 1)(x2 + x – 6)
= (x – 1)[x2 + 3x – 2x – 6]
= (x – 1)[x(x + 3) – 2(x + 3)]
= (x – 1)(x – 2)(x + 3)
Question 6.
Factorize each of the following polynomials.
x3 + 13x2 + 32x + 20
Answer:
Given,
x3 + 13x2 + 32x + 20,
put x = – 1
then, – 1 + 13 – 32 + 20 = 0
since, this equation is divisible by (x + 1).
according to the question,
x3 + 13x2 + 32x + 20 = (x + 1)(x2 + 12x + 20)
= (x + 1)[x2 + 10x + 2x + 20]
= (x + 1)[x(x + 10) + 2(x + 10)]
= (x + 1)(x + 2)(x + 10)
Question 7.
Factorize each of the following polynomials.
2x3 – 9x2 + 7x + 6
Answer:
Given,
2x3 – 9x2 + 7x + 6,
put x = 2
Then, 16 – 36 + 14 + 6 = 0
since, this equation is divisible by (x – 2).
according to the question,
2x3 – 9x2 + 7x + 6 = (x – 2)(2x2 – 5x – 3)
= (x – 2)[2x2 – 6x + x – 3]
= (x – 2)[2x(x – 3) + 1(x – 3)]
= (x – 2)(x – 3)(2x + 1)
Question 8.
Factorize each of the following polynomials.
x3 – 5x + 4
Answer:
Given,
x3 – 5x + 4,put x = 1
then,1 – 5 + 4 = 0
since, this equation is divisible by (x – 1).
according to the question,
x3 – 5x + 4 = (x – 1)(x2 + x – 4).
Question 9.
Factorize each of the following polynomials.
x3 – 10x2 – x + 10
Answer:
Given,
x3 – 10x2 – x + 10,put x = 1
then, 1 – 10 – 1 + 10 = 0
since, this equation is divisible by (x – 1).
according to the question,
x3 – 10x2 – x + 10 = (x – 1)(x2 – 9x – 10)
= (x – 1)[x2 – 10x + x – 10]
= (x – 1)[x(x – 10) + 1(x – 10)]
= (x – 1)(x + 1)(x – 10)
Question 10.
Factorize each of the following polynomials.
2x3 + 11x2 – 7x – 6
Answer:
Given,
2x3 + 11x2 – 7x – 6,
put x = 1
Then, 2 + 11 – 7 – 6 = 0
since, this equation is divisible by (x – 1).
according to the question,
2x3 + 11x2 – 7x – 6 = (x – 1)(2x2 + 13x + 6)
= (x – 1)[2x2 + 12x + x + 6]
= (x – 1)[2x(x + 6) + 1(x + 6)]
= (x – 1)(2x + 1)(x + 6)
Question 11.
Factorize each of the following polynomials.
x3 + x2 + x – 14
Answer:
Given,
x3 + x2 + x – 14,
put x = 2
then, 8 + 4 + 2 – 14 = 0
since, this equation is divisible by (x – 2).
according to the question,
x3 + x2 + x – 14 = (x – 2)(x2 + 3x + 7).
Question 12.
Factorize each of the following polynomials.
x3 – 5x2 – 2x + 24
Answer:
Given,
x3 – 5x2 – 2x + 24,put x = – 2
then, – 8 – 20 + 4 + 24 = 0
since, this equation is divisible by (x + 2).
according to the question,
x3 – 5x2 – 2x + 24 = (x + 2)(x2 – 7x + 12)
= (x + 2)[x2 – 4x – 3x + 12]
= (x + 2)[x(x – 4) – 3(x – 4)]
= (x + 2)(x – 4)(x – 3)
Exercise 3.6
Question 1.Find the greatest common divisor of
7x2 yz4, 21x2 y5 z3
Answer:Given,
7x2 yz4 = 7x2 yz3 × z
21x2 y5 z3 = 3 × 7x2 y × y4 z3
Greatest common divisor = 7x2yz3
Question 2.Find the greatest common divisor of
x2y, x3y, x2y2
Answer:Given,
x2 y = x × x × y
x3 y = x × x × x × y
x2 y2 = x × x × y × y
Greatest common divisor = x2y
Question 3.Find the greatest common divisor of
25bc4 d3, 35b2c5, 45c3 d
Answer:Given,
25bc4 d3 = 5 × 5 × b × c3 × c × d3
35b2c5 = 5 × 7 × b × 2 × c3 × c2
45c3 d = 5 × 3 × 3 × c3 × d
Greatest common divisor = 5c3
Question 4.Find the greatest common divisor of
35x5 y3 z4, 49x2 yz3, 14xy2 z2
Answer:Given,
35x5 y3 z4 = 5 × 7 × x × x4 × y2 × y × z2 × z2
49x2 yz3 = 7 × 7 × x2 × y × z × z2
14xy2 z2 = 2 × 7 × x × y × y × z2
Greatest common divisor = 7xyz2
Question 5.Find the GCD of the following
x3 – x2 + x – 1, x4 – 1
Answer:x3 – x2 + x – 1, put x = 1
Then,1 – 1 + 1 – 1 = 0
Since, this equation is divisible by x – 1
(x – 1)(x2 + 1) = (x + 1)(x2 + 1)
In second equation,
x4 – 1 = (x2)2 – 1 = (x2 – 1)(x2 + 1) = (x + 1)(x – 1)(x2 + 1)
[using a2 – b2 = (a – b)(a + b)]
Greatest common divisor = (x + 1)(x2 + 1)
Question 6.Find the GCD of the following
c2 – d2, – c(c – d)
Answer:Given,
c2 – d2 = (c + d) × (c – d)
– c(c – d) = – c × (c – d)
Greatest common divisor = (c – d)
Question 7.Find the GCD of the following
x4 – 27a3 x,(x – 3a)2
Answer:Given,
x4 – 27a3 x = x[x3 – (3a)3]
= x[(x – 3a)(x2 + 9a2 + 3ax)]
= x(x – 3a)(x2 + 9a2 + 3ax)
(x – 3a)2 = (x – 3a)(x – 3a)
Greatest common divisor = (x – 3a)
Question 8.Find the GCD of the following
m2 – 3m – 18, m2 + 5m + 6
Answer:Given,
m2 – 3m – 18 = m2 – 6m + 3m – 18 = m(m – 6) + 3(m – 6)
= (m + 3)(m – 6)
m2 + 5m + 6 = m2 + 3m + 2m + 6 = m(m + 3) + 2(m + 3)
= (m + 3)(m + 2)
Greatest common divisor = (m + 3)
Question 9.Find the GCD of the following
x2 + 14x + 33, x3 + 10x2 – 11x
Answer:Given,
X2 + 14x + 33 = x2 + 3x + 11x + 33 = x(x + 3) + 11(x + 3)
= (x + 11)(x + 3)
X3 + 10x2 – 11x = x(x2 + 10x – 11) = x(x2 + 11x – x – 11)
= x[x(x + 11) – 1(x + 11)]
= x(x + 11)(x – 1)
Greatest common divisor = (x + 11)
Question 10.Find the GCD of the following
x2 + 3xy + 2y2, x2 + 5xy + 6y2
Answer:Given,
X2 + 3xy + 2y2 = x2 + xy + 2xy + 2y2
= x(x + y) + 2y(x + y)
= (x + 2y)(x + y)
x2 + 5xy + 6y2 = x2 + 3xy + 2xy + 6y2
= x(x + 3y) + 2y(x + 3y)
= (x + 2y)(x + 3y)
Greatest common divisor = (x + 2y)
Question 11.Find the GCD of the following
2x2 – x – 1,4x2 + 8x + 3
Answer:Given,
2x2 – x – 1 = 2x2 – 2x + x – 1 = 2x(x – 1) + 1(x – 1)
= (2x + 1)(x – 1)
4x2 + 8x + 3 = 4x2 + 2x + 6x + 3 = 2x(x + 1) + 3(2x + 1)
= (2x + 1)(2x + 3)
Greatest common divisor = (2x + 1)
Question 12.Find the GCD of the following
x2 – x – 2,x2 + x – 6,3x2 – 13x + 14
Answer:Given,
x2 – x – 2 = x2 – 2x + x – 2 = x(x – 2) + 1(x – 2) = (x + 1)(x – 2)
x2 + x – 6 = x2 + 3x – 2x – 6
= x(x + 3) – 2(x + 3) = (x – 2)(x + 3)
3x2 – 13x + 14 = 3x2 – 6x – 7x + 14 = 3x(x – 2) – 7(x – 2)
= (x – 2)(3x – 7)
Greatest common divisor = (x – 2)
Question 13.Find the GCD of the following
24(6x4 – x3 – 2x2),20(2x6 + 3x5 + x4)
Answer:Given,
24(6x4 – x3 – 2x2) = 2 × 2 × 2 × 3[x2(6x2 – x – 2)]
= 2 × 2 × 2 × 3[x2(6x2 – 4x + 3x – 2)]
= 2 × 2 × 2 × 3 × x2 [2x(3x – 2) + 1(3x – 2)]
= 2 × 2 × 2 × 3 × x2 (2x + 1)(3x – 2)
20(2x6 + 3x5 + x4) = 2 × 2 × 5[x4(2x2 + 3x + 1)]
= 2 × 2 × 5 × x2 × x2[2x(x + 1) + 1(x + 1)]
= 2 × 2 × 5 × x2 × x2(2x + 1)(x + 1)
Greatest common divisor = 2 × 2 × x2 × (2x + 1) = 4x2(2x + 1)
Question 14.Find the GCD of the following
(a – 1)5 (a + 3)2,(a – 2)2 (a – 1)3(a + 3)4
Answer:Given,
(a – 1)5 (a + 3)2 = (a – 1)3(a – 1)2(a + 3)2
(a – 2)2(a – 1)3(a + 3)4 = (a – 2)2(a – 1)3(a + 3)2(a + 3)2
Greatest common divisor = (a – 1)3(a + 3)2
Question 15.Find the GCD of the following pairs of polynomials using division algorithm.
