Real Number System Class 9th Mathematics Term 3 Tamilnadu Board Solution

Class 9^th Mathematics Term 3 Tamilnadu Board Solution

Exercise 1.1

1. root 8 x root 6 Identify which of the following are surds and which are not…
2. root 90 Identify which of the following are surds and which are not with…
3. root 180 x root 5 Identify which of the following are surds and which are not…
4. 4 root 5 / root 8 Identify which of the following are surds and which are not…
5. cube root 4 x cube root 16 Identify which of the following are surds and which…
6. (10+ 3)(2 + 5) Simplify
7. (root 5 + root 3)^2 Simplify
8. (root 13 - root 2) (root 13 + root 2) Simplify
9. (8 + root 3) (8 - root 3) Simplify
10. 5 root 75+8 root 108 - 1/2 root 48 Simplify the following.
11. 7 root 2+6 cube root 16 - cube root 54 Simplify the following.
12. 4 root 72+8 root 50-7 root 128 Simplify the following.
13. 2 cube root 40+3 cube root 625-4 cube root 320 Simplify the following.…
14. cube root 108 Express the following surds in its simplest form.
15. root 98 Express the following surds in its simplest form.
16. root 192 Express the following surds in its simplest form.
17. cube root 625 Express the following surds in its simplest form.
18. 6 root 5 Express the following as pure surds.
19. 5 cube root 4 Express the following as pure surds.
20. 3 root [4]5 Express the following as pure surds.
21. 3/4 root 8 Express the following as pure surds.
22. root 5 x root 18 Simplify the following.
23. cube root 7 x cube root 8 Simplify the following.
24. root [4]8 x root [4]12 Simplify the following.
25. cube root 3 x cube root 5 Simplify the following.
26. root 2 cube root 3 Which is greater ?
27. Which is greater ?
28. root 3 arroot [4]10 Which is greater ?
29. root [4]5 , root 3 cube root 4 Arrange in descending and ascending order.…
30. cube root 2 , cube root 4 , root [4]4 Arrange in descending and ascending…
31. cube root 2 , root [9]4 , root [6]3 Arrange in descending and ascending order.…

Exercise 1.2

1. 3 root 2 Write the rationalizing factor of the following.
2. root 7 Write the rationalizing factor of the following.
3. root 75 Write the rationalizing factor of the following.
4. 2 cube root 5 Write the rationalizing factor of the following.
5. 5-4 root 3 Write the rationalizing factor of the following.
6. root 2 + root 3 Write the rationalizing factor of the following.
7. root 5 - root 2 Write the rationalizing factor of the following.
8. 2 + root 3 Write the rationalizing factor of the following.
9. 3/root 5 Rationalize the denominator of the following
10. 2/3 root 3 Rationalize the denominator of the following
11. 1/root 12 Rationalize the denominator of the following
12. 2 root 7/root 11 Rationalize the denominator of the following
13. 3 cube root 5/cube root 9 Rationalize the denominator of the following…
14. 1/11 + root 3 Simplify by rationalizing the denominator.
15. 1/9+3 root 3 Simplify by rationalizing the denominator.
16. 1/root 11 + root 13 Simplify by rationalizing the denominator.
17. root 5+1/root 5-1 Simplify by rationalizing the denominator.
18. 3 - root 3/2+5 root 3 Simplify by rationalizing the denominator.
19. 1/root 2 Find the values of the following upto 3 decimal places. Given that √2≈…
20. 6/root 3 Find the values of the following upto 3 decimal places. Given that √2≈…
21. 5 - root 3/root 3 Find the values of the following upto 3 decimal places. Given…
22. root 10 - root 5/root 2 Find the values of the following upto 3 decimal places.…
23. 3 - root 5/3+2 root 5 Find the values of the following upto 3 decimal places.…
24. root 5 + root 2/root 5 - root 2 Find the values of the following upto 3 decimal…
25. root 3+1/root 3-1 Find the values of the following upto 3 decimal places. Given…
26. 1/root 10 + root 5 Find the values of the following upto 3 decimal places.…
27. If 5 + root 6/5 - root 6 = a+b root 6 find the values of a and b.…
28. If (root 3+1)^2/4-2 root 3 = a+b root 3 find the values of a and b.…
29. If root 5+1/root 3-1 + root 5-1/root 5+1 = a+b root 5 find the values of a and…
30. If 4 + root 5/4 - root 3 - 4 - root 5/4 + root 5 = a+b root 5 find the values of…
31. If x = 2 + root 3 find the values of x^2 + 1/x^2
32. x = root 3+1 , find the values of (x - 2/x)^2

