# Concise Rational and Irrational Numbers ICSE Class-9 Selina Solutions

Concise Rational and Irrational Numbers ICSE Class-9 Mathematics Selina Solutions Chapter-1 . We provide step by step Solutions of Exercise / lesson-1 Rational and Irrational Numbers for ICSE Class-9 Concise Selina Mathematics by RK Bansal. Our Solutions contain all type Questions with Exe-1 A, Exe-1 B and Exe-1 C ,  to develop skill and confidence. Visit official Website CISCE for detail information about ICSE Board Class-9.

## Concise Rational and Irrational Numbers ICSE Class-9 Mathematics Selina Solutions Chapter-1

–: Select Topics :–

Exe-1 A,

Exe-1 B,

Exe-1 C ,

Note:- Before viewing Solution of Chapter -1 Rational and Irrational Numbers Class-9 of Concise Selina Solutions.  Read the Chapter Carefully then solve all example given in Exercise-1.1, Exercise-1.2, Exercise-1.3. For more practice on The Rational and Irrational Numbers Visit ML Aggarwal Rational and Irrational Numbers Chapter Also.

### Exercise – 1(A) Rational and Irrational Numbers for ICSE Class-9 Maths Concise Selina Solutions

#### Question 1

Is zero a rational number ? Can it be written in the form , where p and q are integers and q≠0 ?

Yes, zero is a rational number.

As it can be written in the form of , where p and q are integers and q≠0 ?

⇒ 0 =

#### Question 2

Are the following statement true or false ? Give reason for your answer.

1. Every whole number is a natural number.
2. Every whole number is a rational number.
3. Every integer is a rational number.
4. Every rational number is a whole number.

i) False, zero is a whole number but not a natural number.
ii) True, Every whole can be written in the formof, where p and q are integers and q≠0.
iii) True, Every integer can be written in the form of , where p and q are integers and q≠0.
iv) False.
Example :  is a rational number, but not a whole number.

#### Question 3

Arrange  in ascending order of their magnitudes.
Also, find the difference between the largest and smallest of these rational numbers. Express this difference as a decimal fraction correct to one decimal place.

#### Question 4

Arrange  in descending order of their magnitudes.
Also, find the sum of the lowest and largest of these fractions. Express the result obtained as a decimal fraction correct to two decimal places.

#### Question 5.1

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

Given number is 716

Since 16 = 2 x 2 x 2 x 2 = 24 = 24 x 50
I.e. 16 can be expressed as 2m x 5n

∴  is convertible into the terminating decimal.

#### Question 5.2

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

Given number is

Since 125 = 5 x 5 x 5 = 53 = 20 x 53
I.e. 125 can be expressed as 2m x 5n

∴  is convertible into the terminating decimal.

#### Question 5.3

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

Given number is

Since 14 = 2 x 7 = 21 x 71
I.e. 14 cannot be expressed as 2m x 5n

∴  is not convertible into the terminating decimal.

#### Question 5.4

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

Given number is

Since, 45 = 3 x 3 x 5 = 32 x 51
I.e. 45 cannot be expressed as 2m x 5n

∴  is not convertible into the terminating decimal.

#### Question 5.5

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

Given number is

Since 50 = 2 x 5 x 5 = 21 x 52
I.e. 50 can be expressed as 2m x 5n

∴  is convertible into the terminating decimal.

#### Question 5.6

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

Given number is

Since 40 = 2 x 2 x 2 x 5 = 23 x 51
I.e. 40 can be expressed as 2m x 5n

∴  is convertible into the terminating decimal.

#### Question 5.7

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

Given number is

Since 75 = 3 x 5 x 5 = 31 x 52
i.e. 75 cannot be expressed as 2m x 5n

∴  is not convertible into the terminating decimal.

#### Question 5.8

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

Given number is

Since 250 = 2 x 5 x 5 x 5 = 21 x 53
i.e. 250 can be expressed as 2m x 5n.1

∴ is convertible into the terminating decimal.

End of Exe-1 A Concise Rational and Irrational Numbers for ICSE Class-9

### ICSE Class-9 Mathematics Concise Selina Solutions Exercise – 1(B) Rational and Irrational Numbers

#### Question 1.1

State, whether the following numbers is rational or not : ( 2 + √2 )2

( 2 + √2 ) = 22 + 2 ( 2 ) ( √2 ) + ( √2 )2
= 4 + 4√2 + 2
= 6 + 4√2

Irrational number.

#### Question 1.2

State, whether the following numbers is rational or not : ( 3 – √3 )2

( 3 – √3 ) = 32 + 2 ( 3 ) ( √3 ) + ( √3 )2
= 9 – 6√3 + 3
= 12 – 6√3  = 6 ( 2 – √3 )
Irrational number.