x3 – 9x2 + 23x – 15, 4x2 – 16x + 12
Answer:x3 – 9x2 + 23x – 15
put x = 1 in polynomials equations is 1 – 9(1) + 23 – 15 = 0
This equation is divisible by (x – 1),
then using division algorithm method.
Since,
(x – 1)(x2 – 8x + 15) = (x – 1)[x2 – 5x – 3x + 15]
= (x – 1)[x(x – 5) – 3(x – 5)] = (x – 1)(x – 3)(x – 5)
4x2 – 16x + 12 = 4x2 – 12x – 4x + 12 = 4x(x – 3) – 4(x – 3)
= 4(x – 1)(x – 3)
Greatest common divisor = (x – 1)(x – 3) = x2 – 4x + 3
Question 16.Find the GCD of the following pairs of polynomials using division algorithm.
3x3 + 18x2 + 33x + 18, 3x2 + 13x + 10
Answer:According to the question,
3x3 + 18x2 + 33x + 18 then put x = – 1 in that equation
put x = – 1
then, – 3 + 18 – 33 + 18 = 0
Then the equation is divisible by x + 1, using division algorithm method.
(x + 1)(3x2 + 15x + 18) = (x + 1)[3x2 + 9x + 6x + 18]
= (x + 1)[3x(x + 3) + 6(x + 3)]
= 3(x + 1)(x + 2)(x + 3)
3x2 + 13x + 10 = 3x2 + 3x + 10x + 10 = 3x(x + 1) + 10(x + 1)
= (x + 1)(3x + 10)
Greatest common divisor = (x + 1)
Question 17.Find the GCD of the following pairs of polynomials using division algorithm.
2x3 + 2x2 + 2x + 2, 6x3 + 12x2 + 6x + 12
Answer:2x3 + 2x2 + 2x + 2 then put x = – 1
Since, – 2 + 2 – 2 + 2 = 0
Then the equation is divisible by x + 1, using division algorithm method.
(x + 1)(2x2 + 2) = 2(x + 1)(x2 + 1)
In equation second,
6x3 + 12x2 + 6x + 12,put x = – 2
Then, – 48 + 48 – 12 + 12 = 0
Then the equation is divisible by x + 2, using division algorithm method.
(x + 2)(6x2 + 6) = 2 × 3(x + 2)(x2 + 1)
Greatest common divisor = 2(x2 + 1)
Question 18.Find the GCD of the following pairs of polynomials using division algorithm.
x3 – 3x2 + 4x – 12, x4 + x3 + 4x2 + 4x
Answer:According to the question,
x3 – 3x2 + 4x – 12,then put x = 3
then,27 – 27 + 12 – 12 = 0
Then the equation is divisible by x – 3, using division algorithm method.
(x – 3)(x2 + 4) = (x – 3)(x2 + 4)
In equation second,
x4 + x3 + 4x2 + 4x,put x = – 1
then, 1 – 1 + 4 – 4 = 0,
Then the equation is divisible by x + 1, using division algorithm method.
(x + 1)(x3 + 4x) = x(x + 1)(x2 + 4)
Greatest common divisor = (x2 + 4)
Find the greatest common divisor of
7x2 yz4, 21x2 y5 z3
Answer:
Given,
7x2 yz4 = 7x2 yz3 × z
21x2 y5 z3 = 3 × 7x2 y × y4 z3
Greatest common divisor = 7x2yz3
Question 2.
Find the greatest common divisor of
x2y, x3y, x2y2
Answer:
Given,
x2 y = x × x × y
x3 y = x × x × x × y
x2 y2 = x × x × y × y
Greatest common divisor = x2y
Question 3.
Find the greatest common divisor of
25bc4 d3, 35b2c5, 45c3 d
Answer:
Given,
25bc4 d3 = 5 × 5 × b × c3 × c × d3
35b2c5 = 5 × 7 × b × 2 × c3 × c2
45c3 d = 5 × 3 × 3 × c3 × d
Greatest common divisor = 5c3
Question 4.
Find the greatest common divisor of
35x5 y3 z4, 49x2 yz3, 14xy2 z2
Answer:
Given,
35x5 y3 z4 = 5 × 7 × x × x4 × y2 × y × z2 × z2
49x2 yz3 = 7 × 7 × x2 × y × z × z2
14xy2 z2 = 2 × 7 × x × y × y × z2
Greatest common divisor = 7xyz2
Question 5.
Find the GCD of the following
x3 – x2 + x – 1, x4 – 1
Answer:
x3 – x2 + x – 1, put x = 1
Then,1 – 1 + 1 – 1 = 0
Since, this equation is divisible by x – 1
(x – 1)(x2 + 1) = (x + 1)(x2 + 1)
In second equation,
x4 – 1 = (x2)2 – 1 = (x2 – 1)(x2 + 1) = (x + 1)(x – 1)(x2 + 1)
[using a2 – b2 = (a – b)(a + b)]
Greatest common divisor = (x + 1)(x2 + 1)
Question 6.
Find the GCD of the following
c2 – d2, – c(c – d)
Answer:
Given,
c2 – d2 = (c + d) × (c – d)
– c(c – d) = – c × (c – d)
Greatest common divisor = (c – d)
Question 7.
Find the GCD of the following
x4 – 27a3 x,(x – 3a)2
Answer:
Given,
x4 – 27a3 x = x[x3 – (3a)3]
= x[(x – 3a)(x2 + 9a2 + 3ax)]
= x(x – 3a)(x2 + 9a2 + 3ax)
(x – 3a)2 = (x – 3a)(x – 3a)
Greatest common divisor = (x – 3a)
Question 8.
Find the GCD of the following
m2 – 3m – 18, m2 + 5m + 6
Answer:
Given,
m2 – 3m – 18 = m2 – 6m + 3m – 18 = m(m – 6) + 3(m – 6)
= (m + 3)(m – 6)
m2 + 5m + 6 = m2 + 3m + 2m + 6 = m(m + 3) + 2(m + 3)
= (m + 3)(m + 2)
Greatest common divisor = (m + 3)
Question 9.
Find the GCD of the following
x2 + 14x + 33, x3 + 10x2 – 11x
Answer:
Given,
X2 + 14x + 33 = x2 + 3x + 11x + 33 = x(x + 3) + 11(x + 3)
= (x + 11)(x + 3)
X3 + 10x2 – 11x = x(x2 + 10x – 11) = x(x2 + 11x – x – 11)
= x[x(x + 11) – 1(x + 11)]
= x(x + 11)(x – 1)
Greatest common divisor = (x + 11)
Question 10.
Find the GCD of the following
x2 + 3xy + 2y2, x2 + 5xy + 6y2
Answer:
Given,
X2 + 3xy + 2y2 = x2 + xy + 2xy + 2y2
= x(x + y) + 2y(x + y)
= (x + 2y)(x + y)
x2 + 5xy + 6y2 = x2 + 3xy + 2xy + 6y2
= x(x + 3y) + 2y(x + 3y)
= (x + 2y)(x + 3y)
Greatest common divisor = (x + 2y)
Question 11.