Exercise 1.3

1. Using division algorithm, find the quotient and remainder of the following pairs.(i) 10, 3…

Exercise 1.4

1. Which one of the following is not a surd?A. cube root 8 B. cube root 30 C. root…
2. The simplest form of root 50 isA. 5 root 10 B. 5 root 2 C. 10 root 5 D. 25 root…
3. root [4]11 is equal toA. root [8] 11^2 B. root [8] 11^4 C. root [8] 11^8 D. root…
4. 2/root 2 is equal toA. 2 root 2 B. root 2 C. root 2/2 D. 2
5. The rationlizing factor of 5/cube root 3 isA. cube root 6 B. cube root 3 C. cube…
6. Which one of the following is not true?A. root 2 is an irrational number B. root…
7. The order and radicand of the surd root [8]12 are respectivelyA. 8,12 B. 12,8 C.…
8. The surd having radicand 9 and order 3 isA. root [9]3 B. cube root 27 C. cube…
9. 5 cube root 3 represents the pure surdA. cube root 15 B. cube root 375 C. cube…
10. Which one of the following is not true?A. root 2 is an irrational number B. If…
11. Which one of the following is not true?A. When x is not a perfect square, root…
12. (root 5-2) (root 5+2) is equal toA. 1 B. 3 C. 23 D. 21

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Exercise 1.1

Question 1.

Identify which of the following are surds and which are not with reasons

Answer:

Given, √8 × √6

Need to find √8 × √6 is surd or not

⇒ we know √a × √b = 

⇒ √8 × √6 can be written as 

⇒ 

= 4√3, which is irrational number

⇒ since, 4√3 cannot be expressed as squares or cubes of any rational numbers

⇒ Hence, √8 × √6 is surd

Question 2.

Identify which of the following are surds and which are not with reasons

Answer:

Given, 

Need to find  is surd or not

⇒ we know √a × √b = 

⇒  can be written as 

⇒ 

⇒ 3 , which is irrational numbers

since, 3cannot be expressed as squares or cubes of any rational numbers

⇒ Hence, it is surd.

Question 3.

Identify which of the following are surds and which are not with reasons

Answer:

Given,  × √5

Need to find  × √5 is surd or not

⇒ we know √a × √b = 

⇒  × √5 can be written as 

⇒ 

⇒ 

⇒ 

= 2 × 3 × 5 = 30 which is not a irrational number as it can be expressed in squares form

Hence, it is not a surd

Question 4.

Identify which of the following are surds and which are not with reasons

Answer:

Given, 4√5 ÷ √8

Need to find 4√5 ÷ √8 is surd or not

⇒ we know √a ÷ √b = 

⇒ 4√5 ÷ √8 can be written as 

⇒ 

= 

= 

= is irrational number

since, cannot be expressed as squares or cubes of any rational numbers

⇒ Hence, it is surd.

Question 5.

Identify which of the following are surds and which are not with reasons

Answer: 

Given, ∛4 × 

Need to find ∛4 ×  is surd or not

⇒ we know √a × √b = 

⇒ ∛4 ×  can be written as 

⇒ 

= 2× 2 × 2 = 8 is not irrational number as it can be expressed in cubes form

⇒ Hence, it is not a surd

Question 6.

Simplify
(10+ √3)(2 + √5)

Answer:

Given, (10+ √3)(2 + √5)

Need to simplify it

⇒ the given expression can be written in expanded form

⇒ 20+10√5 + 2√3 +(√3 × √5)

⇒ We know √a × √b = 

= 20+10√5 + 2√3 + 

Hence, (10+ √3)(2 + √5) is simplified into 20+10√5 + 2√3 + 

Question 7.

Simplify

Answer:

Given, (√5+√3)²

Need to simplify it

⇒ we know that (a+b)² = a²+2ab+b²

⇒ simplifying the given expression we get

⇒ (√5)²+2(√5)(√3)+ (√3)²

= 5+2+3

= 8+2

Hence, (√5+√3)² is simplified into 8+2

Question 8.