Find the GCD of the following
2x2 – x – 1,4x2 + 8x + 3
Answer:
Given,
2x2 – x – 1 = 2x2 – 2x + x – 1 = 2x(x – 1) + 1(x – 1)
= (2x + 1)(x – 1)
4x2 + 8x + 3 = 4x2 + 2x + 6x + 3 = 2x(x + 1) + 3(2x + 1)
= (2x + 1)(2x + 3)
Greatest common divisor = (2x + 1)
Question 12.
Find the GCD of the following
x2 – x – 2,x2 + x – 6,3x2 – 13x + 14
Answer:
Given,
x2 – x – 2 = x2 – 2x + x – 2 = x(x – 2) + 1(x – 2) = (x + 1)(x – 2)
x2 + x – 6 = x2 + 3x – 2x – 6
= x(x + 3) – 2(x + 3) = (x – 2)(x + 3)
3x2 – 13x + 14 = 3x2 – 6x – 7x + 14 = 3x(x – 2) – 7(x – 2)
= (x – 2)(3x – 7)
Greatest common divisor = (x – 2)
Question 13.
Find the GCD of the following
24(6x4 – x3 – 2x2),20(2x6 + 3x5 + x4)
Answer:
Given,
24(6x4 – x3 – 2x2) = 2 × 2 × 2 × 3[x2(6x2 – x – 2)]
= 2 × 2 × 2 × 3[x2(6x2 – 4x + 3x – 2)]
= 2 × 2 × 2 × 3 × x2 [2x(3x – 2) + 1(3x – 2)]
= 2 × 2 × 2 × 3 × x2 (2x + 1)(3x – 2)
20(2x6 + 3x5 + x4) = 2 × 2 × 5[x4(2x2 + 3x + 1)]
= 2 × 2 × 5 × x2 × x2[2x(x + 1) + 1(x + 1)]
= 2 × 2 × 5 × x2 × x2(2x + 1)(x + 1)
Greatest common divisor = 2 × 2 × x2 × (2x + 1) = 4x2(2x + 1)
Question 14.
Find the GCD of the following
(a – 1)5 (a + 3)2,(a – 2)2 (a – 1)3(a + 3)4
Answer:
Given,
(a – 1)5 (a + 3)2 = (a – 1)3(a – 1)2(a + 3)2
(a – 2)2(a – 1)3(a + 3)4 = (a – 2)2(a – 1)3(a + 3)2(a + 3)2
Greatest common divisor = (a – 1)3(a + 3)2
Question 15.
Find the GCD of the following pairs of polynomials using division algorithm.
x3 – 9x2 + 23x – 15, 4x2 – 16x + 12
Answer:
x3 – 9x2 + 23x – 15
put x = 1 in polynomials equations is 1 – 9(1) + 23 – 15 = 0
This equation is divisible by (x – 1),
then using division algorithm method.
Since,
(x – 1)(x2 – 8x + 15) = (x – 1)[x2 – 5x – 3x + 15]
= (x – 1)[x(x – 5) – 3(x – 5)] = (x – 1)(x – 3)(x – 5)
4x2 – 16x + 12 = 4x2 – 12x – 4x + 12 = 4x(x – 3) – 4(x – 3)
= 4(x – 1)(x – 3)
Greatest common divisor = (x – 1)(x – 3) = x2 – 4x + 3
Question 16.
Find the GCD of the following pairs of polynomials using division algorithm.
3x3 + 18x2 + 33x + 18, 3x2 + 13x + 10
Answer:
According to the question,
3x3 + 18x2 + 33x + 18 then put x = – 1 in that equation
put x = – 1
then, – 3 + 18 – 33 + 18 = 0
Then the equation is divisible by x + 1, using division algorithm method.
(x + 1)(3x2 + 15x + 18) = (x + 1)[3x2 + 9x + 6x + 18]
= (x + 1)[3x(x + 3) + 6(x + 3)]
= 3(x + 1)(x + 2)(x + 3)
3x2 + 13x + 10 = 3x2 + 3x + 10x + 10 = 3x(x + 1) + 10(x + 1)
= (x + 1)(3x + 10)
Greatest common divisor = (x + 1)
Question 17.
Find the GCD of the following pairs of polynomials using division algorithm.
2x3 + 2x2 + 2x + 2, 6x3 + 12x2 + 6x + 12
Answer:
2x3 + 2x2 + 2x + 2 then put x = – 1
Since, – 2 + 2 – 2 + 2 = 0
Then the equation is divisible by x + 1, using division algorithm method.
(x + 1)(2x2 + 2) = 2(x + 1)(x2 + 1)
In equation second,
6x3 + 12x2 + 6x + 12,put x = – 2
Then, – 48 + 48 – 12 + 12 = 0
Then the equation is divisible by x + 2, using division algorithm method.
(x + 2)(6x2 + 6) = 2 × 3(x + 2)(x2 + 1)
Greatest common divisor = 2(x2 + 1)
Question 18.
Find the GCD of the following pairs of polynomials using division algorithm.
x3 – 3x2 + 4x – 12, x4 + x3 + 4x2 + 4x
Answer:
According to the question,
x3 – 3x2 + 4x – 12,then put x = 3
then,27 – 27 + 12 – 12 = 0
Then the equation is divisible by x – 3, using division algorithm method.
(x – 3)(x2 + 4) = (x – 3)(x2 + 4)
In equation second,
x4 + x3 + 4x2 + 4x,put x = – 1
then, 1 – 1 + 4 – 4 = 0,
Then the equation is divisible by x + 1, using division algorithm method.
(x + 1)(x3 + 4x) = x(x + 1)(x2 + 4)
Greatest common divisor = (x2 + 4)
Exercise 3.7
Question 1.Find the LCM of the following
x3 y2 , xyz
Answer:Given terms: –
x3, y2 , xyz
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be divided completely;
x3 = x × x × x
y2 = y × y
xyz = x × y × z
⇒ first find the common factors in all terms
Common factor = x × y
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x × y) × [(x2)(y)(z)]
= x3y2z
Conclusion: –
The LCM of given terms [x3 , y2 , xyz] is x3y2z
Question 2.Find the LCM of the following
3x2yz, 4x3y3
Answer:Given terms: –
3x2yz, 4x3y3
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
3x2yz, = 3 × x × x × y × z
4x3y3 = 4 × x × x × x × y × y × y
⇒ first find the common factors in all terms
Common factor = x × x × y
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x × y × x) × [(3yz)(4xy2)]
= 12x3y3z
Conclusion: –
The LCM of given terms [3x2yz, 4x3y3] is 12x3y3z
Question 3.Find the LCM of the following
a2bc, b2ca, c2ab
Answer:Given terms: –
a2bc, b2ca, c2ab
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
a2bc = a × a × b × c
b2ca = a × b × b × c
c2ab = a × b × c × c
⇒ first find the common factors in all terms
Common factor = a × b × c
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= ( a × b × c) × [(a)(b)(c)]
= a2b2c2
Conclusion: –
The LCM of given terms [a2bc, b2ca, c2ab] is a2b2c2
Question 4.Find the LCM of the following
66a4b2c3, 44a3b4c2, 24a2b3c4
Answer:Given terms: –
66a4b2c3, 44a3b4c2, 24a2b3c4
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
66a4b2c3 = 3 × 2 × 11 × a × a × a × a × b × b × c × c × c
44a3b4c2 = 2 × 2 × 11 × a × a × a × b × b × b × b × c × c
24a2b3c4 = 2 × 2 × 2 × 3 × a × a × b × b × b × c × c × c × c
⇒ first find the common factors in all terms
Common factor in all terms = 2 × a × a × b × b × c × c
Common factors from any 2 terms
2a2b2c2 × [(3 × 11 × a × a × c)( 2 × 11 × a × b × b)( 2 × 2 × 3 × b × c × c)]
2a2b2c2 × (3 × 11 × 2 × a × b × c)[(a)(b)(2c)]
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= 2a2b2c2 × (66abc) × (2abc)
(66 × 2 × 2)( a2b2c2 × abc × abc )
264 a4b4c4
Conclusion: –
The LCM of given terms [66a4b2c3, 44a3b4c2, 24a2b3c4] is 264 a4b4c4
Question 5.Find the LCM of the following
am + 1, am + 2, am + 3
Answer:Given terms: –
am + 1, am + 2, am + 3
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
am + 1 = am × a
am + 2 = am × a × a
am + 3 = am × a × a × a
⇒ first find the common factors in all terms
Common factor = am × a
Common factor from any 2 terms
= ( am × a) × [(1)(a)(a2)]
= ( am × a) × a[(1)(1)(a)]
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= ( am × a2)(a)
= ama3
Conclusion: –
The LCM of given terms [am + 1, am + 2, am + 3] is ama3
Question 6.Find the LCM of the following
x2y + xy2, x2 + xy
Answer:Given terms: –
x2y + xy2, x2 + xy
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
x2y + xy2 = x × y × x + x × y × y
= x × y × (x + y)
x2 + xy = x × x + x × y
= x × (x + y)
⇒ first find the common factors in all terms
Common factor = (x + y) × x
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x + y) × x × (x)
= x3 + yx2
Conclusion: –
The LCM of given terms [x2y + xy2, x2 + xy] is x3 + yx2
Question 7.Find the LCM of the following
3(a – 1), 2(a – 1)2, (a2 – 1)
Answer:Given terms: –
3(a – 1), 2(a – 1)2, (a2 – 1)
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
3(a – 1) = 3 × (a – 1)
2(a – 1)2 = 2 × (a – 1) × (a – 1)
(a2 – 1) = (a2 – 12) = (a – 1) × (a + 1)
⇒ first find the common factors in all terms
Common factor = (a – 1)
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (a – 1) × [(3) × (2(a – 1)) × (a + 1)]
= 6(a + 1)(a – 1)2
Conclusion: –
The LCM of given terms [3(a – 1), 2(a – 1)2, (a2 – 1)] is
6(a + 1)(a – 1)2
Question 8.Find the LCM of the following
2x2 – 18, 5x2y + 15xy2, x3 + 27y3
Answer:Given terms: –
2x2 – 18, 5x2y + 15xy2, x3 + 27y3
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
2x2 – 18y2 = 2 × (x2 – 9y2) = 2(x2 – (3y)2) = 2 × (x – 3y) × (x + 3y)
5x2y + 15xy2 = 5 × x × y × (x + 3y)
x3 + 27y3 = (x3 + (3y)3) = (x + 3y)(x2 – 3xy + 9y2)
⇒ first find the common factors in all terms
Common factor = (x + 3y)
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x + 3y) × [2 × (x – 3y) × 5xy × ( x2 – 3xy + 9y2)]
= 10xy(x + 3y)(x – 3y)( x2 – 3xy + 9y2)]
Conclusion: –
The LCM of given terms [2x2 – 18, 5x2y + 15xy2, x3 + 27y3] is
10xy(x + 3y)(x – 3y)( x2 – 3xy + 9y2)]
Question 9.Find the LCM of the following
(x + 4)2 (x – 3)3, (x – 1)(x + 4)(x – 3)2
Answer:Given terms: –
(x + 4)2 (x – 3)3, (x – 1)(x + 4)(x – 3)2
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be divided completely;
(x + 4)2 (x – 3)3 = (x + 4) × (x + 4) × (x – 3) × (x – 3) × (x – 3)
(x – 1)(x + 4)(x – 3)2 = (x – 1) × (x + 4) × (x – 3) × (x – 3)
⇒ first find the common factors in all terms
Common factor = (x + 4) × (x – 3) × (x – 3)
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x + 4)(x – 3)(x – 3) × [(x + 4) × (x – 3) × (x – 1)]
= (x + 4)2 (x – 3)3(x – 1)
Conclusion: –
The LCM of given terms [(x + 4)2 (x – 3)3, (x – 1)(x + 4)(x – 3)2] is
(x + 4)2 (x – 3)3(x – 1)
Question 10.Find the LCM of the following
10(9x2 + 6xy + y2), 12(3x2 – 5xy – 2y2), 14(6x4 + 2x3)
Answer:Given terms: –
10(9x2 + 6xy + y2), 12(3x2 – 5xy – 2y2), 14(6x4 + 2x3)
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
10(9x2 + 6xy + y2) = 2 × 5 × ((3x)2 + 2 × 3x × y + y2)
= 2 × 5 × (3x + y)2
= 2 × 5 × (3x + y) × (3x + y)
12(3x2 – 5xy – 2y2) = 2 × 2 × 3 × (3x2 – 6xy + xy – 2y2)
= 2 × 2 × 3 × (3x(x – 2y) + y(x – 2y))
= 2 × 2 × 3 × (x – 2y) × (3x + y)
14(6x4 + 2x3) = 2 × 7 × 2 × x × x × x × (3x + 1)
⇒ first find the common factors in all terms
Common factor = 2
Common factors in any 2 terms
2 × [(5(3x + 4)2)(2 × 3 × (x – 2y)(3x + y))(7 × 2 × x3 × (3x + 1))]
2 × 2 × (3x + y)[(5(3x + 4))(3 × (x – 2y))(7 × x3 × (3x + 1))]
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= 2 × 2 × 5 × 3 × 7 × x3 × (3x + y)(3x + y)(x – 2y)(3x + 1)
= 420x3(3x + y)2(x – 2y)(3x + 1)
Conclusion: –
The LCM of given terms [10(9x2 + 6xy + y2), 12(3x2 – 5xy – 2y2), 14(6x4 + 2x3)] is 420x3(3x + y)2(x – 2y)(3x + 1)
Find the LCM of the following
x3 y2 , xyz
Answer:
Given terms: –
x3, y2 , xyz
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be divided completely;
x3 = x × x × x
y2 = y × y
xyz = x × y × z
⇒ first find the common factors in all terms
Common factor = x × y
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x × y) × [(x2)(y)(z)]
= x3y2z
Conclusion: –
The LCM of given terms [x3 , y2 , xyz] is x3y2z
Question 2.
Find the LCM of the following
3x2yz, 4x3y3
Answer:
Given terms: –
3x2yz, 4x3y3
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
3x2yz, = 3 × x × x × y × z
4x3y3 = 4 × x × x × x × y × y × y
⇒ first find the common factors in all terms
Common factor = x × x × y
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x × y × x) × [(3yz)(4xy2)]
= 12x3y3z
Conclusion: –
The LCM of given terms [3x2yz, 4x3y3] is 12x3y3z
Question 3.
Find the LCM of the following
a2bc, b2ca, c2ab
Answer:
Given terms: –
a2bc, b2ca, c2ab
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
a2bc = a × a × b × c
b2ca = a × b × b × c
c2ab = a × b × c × c
⇒ first find the common factors in all terms
Common factor = a × b × c
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= ( a × b × c) × [(a)(b)(c)]
= a2b2c2
Conclusion: –
The LCM of given terms [a2bc, b2ca, c2ab] is a2b2c2
Question 4.
Find the LCM of the following
66a4b2c3, 44a3b4c2, 24a2b3c4
Answer:
Given terms: –
66a4b2c3, 44a3b4c2, 24a2b3c4
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
66a4b2c3 = 3 × 2 × 11 × a × a × a × a × b × b × c × c × c
44a3b4c2 = 2 × 2 × 11 × a × a × a × b × b × b × b × c × c
24a2b3c4 = 2 × 2 × 2 × 3 × a × a × b × b × b × c × c × c × c
⇒ first find the common factors in all terms
Common factor in all terms = 2 × a × a × b × b × c × c
Common factors from any 2 terms
2a2b2c2 × [(3 × 11 × a × a × c)( 2 × 11 × a × b × b)( 2 × 2 × 3 × b × c × c)]
2a2b2c2 × (3 × 11 × 2 × a × b × c)[(a)(b)(2c)]
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= 2a2b2c2 × (66abc) × (2abc)
(66 × 2 × 2)( a2b2c2 × abc × abc )
264 a4b4c4
Conclusion: –
The LCM of given terms [66a4b2c3, 44a3b4c2, 24a2b3c4] is 264 a4b4c4
Question 5.
Find the LCM of the following
am + 1, am + 2, am + 3
Answer:
Given terms: –
am + 1, am + 2, am + 3
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
am + 1 = am × a
am + 2 = am × a × a
am + 3 = am × a × a × a
⇒ first find the common factors in all terms
Common factor = am × a
Common factor from any 2 terms
= ( am × a) × [(1)(a)(a2)]
= ( am × a) × a[(1)(1)(a)]
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= ( am × a2)(a)
= ama3
Conclusion: –
The LCM of given terms [am + 1, am + 2, am + 3] is ama3
Question 6.
Find the LCM of the following
x2y + xy2, x2 + xy
Answer:
Given terms: –
x2y + xy2, x2 + xy
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
x2y + xy2 = x × y × x + x × y × y
= x × y × (x + y)
x2 + xy = x × x + x × y
= x × (x + y)
⇒ first find the common factors in all terms
Common factor = (x + y) × x
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x + y) × x × (x)
= x3 + yx2
Conclusion: –
The LCM of given terms [x2y + xy2, x2 + xy] is x3 + yx2
Question 7.