Simplify

Answer:

Given, ( – √2) ( + √2)

Need to simplify it

⇒ we know that (a–b)(a+b) = a² – b²

⇒ the given expression can be written in this form

⇒ ()² –(√2)²

= 13 –2

= 11

Hence, ( – √2) ( + √2) is simplified into 11

Question 9.

Simplify

Answer:

Given, (8+√3) (8 – √3)

Need to simplify it

⇒ we know that (a–b)(a+b) = a² – b²

⇒ the given expression can be written in this form

⇒ (8)² –(√3)²

= 64–3

= 61

Hence, (8+√3) (8 – √3) is simplified into 61

Question 10.

Simplify the following.

Answer: 

Given, 5 + 8 – 

Need to simplify it

⇒ the given expression is written as follows

⇒ 5 + 8 – 

= 25√3 + 48√3 – 2√3

= (25 +48 –2)√3

= 71√3

Hence, 5 + 8 – is simplified into 71√3

Question 11.

Simplify the following.

Answer:

Given, 7∛2 + 6 – 

Need to simplify it

⇒ the given expression is written as follows

⇒ 7∛2 + 6 – 

= 7∛2 + 12∛2 – 3∛2

= 16∛2

Hence, 7∛2 + 6 –  is simplified into 16∛2

Question 12.

Simplify the following.

Answer:

Given, 4√72 – √50 – 7√128

Need to simplify it

⇒ the given expression is written as follows

⇒ 4 –  – 7

= 24√2 –5√2 –56√2

= (24–5–56)√2

= –37√2

Hence, 4 –  – 7is simplified into –37√2

Question 13.

Simplify the following.

Answer:

Given

Need to simplify it

⇒ the given expression is written as follows

⇒ 2 + 3 – 4

⇒ 4∛5 +15∛5 – 16∛5

= (4 + 15 –16)∛5

= 3∛5

Hence, 2 + 3 – 4is simplified into 3∛5

Question 14.

Express the following surds in its simplest form.

Answer:

Given, 

Need to simplify it

⇒ the given number can be written as follows

⇒ 

= 3∛4

Hence, is simplified into 3∛4

Question 15.

Express the following surds in its simplest form.

Answer:

Given, 

Need to simplify it

⇒ the given number can be written as follows

⇒ 

= 7√2

Hence, is simplified into 7√2

Question 16.

Express the following surds in its simplest form.

Answer:

Given, 

Need to simplify it

⇒ the given number can be written as follows

⇒ 

= 8√3

Hence, is simplified into 8√3

Question 17.

Express the following surds in its simplest form.

Answer:

Given, 

Need to simplify it

⇒ the given number can be written as follows

⇒ 

= 5∛5

Hence, is simplified into 5∛5

Question 18.

Express the following as pure surds.

Answer:

Given, 6√5

Need to express it as pure surd

⇒ 6√5 can be expressed as (√6)² .√5

⇒ 

= 

= 

∴  is pure surd

∵ a surd with rational coefficient as unity is pure surd

Hence, 6√5 is expressed as pure surd

Question 19.

Express the following as pure surds.

Answer:

Given, 5∛4

Need to express it as pure surd

⇒ 5∛4 can be expressed as (∛5)³.∛4

⇒ 

= 

= 

∵ a surd with rational coefficient as unity is pure surd

∴  is a pure surd

Hence, 5∛4 is expressed as pure surd

Question 20.

Express the following as pure surds.

Answer:

Given, 3∜5

Need to express it as pure surd

⇒ 3∜5 can be written as 

⇒ 

⇒ 

= 

∵ a surd with rational coefficient as unity is pure surd

∴  is a pure surd

Hence, 3∜5 is expressed as pure surd

Question 21.

Express the following as pure surds.

Answer:

Given, 

Need to express it as pure surd

⇒  can be expressed as follows

⇒ 

⇒ 
= 

= 

∵ a surd with rational coefficient as unity is pure surd

∴  is pure surd

Hence,  is expressed as pure surd

Question 22.

Simplify the following.

Answer:

Given, √5 ×

Need to simplify it

⇒ we know √a × √b = 

⇒ √5 ×  can be written as 

⇒ 

= 

= 3

Hence, √5 × is simplified into 3

Question 23.

Simplify the following.