Find the LCM of the following
3(a – 1), 2(a – 1)2, (a2 – 1)
Answer:
Given terms: –
3(a – 1), 2(a – 1)2, (a2 – 1)
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
3(a – 1) = 3 × (a – 1)
2(a – 1)2 = 2 × (a – 1) × (a – 1)
(a2 – 1) = (a2 – 12) = (a – 1) × (a + 1)
⇒ first find the common factors in all terms
Common factor = (a – 1)
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (a – 1) × [(3) × (2(a – 1)) × (a + 1)]
= 6(a + 1)(a – 1)2
Conclusion: –
The LCM of given terms [3(a – 1), 2(a – 1)2, (a2 – 1)] is
6(a + 1)(a – 1)2
Question 8.
Find the LCM of the following
2x2 – 18, 5x2y + 15xy2, x3 + 27y3
Answer:
Given terms: –
2x2 – 18, 5x2y + 15xy2, x3 + 27y3
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
2x2 – 18y2 = 2 × (x2 – 9y2) = 2(x2 – (3y)2) = 2 × (x – 3y) × (x + 3y)
5x2y + 15xy2 = 5 × x × y × (x + 3y)
x3 + 27y3 = (x3 + (3y)3) = (x + 3y)(x2 – 3xy + 9y2)
⇒ first find the common factors in all terms
Common factor = (x + 3y)
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x + 3y) × [2 × (x – 3y) × 5xy × ( x2 – 3xy + 9y2)]
= 10xy(x + 3y)(x – 3y)( x2 – 3xy + 9y2)]
Conclusion: –
The LCM of given terms [2x2 – 18, 5x2y + 15xy2, x3 + 27y3] is
10xy(x + 3y)(x – 3y)( x2 – 3xy + 9y2)]
Question 9.
Find the LCM of the following
(x + 4)2 (x – 3)3, (x – 1)(x + 4)(x – 3)2
Answer:
Given terms: –
(x + 4)2 (x – 3)3, (x – 1)(x + 4)(x – 3)2
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be divided completely;
(x + 4)2 (x – 3)3 = (x + 4) × (x + 4) × (x – 3) × (x – 3) × (x – 3)
(x – 1)(x + 4)(x – 3)2 = (x – 1) × (x + 4) × (x – 3) × (x – 3)
⇒ first find the common factors in all terms
Common factor = (x + 4) × (x – 3) × (x – 3)
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= (x + 4)(x – 3)(x – 3) × [(x + 4) × (x – 3) × (x – 1)]
= (x + 4)2 (x – 3)3(x – 1)
Conclusion: –
The LCM of given terms [(x + 4)2 (x – 3)3, (x – 1)(x + 4)(x – 3)2] is
(x + 4)2 (x – 3)3(x – 1)
Question 10.
Find the LCM of the following
10(9x2 + 6xy + y2), 12(3x2 – 5xy – 2y2), 14(6x4 + 2x3)
Answer:
Given terms: –
10(9x2 + 6xy + y2), 12(3x2 – 5xy – 2y2), 14(6x4 + 2x3)
Formula used: –
LCM = Least Common Multiple
Means it is the lowest term by which every element must be
divided completely;
10(9x2 + 6xy + y2) = 2 × 5 × ((3x)2 + 2 × 3x × y + y2)
= 2 × 5 × (3x + y)2
= 2 × 5 × (3x + y) × (3x + y)
12(3x2 – 5xy – 2y2) = 2 × 2 × 3 × (3x2 – 6xy + xy – 2y2)
= 2 × 2 × 3 × (3x(x – 2y) + y(x – 2y))
= 2 × 2 × 3 × (x – 2y) × (3x + y)
14(6x4 + 2x3) = 2 × 7 × 2 × x × x × x × (3x + 1)
⇒ first find the common factors in all terms
Common factor = 2
Common factors in any 2 terms
2 × [(5(3x + 4)2)(2 × 3 × (x – 2y)(3x + y))(7 × 2 × x3 × (3x + 1))]
2 × 2 × (3x + y)[(5(3x + 4))(3 × (x – 2y))(7 × x3 × (3x + 1))]
⇒ then multiply the remaining factors of terms in common
factor to get the LCM
= 2 × 2 × 5 × 3 × 7 × x3 × (3x + y)(3x + y)(x – 2y)(3x + 1)
= 420x3(3x + y)2(x – 2y)(3x + 1)
Conclusion: –
The LCM of given terms [10(9x2 + 6xy + y2), 12(3x2 – 5xy – 2y2), 14(6x4 + 2x3)] is 420x3(3x + y)2(x – 2y)(3x + 1)
Exercise 3.8
Question 1.Find the LCM of each pair of the following polynomials.
x2 – 5x + 6, x2 + 4x – 12 whose GCD is x – 2.
Answer:Given: –
Polynomials x2 – 5x + 6, x2 + 4x – 12
And GCD[Greatest Common Divisor] = (x – 2)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
Product of 2 polynomial = (x2 – 5x + 6) × (x2 + 4x – 12)
= ( x2 – 2x – 3x + 6)(x2 + 6x – 2x – 12)
= (x(x – 2) – 3(x – 2))(x(x + 6) – 2(x + 6))
= (x – 3)(x – 2)(x – 2)(x + 6)
Product of 2 polynomial = LCM × GCD
LCM = ![](data:image/png;base64,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)
LCM = ![](data:image/png;base64,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)
LCM = (x – 3)(x – 2)(x + 6)
Conclusion: –
The LCM of polynomial [x2 – 5x + 6, x2 + 4x – 12] is
(x – 3)(x – 2)(x + 6)
Question 2.Find the LCM of each pair of the following polynomials.
x4 + 3 x3 + 6 x2 + 5x + 3, x4 + 2 x2 + x + 2 whose GCD is x2 + x + 1
Answer:Given: –
Polynomials x4 + 3 x3 + 6 x2 + 5x + 3 , x4 + 2 x2 + x + 2
And GCD[Greatest Common Divisor] = (x2 + x + 1)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
Product of 2 polynomial = (x4 + 3x3 + 6x2 + 5x + 3) × (x4 + 2x2 + x + 2)
Product of 2 polynomial = LCM × GCD
LCM = ![](data:image/png;base64,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)
LCM = ![](data:image/png;base64,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)
LCM = ![](data:image/png;base64,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)
LCM = (x2 + 2x + 3)(x4 + 2x2 + x + 2)
Conclusion: –
The LCM of polynomial [x4 + 3 x3 + 6 x2 + 5x + 3, x4 + 2 x2 + x + 2] is
(x2 + 2x + 3)(x4 + 2x2 + x + 2)
Question 3.Find the LCM of each pair of the following polynomials.
2x3 + 15x2 + 2x – 35, x3 + 8x2 + 4x – 21 whose GCD is x + 7.
Answer:Given: –
Polynomials 2x3 + 15x2 + 2x – 35 , x3 + 8x2 + 4x – 21
And GCD[Greatest Common Divisor] = (x + 7)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
Product of 2 polynomial = (2x3 + 15x2 + 2x – 35) × (x3 + 8x2 + 4x – 21)
Product of 2 polynomial = LCM × GCD
LCM = ![](data:image/png;base64,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)
LCM = ![](data:image/png;base64,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)
LCM = ![](data:image/png;base64,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)
LCM = (2x2 + x – 5)(x3 + 8x2 + 4x – 21)
Conclusion: –
The LCM of given polynomials [2x3 + 15x2 + 2x – 35 , x3 + 8x2 + 4x – 21] is (2x2 + x – 5)(x3 + 8x2 + 4x – 21)
Question 4.Find the LCM of each pair of the following polynomials.
2x3 – 3x2 – 9x + 5, 2x4 – x3 – 10x2 – 11x + 8 whose GCD is 2x – 1
Answer:Given: –
Polynomials 2x3 – 3x2 – 9x + 5, 2x4 – x3 – 10x2 – 11x + 8
And GCD[Greatest Common Divisor] = (x + 7)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
Product of 2 polynomial = (2x3 – 3x2 – 9x + 5) × (2x4 – x3 – 10x2 – 11x + 8)
Product of 2 polynomial = LCM × GCD
LCM = ![](data:image/png;base64,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)
LCM = ![](data:image/png;base64,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)
LCM = ![](data:image/png;base64,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)
LCM = (x3 – 5x – 8)( 2x3 – 3x2 – 9x + 5)
Conclusion: –
The LCM of given polynomials [2x3 – 3x2 – 9x + 5, 2x4 – x3 – 10x2 – 11x + 8] is (x3 – 5x – 8)( 2x3 – 3x2 – 9x + 5)
Question 5.Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(x + 1)2 (x + 2)2, (x + 1) (x + 2), (x + 1)2 (x + 2)
Answer:Given: –
Polynomials p(x) = (x + 1)2 (x + 2)
And GCD[Greatest Common Divisor] = (x + 1) (x + 2)
And LCM[Lowest Common Multiple] = (x + 1)2 (x + 2)2
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (x + 1)2 × (x + 2)2 × (x + 1) × (x + 2)
p(x) × q(x) = LCM × GCD
q(x) = ![](data:image/png;base64,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)
q(x) = ![](data:image/png;base64,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)
q(x) = ![](data:image/png;base64,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)
q(x) = (x + 1)(x + 2)2
Conclusion: –
The other polynomial term q(x) is (x + 1)(x + 2)2
Question 6.Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(4x + 5)3 (3x – 7)3, (4x + 5) (3x – 7)2, (4x + 5)3 (3x – 7)2
Answer:Given: –
Polynomials p(x) = (4x + 5)3 (3x – 7)2
And GCD[Greatest Common Divisor] = (4x + 5) (3x – 7)2
And LCM[Lowest Common Multiple] = (4x + 5)3 (3x – 7)3
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (4x + 5)3 × (3x – 7)3 × (4x + 5) × (3x – 7)2
= (4x + 5)4(3x – 7)5
p(x) × q(x) = LCM × GCD
q(x) = ![](data:image/png;base64,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)
q(x) = ![](data:image/png;base64,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)
q(x) = ![](data:image/png;base64,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)
q(x) = (4x + 5)(3x – 7)3
Conclusion: –
The other polynomial term q(x) is (4x + 5)(3x – 7)3
Question 7.Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(x4 – y4) (x4 + x2y2 + y4), x2 – y2, x4 – y4.
Answer:Given: –
Polynomials p(x) = x4 – y4
And GCD[Greatest Common Divisor] = x2 – y2
And LCM[Lowest Common Multiple] = (x4 – y4)(x4 + x2y2 + y4)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (x4 – y4) (x4 + x2y2 + y4) × (x2 – y2)
p(x) × q(x) = LCM × GCD
q(x) = ![](data:image/png;base64,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)
q(x) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMIAAAAmCAMAAACh6rSXAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgBmOjoAOjo6OjqQOma2OpDbWGY6ZgAAZgA6ZgBmZjoAZjpmZmaQZma2ZpDbZra2Zrb/kDoAkDo6kDpmkLb/kNv/tmYAtmY6tmZmvJA6ttv/tv//25A625Bm27Zm27aQ2/+22////7Zm/7aQ/9uQ/9u2//+2///bbvufEQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACj0lEQVRYR+1Y23LjIAzF3Ta7Ti/ZdnfTe+304mzr1LX//+eKjQAJUAZ3Ji2dMTN5AKGjcwSGCCGmll4GygXDiTUI3tLdZNnBkwakPT9MEIdz4sHWR4wE1iB4i2j/PLZ/DSDteRLCOJyTM/68fATAzaIKSxgM6/NaB45wUVPbf2YVvB5RoULrEBbfhbBOPTTMt6zbZV0t7nFMcABDcwwaIlyUZ7M0qr0eUaBDqxA4kRTCOqnxYb7hJT0z2X4EJGgDQHMu3fUV3R+b87q5MEO45041ofsQCF9QCAuvx/v5/e9/vlc0+awWTZ5hFmW2V2yyfkgZmvmgj3MxvADv5VSmZCEA5VL1VDNTTQgI3YdA+ABhqbtgzRywNrO6OsFrrnzeTmXIlf08upvChA+42NQSo4tCJARCKJhoSjod1b5dcNH2+ZLtSlSzuru1O0uBA1PiIr864zQsFMZzUOhUP0QIfwslzWd9KLeR39qzAn9aRILvgjY4MTooZBWEHwJgYimBhNe77vpAH61YSfkTn4p4IwVcrATHSFGoBOGFUDDRlNS3uc5/dWUWWocmxxcF+pxDLkaCa6QojgQ/xDhK5pQJ7KJ+iF5N+lBVB1N8oyiOnx8CMhVPqTphuawKdByh83qLSwDLQaEzgiFGU8K3OcUvMywv+AcjYikoiuMQDvEBShFEpik7zIC6xL5z22FyJugpA1MGZAa4NwH+rSCJtKH7iHsTGMbRFZkEb0sC/SvnnhFgHBeWKYlAvLhnBDPOvJN8tRxU8D64zwhQ8P7W49v/in6VElrwOs8IpkCGcVU6pdZGFrzOa00SakYWvOlKiC54U9xIH6nBk9g+iMTogjc1AZLP6II3QQ1TwZvgonwWpXdGPWhdx161KwAAAABJRU5ErkJggg==)
q(x) = ![](data:image/png;base64,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)
q(x) = (x4 + x2y2 + y4)(x2 – y2)
Conclusion: –
The other polynomial term q(x) is (x4 + x2y2 + y4)(x2 – y2)
Question 8.Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(x3 – 4x) (5x + 1), (5 x2 + x), (5 x3 – 9 x2 – 2x).
Answer:Given: –
Polynomials p(x) = (5x3 – 9x2 – 2x)
And GCD[Greatest Common Divisor] = (5x2 + x)
And LCM[Lowest Common Multiple] = (x3 – 4x)(5x + 1)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (x3 – 4x) (5x + 1) × (5x2 + x)
= x(x2 – 4)(5x + 1) × x(5x + 1)
= x2(x + 2)(x – 2)(5x + 1)(5x + 1)
p(x) = (5 x3 – 9 x2 – 2x)
= x(5x2 – 9x – 2)
= x(5x2 – 10x + x – 2)
= x[5x(x – 2) + 1(x – 2)]
= x(5x + 1)(x – 2)
p(x) × q(x) = LCM × GCD
q(x) = ![](data:image/png;base64,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)
q(x) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJkAAAAmCAMAAADgHskoAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjqQOmZmOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZjqQZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkLb/kNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/7aQ/9uQ/9u2//+2///bpV5engAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC1ElEQVRYR+1X7VLbMBCUUwgxUBoCaaGOQ9uEghPa2jXS+z9aT98nWXZk7B/MNJ5hhslptaeVLO8ScnwGKcAekuT0edAU8WC27kFGb5/ochU/+aCRguy6e4qXuycz4OVj2T54d2+KGPNmAP3cskGKqLCN00XyyeusTpGIte6bY4okSSabtrbY4wxKnYD6rlUFgTNgQuiXkuWuwGx9ibdXrUJgiua+s1z/trta8s5IB6C6L+uvcmEWpxcqlg5/+3SyqdPpn2XjUBYZJ1P1ktQXQn/BJzszJZ+hEJ01AYZsAZorGUxniEi1W03LYh7QtppLkK6zNd8++RvfzbkthTsLAMxk9iBYzSyRFrI4UcLy4XDcxLOCI8ry7BvfOlmXU5iJxCHUUIOSjFKzEMAg5EAXZ4g0y+5yGpCMy5IkvKLqHpE4lQiKz0uwM3mMG2QIZ4hUZ39/sPzU3h1WZ6+ONof/yzXDUMTAtmd8qTCK3oLwoK0GOAjJZHF6NraWp3mXzthWaNN4fqUfnm0dHWj2CJdG5kItg1AbZgYAx3ASBQiRGZxppL4wb0+gqcBP+hKQr2jEA4Cf8NKoFUUA1BD59s+jAfbijMT0BqhWFC7uS8NP7shfp1ZFEFG0aseBoIC8U9/hc9ydowLvWoHuNNMrfgxbZ+M27k4z3fFjzIs06PvdNFMvkwnyeMSNH7/BLtsvGnL4wwSTWcHz/V6aoYtMmCL9uPGD3nwnexRnUCoa1hry/cbGe2mmArNEttfayKP4YXo93/h5YlhXgHZ8v7Lx1Esz9fmGvOYzXUXxQ9Pz5bl5YnBjnu93bLw18eAVr3jmxokCW/y9OGZOnhixsxYbrxnoDZjEcKLQLtDJEyN11mHjFcMrl6wlUVSgWGVDg8wTIzwHbLxgYPlJ1poo5LaukM2PtesHuu9t4w+pMdqt0ScrHGqK18e7aQmJzgoxjY35dYrh+2/G/AM5cHCO7oVgSAAAAABJRU5ErkJggg==)
q(x) = ![](data:image/png;base64,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)
q(x) = x(x + 2)(5x + 1)
Conclusion: –
The other polynomial term q(x) is x(x + 2)(5x + 1)
Question 9.Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(x – 1) (x – 2) (x2 – 3x + 3), (x – 1), (x3 – 4 x2 + 6x – 3).