Answer:

Given, ∛7 × ∛8

Need to simplify it

⇒ we know √a × √b = 

⇒ ∛7 × ∛8 can be expressed as 

⇒ 

= 2∛7

Hence, ∛7 × ∛8 is simplified into 2∛7

Question 24.

Simplify the following.

Answer:

Given,∜8 × 

Need to simplify it

⇒ we know √a × √b = 

⇒ ∜8 ×  can be expressed as 

⇒ 

⇒ 

= 

= 2∜6

Hence, ∜8 × is simplified into 2∜6

Question 25.

Simplify the following.

Answer:

Given, ∛3 × 

Need to simplify it

⇒ we know √a × √b = 

⇒ ∛3 × can be expressed as 

⇒ 

= 

= 

= 

= 

Hence, ∛3 ×  is simplified into 

Question 26.

Which is greater ?

Answer:

Given, √2 or ∛3

Need to find the greater number

⇒ The order of the given irrational number is 2 and 3

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 2 and 3 is 6

⇒ now, each irrational number is converted into order of 6

⇒ √2 = 

= 

= 

= 

and

⇒ ∛3 = 

= 

= 

= 

⇒  is greater than 

∴ ∛3 > √2

Hence, ∛3 is greater than √2

Question 27.

Which is greater ?

Answer:

Given, ∛3 or ∜4

Need to find the greater number

⇒ The order of the given irrational number is 3 and 4

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 3 and 4 is 12

⇒ now, each irrational number is converted into order of 12

⇒ ∛3 = 

= 

= 

= 

And

⇒ ∜4 = 

= 

= 

= 

= 

∴  is greater than 

⇒ ∛3 is greater than ∜4

∴ ∛3 > ∜4

Hence, ∛3 is greater than ∜4

Question 28.

Which is greater ?

Answer:

Given, √3 or 

Need to find the greater number

⇒ The order of the given irrational number is 2 and 4

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 2 and 4 is 4

⇒ now, each irrational number is converted into order of 4

⇒ √3 = 

= 

= 

= 

And

⇒ 

∴ the greater number between  and  is 

⇒  > 

∴  > √3

Hence, is greater than √3

Question 29.

Arrange in descending and ascending order.

Answer:

Given, ∜5, √3 , ∛4

Need to arrange the given numbers in ascending and descending order

⇒ The order of the given irrational number is 4, 2 and 3 respectively.

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 4,2 and 3 is 12

⇒ now, each irrational number is converted into order of 12

⇒ ∜5 = 

= 

= 

= 

= 

And

⇒ √3 = 

= 

= 

= 

And

⇒ ∛4 = 

= 

= 

= 

∴ Ascending order is ∜5, ∛4 , √3

∴ Descending order is √3, ∛4 , ∜5

Question 30.

Arrange in descending and ascending order.

Answer:

Given, ∛2, ∛4, ∜4

Need to arrange the given numbers in ascending and descending order

⇒ The order of the given irrational number is 3, 3 and 4 respectively.

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 3,3 and 4 is 12

⇒ now, each irrational number is converted into order of 12

⇒ ∛2 = 

= 

= 

= 

= 

And

⇒ ∛4 = 

= 

= 

= 

And

⇒ ∜4 = 

= 

= 

= 

∴ Ascending order is ∛2, ∜4, ∛4

∴ Descending order is ∛4, ∜4, ∛2

Question 31.

Arrange in descending and ascending order.

Answer:

Given, ∛2, , 

Need to arrange the given numbers in ascending and descending order

⇒ The order of the given irrational number is 3, 9 and 6 respectively.

⇒ now, we have to convert each irrational number into irrational number with same order

⇒ First we need to do the LCM of 3, 9 and 6 is 18

⇒ now, each irrational number is converted into order of 18

⇒ ∛2 = 

= 

= 

= 

And

⇒  = 

= 

= 

= 

And

⇒  = 

= 

= 

= 

∴ Ascending order is  ,  , ∛2

∴ Descending order is ∛2 ,

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Exercise 1.2

Question 1.

Write the rationalizing factor of the following.

Answer:

Given, 3√2

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ 3√2 × √2 = (3)(√2)² = 6 is rational number

Hence, rationalizing factor of 3√2 is √2

Question 2.

Write the rationalizing factor of the following.

Answer:

Given, √7

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ √7 × √7 = 7

∴ √7 is rationalizing factor

Question 3.