Answer:Given: –
Polynomials p(x) = (x3 – 4 x2 + 6x – 3)
And GCD[Greatest Common Divisor] = (x – 1)
And LCM[Lowest Common Multiple] = (x – 1)(x – 2)(x2 – 3x + 3)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (x – 1) (x – 2) (x2 – 3x + 3) × (x – 1)
= (x – 1)2 (x – 2) (x2 – 3x + 3)
p(x) × q(x) = LCM × GCD
q(x) = ![](data:image/png;base64,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)
q(x) =
![](data:image/png;base64,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)
q(x) =
![](data:image/png;base64,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)
q(x) =
![](data:image/png;base64,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)
q(x) = (x – 1)(x – 2)
Conclusion: –
The other polynomial term q(x) is (x – 1)(x – 2)
Question 10.Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
2(x + 1) (x2 – 4), (x + 1), (x + 1) (x – 2).
Answer:Given: –
Polynomials p(x) = (x + 1)(x – 2)
And GCD[Greatest Common Divisor] = (x + 1)
And LCM[Lowest Common Multiple] = 2(x + 1) (x2 – 4)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = 2(x + 1) (x2 – 4) × (x + 1)
= 2(x + 1)2 (x2 – 22)
= 2(x + 1)2 (x – 2)(x + 2)
p(x) × q(x) = LCM × GCD
q(x) = ![](data:image/png;base64,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)
q(x) = ![](data:image/png;base64,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)
q(x) = ![](data:image/png;base64,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)
q(x) = 2(x + 2)(x + 1)
Conclusion: –
The other polynomial term q(x) is 2(x + 2)(x + 1)
Find the LCM of each pair of the following polynomials.
x2 – 5x + 6, x2 + 4x – 12 whose GCD is x – 2.
Answer:
Given: –
Polynomials x2 – 5x + 6, x2 + 4x – 12
And GCD[Greatest Common Divisor] = (x – 2)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
Product of 2 polynomial = (x2 – 5x + 6) × (x2 + 4x – 12)
= ( x2 – 2x – 3x + 6)(x2 + 6x – 2x – 12)
= (x(x – 2) – 3(x – 2))(x(x + 6) – 2(x + 6))
= (x – 3)(x – 2)(x – 2)(x + 6)
Product of 2 polynomial = LCM × GCD
LCM =
LCM =
LCM = (x – 3)(x – 2)(x + 6)
Conclusion: –
The LCM of polynomial [x2 – 5x + 6, x2 + 4x – 12] is
(x – 3)(x – 2)(x + 6)
Question 2.
Find the LCM of each pair of the following polynomials.
x4 + 3 x3 + 6 x2 + 5x + 3, x4 + 2 x2 + x + 2 whose GCD is x2 + x + 1
Answer:
Given: –
Polynomials x4 + 3 x3 + 6 x2 + 5x + 3 , x4 + 2 x2 + x + 2
And GCD[Greatest Common Divisor] = (x2 + x + 1)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
Product of 2 polynomial = (x4 + 3x3 + 6x2 + 5x + 3) × (x4 + 2x2 + x + 2)
Product of 2 polynomial = LCM × GCD
LCM =
LCM =
LCM =
LCM = (x2 + 2x + 3)(x4 + 2x2 + x + 2)
Conclusion: –
The LCM of polynomial [x4 + 3 x3 + 6 x2 + 5x + 3, x4 + 2 x2 + x + 2] is
(x2 + 2x + 3)(x4 + 2x2 + x + 2)
Question 3.
Find the LCM of each pair of the following polynomials.
2x3 + 15x2 + 2x – 35, x3 + 8x2 + 4x – 21 whose GCD is x + 7.
Answer:
Given: –
Polynomials 2x3 + 15x2 + 2x – 35 , x3 + 8x2 + 4x – 21
And GCD[Greatest Common Divisor] = (x + 7)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
Product of 2 polynomial = (2x3 + 15x2 + 2x – 35) × (x3 + 8x2 + 4x – 21)
Product of 2 polynomial = LCM × GCD
LCM =
LCM =
LCM =
LCM = (2x2 + x – 5)(x3 + 8x2 + 4x – 21)
Conclusion: –
The LCM of given polynomials [2x3 + 15x2 + 2x – 35 , x3 + 8x2 + 4x – 21] is (2x2 + x – 5)(x3 + 8x2 + 4x – 21)
Question 4.
Find the LCM of each pair of the following polynomials.
2x3 – 3x2 – 9x + 5, 2x4 – x3 – 10x2 – 11x + 8 whose GCD is 2x – 1
Answer:
Given: –
Polynomials 2x3 – 3x2 – 9x + 5, 2x4 – x3 – 10x2 – 11x + 8
And GCD[Greatest Common Divisor] = (x + 7)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
Product of 2 polynomial = (2x3 – 3x2 – 9x + 5) × (2x4 – x3 – 10x2 – 11x + 8)
Product of 2 polynomial = LCM × GCD
LCM =
LCM =
LCM =
LCM = (x3 – 5x – 8)( 2x3 – 3x2 – 9x + 5)
Conclusion: –
The LCM of given polynomials [2x3 – 3x2 – 9x + 5, 2x4 – x3 – 10x2 – 11x + 8] is (x3 – 5x – 8)( 2x3 – 3x2 – 9x + 5)
Question 5.
Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(x + 1)2 (x + 2)2, (x + 1) (x + 2), (x + 1)2 (x + 2)
Answer:
Given: –
Polynomials p(x) = (x + 1)2 (x + 2)
And GCD[Greatest Common Divisor] = (x + 1) (x + 2)
And LCM[Lowest Common Multiple] = (x + 1)2 (x + 2)2
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (x + 1)2 × (x + 2)2 × (x + 1) × (x + 2)
p(x) × q(x) = LCM × GCD
q(x) =
q(x) =
q(x) =
q(x) = (x + 1)(x + 2)2
Conclusion: –
The other polynomial term q(x) is (x + 1)(x + 2)2
Question 6.
Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(4x + 5)3 (3x – 7)3, (4x + 5) (3x – 7)2, (4x + 5)3 (3x – 7)2
Answer:
Given: –
Polynomials p(x) = (4x + 5)3 (3x – 7)2
And GCD[Greatest Common Divisor] = (4x + 5) (3x – 7)2
And LCM[Lowest Common Multiple] = (4x + 5)3 (3x – 7)3
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (4x + 5)3 × (3x – 7)3 × (4x + 5) × (3x – 7)2
= (4x + 5)4(3x – 7)5
p(x) × q(x) = LCM × GCD
q(x) =
q(x) =
q(x) =
q(x) = (4x + 5)(3x – 7)3
Conclusion: –
The other polynomial term q(x) is (4x + 5)(3x – 7)3
Question 7.
Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(x4 – y4) (x4 + x2y2 + y4), x2 – y2, x4 – y4.
Answer:
Given: –
Polynomials p(x) = x4 – y4
And GCD[Greatest Common Divisor] = x2 – y2
And LCM[Lowest Common Multiple] = (x4 – y4)(x4 + x2y2 + y4)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (x4 – y4) (x4 + x2y2 + y4) × (x2 – y2)
p(x) × q(x) = LCM × GCD
q(x) =
q(x) =
q(x) =
q(x) = (x4 + x2y2 + y4)(x2 – y2)
Conclusion: –
The other polynomial term q(x) is (x4 + x2y2 + y4)(x2 – y2)
Question 8.
Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(x3 – 4x) (5x + 1), (5 x2 + x), (5 x3 – 9 x2 – 2x).
Answer:
Given: –
Polynomials p(x) = (5x3 – 9x2 – 2x)
And GCD[Greatest Common Divisor] = (5x2 + x)
And LCM[Lowest Common Multiple] = (x3 – 4x)(5x + 1)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (x3 – 4x) (5x + 1) × (5x2 + x)
= x(x2 – 4)(5x + 1) × x(5x + 1)
= x2(x + 2)(x – 2)(5x + 1)(5x + 1)
p(x) = (5 x3 – 9 x2 – 2x)
= x(5x2 – 9x – 2)
= x(5x2 – 10x + x – 2)
= x[5x(x – 2) + 1(x – 2)]
= x(5x + 1)(x – 2)
p(x) × q(x) = LCM × GCD
q(x) =
q(x) =
q(x) =
q(x) = x(x + 2)(5x + 1)
Conclusion: –
The other polynomial term q(x) is x(x + 2)(5x + 1)
Question 9.
Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
(x – 1) (x – 2) (x2 – 3x + 3), (x – 1), (x3 – 4 x2 + 6x – 3).
Answer:
Given: –
Polynomials p(x) = (x3 – 4 x2 + 6x – 3)
And GCD[Greatest Common Divisor] = (x – 1)
And LCM[Lowest Common Multiple] = (x – 1)(x – 2)(x2 – 3x + 3)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = (x – 1) (x – 2) (x2 – 3x + 3) × (x – 1)
= (x – 1)2 (x – 2) (x2 – 3x + 3)
p(x) × q(x) = LCM × GCD
q(x) =
q(x) =
q(x) =
q(x) =
q(x) = (x – 1)(x – 2)
Conclusion: –
The other polynomial term q(x) is (x – 1)(x – 2)
Question 10.
Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.
2(x + 1) (x2 – 4), (x + 1), (x + 1) (x – 2).
Answer:
Given: –
Polynomials p(x) = (x + 1)(x – 2)
And GCD[Greatest Common Divisor] = (x + 1)
And LCM[Lowest Common Multiple] = 2(x + 1) (x2 – 4)
Formula used: –
The product of 2 polynomial is equal to product of their LCM
and GCD.
Product of 2 polynomial = LCM × GCD
p(x) × q(x) = LCM × GCD
LCM × GCD = 2(x + 1) (x2 – 4) × (x + 1)
= 2(x + 1)2 (x2 – 22)
= 2(x + 1)2 (x – 2)(x + 2)
p(x) × q(x) = LCM × GCD
q(x) =
q(x) =
q(x) =
q(x) = 2(x + 2)(x + 1)
Conclusion: –
The other polynomial term q(x) is 2(x + 2)(x + 1)
Exercise 3.9
Question 1.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
[dividing numerator and denominator by 3x]
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 2.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEgAAAAtCAMAAAA6Ct2cAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkNv/tmYAtpBmttv/tv//25A627Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bK9T2NwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABq0lEQVRIS+WW61LDIBCFoVrjrdVYq5WqjS1VS4T3fzxXAoGEa2xnHKf8aZNhP84uCycI/YfxOcN4egCh/O4VbUer35DE29WmG1ef54LoZN2GCnKz661Pb9OC3u/lYvXlk5oriBO17ZdIED1bL/A1wydNGq16pl4YDdQptQOqJ2uq4+hYJiTI2c9PhfF4RzGGlRlwWFeBqwihFqQk1UUTU8FGyfm8xDAGgFRtmNppQUBRqMZRRaiSOWkQqotrbxtC0mp0m6tNDVWySOaZhrswoUiCWkV8/uDsn97JISCxXPFSJusZOTVSuyYqKDTDgWp7QboSvJRRTR8JgkegCAdIDkg8F7ADp492a7udHcgv9Fp1NogJdU8e0NwUntOfh5CzzOmHAjn3UT7Ivo/yo450Znu09/xzpOU7TNraE/alieVF3xJtpHLZjFXowvFWE2VcNkliU9+dqMJsl/WSjNny+UaQxUv/08JEGevxk7TZgnXDaCzdOxIg22wjqQE6AbLN9qPQJudz2hQIRcy2k2IKFDPbIaCo2Q4Axc22C4p99SbM1oBsl0229t9N+AbM3CgzZlRD6wAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
The like terms are cancelled.
which is the required answer.
Question 3.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
=x – 1 which is the required answer.
Question 4.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADsAAAAtCAMAAAAndOMHAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOmY6Oma2OpDbZgAAZgA6ZjoAZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtpA6ttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bm/qn2wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABhklEQVRIS+1V2VbCMBBNikpdsLhAwTZqoBpo/v//nMnWzTSxHg88OA89tM2dOwu9l5DTx+6GJs/TypCbguyTchoYUIfrydjjduWjldUDpZeYmVMVV5+do4xeeLGMrsgxm70TwtZYYIrXTlTDR+Y9mytKi+AqC7JDGWtSL8vv0rVzC4uts3t8zmC0Msd8uzS0I4ZsGFzvQ+bAq7IEwzUpc+xAtX236MOgExOthcvcUrjabQEhWkcGfdoF1Y9Pto1ROJvbjR5SU4DclHVmyh/DcuQSeryGSzK4FTQ4LPVvVYeJWRAhOU2AlwbBZnqIFb/4YEJD/X9/RhNwH8zPf5xRF6cp5QMkfCBBcaXUy1FTUubwNpLKb0o8KeD7H1Mor+JqwXCyoekbc4Cbvbddrc8o6O2qG3Mg3D8pzdjHNuYgACoGHmd42Az857Vvn9YcUPbQ2EwnPXOQWxjzy0CRjTtFbNIpuz0baw5wvs8Saw5qRV0TiTUHgFbporMhGWsO4Le3RcRI/vbIF788HGhWxN04AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
The like terms are cancelled.
which is the required answer.
Question 5.Simplify the following into their lowest forms.
(Hint : x4 + x2 + 1 =(x2 + 1)2 – x2)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
=x2 + x + 1 which is the required answer.
Question 6.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution.
Question 7.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution.
Question 8.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
=x + 3
Required solution.
Question 9.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required answer.
Question 10.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
=1 required solution.
Question 11.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The like terms are cancelled.
![](data:image/png;base64,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)
Required solution
Question 12.Simplify the following into their lowest forms.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAO8AAAAwCAMAAADOxt1xAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGY6kNv/tmYAtpBmttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bgpugWQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC7klEQVRoQ+1a2VbbMBCVoC3pQtvQ1S0uNdhE//+F1W47HkszynBiN/ZLOGGuZ+4sWrgIsT0XmAH15/1fLO3m9iGaEnDng02JqerTE5auEN27n96YhDsXDKL7GcX28cu9teveuk9V5XHq8U7K19aeAmukfd7oMlBggDeAWXuNaebnOxnsGhOHEBhcLX+I570DUmCmhbqdbSQKbOoNKu8Norzd7UMT+LqUqwqBq41NI23gBJgNyPsjwKbeAGY+jbVpH91HYTinlpGva2Q0TrTupUTYYe/GhQgbe4Pa+cqNY60/VTVPN+TbmJqqtVicqH1j0GCNf/9J3hJ8VaXr61IKP7G+ojYDHPhmcb4RdJoosH5cSLDQdt4bwKQfy92Hj+H3urv9E7Ic58nU1wQO4SBYv4yTYH4ITvSWqK9m0HODKjxT3wxusKyNCpVzZ43doA3bgugtxffw9VtyZxrwHc6vSOPqm3iWGc1vBtbt4mhRYMfeAL5hvNSv+8M+tcVEvm7lxOHs7tna2CmwuMbRYBNvAF/fcKrWQbUysWDFdh/tiGmcNbUmR/tvxl3YjGiwqTdoLu05SVXyStdXzhFWv3d6AXv13bxgeOLJ4PwCZvkSYHHxp8EAb2CBE0WFBoBwfh7AKQdhPhhUYNI9Z/33I93Ml3X/hWq+fbdlYMvA8jIAneiXF+ULRhQvQv/hDy+YtqW++uL7eamFOVdc6zh7sWVnHWdrRrqIu1OhGqCDZJIs2OiiNARRl6kBgk2yYOOL0hCcN7oawCZZsNElaAhlagCPZMHHF68hlKkBM3/yzEodY8mCn29WQyhUA3gkCz6+WA0hiFTa85yIAB7sjvgC7kDYkWTBxzf0c04McCztQ1EDhhIcr4hQmILIN6M9FKoBc5IjUbIoJAfAcBqCLmoUYkgiQs+XAsOICIUpQGoPZWqA2bSDQneSZFFIDoKhtIdCNYBPsuAjjPkfnKG35YgIhTm4sPvRWrSHwmpusC0Da8rAPxzyg8ojvteBAAAAAElFTkSuQmCC)
The like terms are cancelled.
=x – 2 required solution.
Simplify the following into their lowest forms.
Answer:
[dividing numerator and denominator by 3x]
Question 2.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
which is the required answer.
Question 3.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
=x – 1 which is the required answer.
Question 4.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
which is the required answer.
Question 5.
Simplify the following into their lowest forms. (Hint : x4 + x2 + 1 =(x2 + 1)2 – x2)
Answer:
The like terms are cancelled.
=x2 + x + 1 which is the required answer.
Question 6.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
Required solution.
Question 7.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
Required solution.
Question 8.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
=x + 3
Required solution.
Question 9.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
Required answer.
Question 10.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
=1 required solution.
Question 11.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
Required solution
Question 12.
Simplify the following into their lowest forms.
Answer:
The like terms are cancelled.
=x – 2 required solution.