Write the rationalizing factor of the following.

Answer:

Given, 

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒  can be written as 

⇒ 

∴  × √3

⇒ (5)(3) = 15

Hence, √3 is rationalizing factor

Question 4.

Write the rationalizing factor of the following.

Answer:

Given, 2∛5

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ 2∛5 × 

= 2( )

= 2 ( )

= 2 (5) = 10 is rational number

Hence,  is rationalizing factor

Question 5.

Write the rationalizing factor of the following.

Answer:

Given, 5–4√3

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ (5–4√3)(5+4√3)

⇒ 25+20√3–20√3–(16)(3)

= –23 is rational number

∴ Rationalizing factor of (5–4√3) is (5 + 4√3)

Question 6.

Write the rationalizing factor of the following.

Answer:

Given, √2 +√3

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ (√2 + √3)(√2 – √3)

= 4 –√6 + √6 –3

= 1 is a rational number

∴ Rationalizing factor of (√2 + √3) is (√2 – √3)

Question 7.

Write the rationalizing factor of the following.

Answer:

Given, √5 –√2

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ (√5 – √2)(√5 + √2)

= 25 +  –  – 2

= 23 is a rational number

∴ Rationalizing factor of (√5 – √2) is (√5 + √2)

Question 8.

Write the rationalizing factor of the following.

Answer:

Given, 2 + √3

Need to find the rationalizing factor

⇒ We know that if the product of two surds is rational then each is called a rationalizing factor of each other

⇒ (2 + √3)(2– √3)

= 4 + 2√3 –2√3 –3

= 1 is a rational number

∴ Rationalizing factor of (2+√3) is (2 – √3)

Question 9.

Rationalize the denominator of the following

Answer:

Given, 

Need to rationalize the denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

= 

= 

Hence, rationalizing the denominator of  we get 

Question 10.

Rationalize the denominator of the following

Answer:

Given, 

Need to rationalize the denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

= 

= 

= 

Hence, rationalizing the denominator of we get 

Question 11.

Rationalize the denominator of the following

Answer:

Given, 

Need to rationalize the denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒ here,  can be written as 2√3

⇒  × 

= 

= 

= 

Hence, rationalizing the denominator of we get 

Question 12.

Rationalize the denominator of the following

Answer:

Given, 

Need to rationalize the denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

= 

= 

Hence, rationalizing the denominator of we get 

Question 13.

Rationalize the denominator of the following

Answer:

Given, 

Need to rationalize the denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

= 

⇒ Now, we can write 3 as  = 

∴ 

= 

∵  = 

= 

= 

Hence, rationalizing the denominator of we get 

Question 14.

Simplify by rationalizing the denominator.

Answer:

Given, 

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

= 

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

= 

= 

Hence,  is simplified by rationalizing denominator as 

Question 15.

Simplify by rationalizing the denominator.

Answer:

Given, 

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

= 

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

= 

= 

= 

Hence,  is simplified by rationalizing denominator as 

Question 16.

Simplify by rationalizing the denominator.

Answer:

Given, 

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  can be written as 

⇒  × 

= 

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

= 

= 

Hence,  is simplified by rationalizing denominator as 

Question 17.

Simplify by rationalizing the denominator.

Answer:

Given, 

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

= 

⇒ we know that numerator is in the form of (a+b)² = a²+2ab+b²

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

= 

= 

= 

= 

Hence,  is simplified by rationalizing denominator as 

Question 18.

Simplify by rationalizing the denominator.

Answer:

Given, 

Need to simplify by rationalizing denominator

⇒ To rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

⇒ we know the denominator is in the form of a²–b² = (a+b)(a–b)

= 

= 

= 

= 

= 

Hence,  is simplified by rationalizing denominator as 

Question 19.

Find the values of the following upto 3 decimal places. Given that √2≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162.

Answer:

Given, 

Need to find the values upto 3 decimal places

⇒ substitute the value of √2  1.414

⇒ 

= 0.707

Hence, the value of  is 0.707

Question 20.

Find the values of the following upto 3 decimal places. Given that √2≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162.

Answer:

Given, 

Need to find the values upto 3 decimal places

⇒ substitute the value of √3  1.732

⇒ 

= 3.46

Hence, the value of  is 3.46

Question 21.

Find the values of the following upto 3 decimal places. Given that √2≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162.

Answer:

Given, 

Need to find the values upto 3 decimal places

⇒ substitute the value of √3  1.732

⇒ 

= 1.887

Hence, the value of  is 1.887

Question 22.

Find the values of the following upto 3 decimal places. Given that √2≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162.

Answer:

Given, 

Need to find the values upto 3 decimal places

⇒ Substitute the value of √5  2.236, √10  3.162 and

√2  1.414

⇒ 

= 

= 

= 0.655

Hence, the value of  is 0.655

Question 23.

Find the values of the following upto 3 decimal places. Given that √2≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162.

Answer:

Given, 

Need to find the values upto 3 decimal places

⇒ Substitute the value of √5  2.236, √2  1.414

⇒ 

= 

₌

₌

_{= 0.1022}

Hence, the value of  is _0.1022

Question 24.

Find the values of the following upto 3 decimal places. Given that √2≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162.

Answer:

Given, 

Need to find the values upto 3 decimal places

⇒ Substitute the value of √5  2.236, √2  1.414

⇒ 

= 

= 

= 4.441

Hence, the value of  is 4.441

Question 25.

Find the values of the following upto 3 decimal places. Given that √2≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162.

Answer:

Given, 

Need to find the values upto 3 decimal places

⇒ Substitute the value of √3  1.732

⇒ 

= 

= 

= 3.732

Hence, the value of  is 3.732

Question 26.

Find the values of the following upto 3 decimal places. Given that √2≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162.

Answer:

Given, 

Need to find the values upto 3 decimal places

⇒ Substitute the value of √5  2.236, √10  3.162

⇒ 

= 

= 

= 0.1854

Hence, the value of  is 0.1854

Question 27.

If  find the values of a and b.

Answer:

Given,  = a+b√6

Need to find the value of a and b

⇒ Now, we can find by rationalizing the denominator

⇒ Since, we know to rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

= 

∵ we know that numerator is in the form of (a+b)² = a²+2ab+b²

And denominator in the form of a²–b² = (a+b)(a–b)

⇒ 

= 

=  +

∴ a =  and b = 

Question 28.

If  find the values of a and b.

Answer:

Given,  = a+b√3

Need to find the value of a and b

⇒ Now, we can find by rationalizing the denominator

⇒ Since, we know to rationalize the denominator of  we must multiply the number with its denominator as follows

⇒  × 

= 

= 

= 

= 

= 

= 

= 

= 7+4√ 3

∴ 7+4√3 = a+b√3

Hence, the value of a = 7 and b = 4

Question 29.

If  find the values of a and b.

Answer:

Given,  +  = a+b√5

Need to find the value of a and b

⇒ ⇒ Now, we can find by rationalizing the denominator

⇒ Since, we know to rationalize the denominator of  +  we must multiply the number with its denominator as follows

⇒(  × )+(  × )

=  + 

=  + 

=  + 

= 

= 

= 3

∴ a = 3 b = 0

Question 30.

If  find the values of a and b.

Answer:

Given,  –  = a+b√5

Need to find the value of a and b

⇒ ⇒ Now, we can find by rationalizing the denominator

⇒ Since, we know to rationalize the denominator of  –  we must multiply the number with its denominator as follows

⇒(  × )–(  × )

=  –

=  – 

=  – 

= 

= 

⇒  = a+b√5

Hence, the value of a = 0 and b = 

Question 31.

If  find the values of 

Answer:

Given, x= 2 + √3

Need to find the values of +

⇒ By substituting the given values of x in the equation we get

⇒ x² = (2+√3)² = 4+2(2)(√3)+(3) = 7+4√3

⇒  = 

= 

⇒ By rationalizing method we can write as

⇒  × 

= 

= 

= 

= 7–4√3

⇒ + = 7+4√3+7–4√3

= 14

Hence, the value of + is 14

Question 32.

, find the values of 

Answer:

Given, x = √3 +1

Need to find the value of 

⇒  = 

=  × 

= 

= 

= √3 –1

⇒  = ((√3 +1) –(√3–1))²

= (√3+1–√3+1)²

= 4

Hence, the value of  is 4

---

Exercise 1.3

Question 1.

Using division algorithm, find the quotient and remainder of the following pairs.

(i) 10, 3 (ii) 5, 12 (iii) 27, 3

Answer:

(i) 10,3

We write the given pair in the form a = bq + r, 0 r<b as follows.

10 = 3(3) + 1[3 divides 10 three time and leaves the remainder 1]

quotient = 3; remainder = 1

(ii) 5,12

We write the given pair in the form a = bq + r, 0 r<b as follows.

5 = 12(0) + 5 [12 divides 5 Zero time and leaves the remainder 5]

quotient = 0; remainder = 5

(iii) 27,3

We write the given pair in the form a = bq + r, 0 r<b as follows.

27 = 3(9) + 0 [3 divides 27 Nine time and leaves the remainder 1]

quotient = 3; remainder = 0

---

Exercise 1.4

Question 1.

Which one of the following is not a surd?
A. 

B. 

C. 

D. 

Answer:

As we know

surd is a number in which we cant remove its square root(cube root …etc)

A. =  = 2 is not surd

B.  cant be simplified hence surd

C. cannot be simplified hence surd

 cannot be simplified hence surd

Hence A is the answer.

Question 2.

The simplest form of  is
A. 

B. 

C. 

D. 

Answer:

On simplification

= 

Hence B is the answer

Question 3.

 is equal to
A. 

B. 

C. 

D. 

Answer:

On Simplification

= 

= 

Hence A is the answer

Question 4.

 is equal to
A. 

B. 

C. 

D. 2

Answer:

On rationalizing 

= 

= 

Hence B is the answer

Question 5.

The rationlizing factor of  is
A. 

B. 

C. 

D. 

Answer:

As we know

Prime factorization of 27 = 3

 = 

= 

Hence C is the answer

Question 6.

Which one of the following is not true?
A.  is an irrational number

B.  is an irrational number

C. 0.10110011100011110… is an irrational number

D.  is an irrational number

Answer:

As we know an irrational number are those number which cannot be represented in a simple fraction

A. √2 cannot be represent in simple fraction hence it is an irrational number

Hence A is true

B. √17 cannot be represent in simple fraction hence it is an irrational number

Hence B is true

C. 0.10110011100011110….cannot be represent in simple fraction hence it is a irrational number . Hence C is true

D. on simplification

 = 

Hence is rational number and not irrational.

Hence D is the right answer

Question 7.

The order and radicand of the surd  are respectively
A. 8,12

B. 12,8

C. 16,12

D. 12,16

Answer:

As we know

 here a is the order of surd and n is the radicand

 order = 8

Radicand = 12

A is the answer

Question 8.

The surd having radicand 9 and order 3 is
A. 

B. 

C. 

D. 

Answer:

As we know

 here a is the order of surd and n is the radicand

Hence C is the option

Question 9.

represents the pure surd
A. 

B. 

C. 

D. 

Answer:

Which is B

Hence B is the answer

Question 10.

Which one of the following is not true?
A.  is an irrational number

B. If a is a rational number and  is an irrational number

C. Every surd is an irrational number.

D. The square root of every positive integer is always irrational

Answer:

Option A is incorrect because √2 cannot be written in simple fraction.

hence is irrational number, hence true.

option B is incorrect as it can be written in simple form hence it is a rational number

And also √b cannot be written in simple form hence it is a irrational number

Option C

As we know surd is a number in which we cant remove its square root(cube root …etc) hence cannot represent, in simple fraction hence it is irrational number

Hence C is true

Option D

D is not true as square root of every positive integer is not always irrational

For example, square root of 4 is 2 which is rational number hence D is not true.

Hence D is the answer

Question 11.

Which one of the following is not true?
A. When x is not a perfect square,  is an irrational number

B. The index form of 

C. The radical form of 

D. Every real number is an irrational number

Answer:

As in

Option A

As we know an irrational number are those number which cannot be represented in simple fraction

Hence they are not perfect square

For example 3 is not a perfect square

also is irrational number

hence A is true

option B

index form of number means power form

as 

hence B is true

Option C

Radical form means surd form

Hence C is also true

Option D

It is not true that every real number is a rational number

For example 2 is real number but not irrational number

Hence D is not true

Hence D is the right answer

Question 12.

is equal to
A. 1

B. 3

C. 23

D. 21

Answer:

using identity

(a + b)(a-b) = a² – b²

 = 

= (5-4)

= 1

Hence A is the answer.