#### Question 1.3

State, whether the following numbers is rational or not : ( 5 + √5 )( 5 – √5 )

( 5 + √5 )( 5 – √5 ) = ( 5 )2 – ( √5 )2
= 25 – 5 = 20
Rational Number

#### Question 1.4

State, whether the following numbers is rational or not : ( √3 – √2 )2

( √3 – √2 ) = ( √3 )2 – 2 ( √3 )( √2 ) + ( √2 )2
= 3 – 2√6 + 2
= 5 – 2√6
Irrational Number

#### Question 1.5

State, whether the following numbers is rational or not :

#### Question 1.6

State, whether the following number is rational or not :

#### Question 2.1

Find the square of :

#### Question 2.2

Find the square of : √3 + √2

( √3 + √2 )2 = ( √3 )2 + 2( √3 )( √2 ) + ( √2 )2
= 3 + 2√6 + 2
= 5 + 2√6

#### Question 2.3

Find the square of : √5 – 2

( √5 – 2 )2 = ( √5 )2 – 2( √5 )( 2 ) + ( 2 )2
= 5 – 4√5 + 4
= 9 – 4√5

#### Question 2.4

Find the square of : 3 + 2√5

( 3 + 2√5 )2 = 32 + 2( 3 )( 2√5) + ( 2√5 )2
= 9 + 12√5 + 20
= 29 + 12√5

#### Question 3.1

State, in each case, whether true or false :
√2 + √3 = √5

• True
• False

False

#### Question 3.2

State, in each case, whether true or false :
2√4 + 2 = 6

##### Options
• True
• False

2√4 + 2 = 2 x 2 + 2 = 4 + 2 = 6 which is True.

#### Question 3.3

State, in each case, whether true or false :
3√7 – 2√7 = √7

##### Options
• True
• False

3√7 – 2√7 = √7 – True.

#### Question 3.4

State, in each case, whether true or false :
27 ia an irrational number.

##### Options
• True
• False

False Because 27=0.285714¯ which is recurring and non-terminating and hence it is rational.

#### Question 3.5

State, in each case, whether true or false :
is a rational number.

##### Options
• True
• False

True, because 0.45¯ which is recurring and non-terminating

#### Question 3.6

State, in each case, whether true or false :
All rational numbers are real numbers.

• True
• False

True

#### Question 3.7

state, in each case, whether true or false :
All real numbers are rational numbers.

• True
• False

False

#### Question 3.8

State, in each case, whether true or false :
Some real numbers are rational numbers.

• True
• False

True

#### Question 4.1

Given Universal set is

#### Question 4.3

Given universal set =
{ -6, -5 3/4, -sqrt4, -3/5, -3/8, 0, 4/5, 1, 1 2/3, sqrt8, 3.01, π, 8.47 }
From the given set, find : set of integers

Given Universal set is

{ – 6, -5 3/4, -sqrt4, -3/5, -3/8, 0, 4/5, 1, 1 2/3, sqrt8, 3.01, π, 8.47 }

We need to find the set of integers.
Set of integers consists of zero, the natural numbers and their additive inverses.
The set of integers is Z.
Z = { …… -3, -2, -1, 0, 1, 2, 3,……. }
Here the set of integers is U ∩ Z = { -6, √4, 0, 1 }

#### Question 4.4

We need to find the set of non-negative integers.
Set of non-negative integers consists of zero and the natural numbers.
The set of non-negative integers is Z+ and Z= { 0, 1, 2, 3,…… }
Here the set of integers is U ∩ Z= {0, 1}

#### Question 5

Use method of contradiction to show that √3 and √5 are irrational numbers.

#### Question 6.1

Prove that the following number is irrational: √3 + √2

3 + √2
Let √3 + √2 be a rational number.
⇒  √3 + √2 = x
Squaring on both the sides, we get
( √3 + √2 )2 = x2
⇒ 3 + 2 + 2 x √3 x √2 = x2

⇒ x2 is an irrational number.

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that √3 + √2 is a rational number is wrong.

∴ √3 + √2 is an irrational number.

#### Question 6.2

Prove that the following number is irrational:  3 – √2

3 – √2
Let  3 – √2 be a rational number.
⇒  3 – √2 = x
Squaring on both the sides, we get
(  3 – √2 )2 = x2
⇒ 9 + 2 – 2 x  3 x √2 = x2
⇒  11 – x2 = 6√2

⇒ 11 – x2 is an irrational number.

⇒ x2 is an irrational number.

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that  3 – √2 is a rational number is wrong.
∴ 3 – √2 is an irrational number.

#### Question 6.3

Prove that the following number is irrational: √5 – 2

√5 – 2
Let √5 – 2 be a rational number.
⇒ √5 – 2 = x
Squaring on both the sides, we get
(√5-2)2=x2

⇒ 5 + 4 – 2 x 2 x √5 = x2

⇒ 9 – x= 4√5

⇒ √5 = 9-x24

Here, x is a rational number.

⇒ 9 – x2 is an irrational number.

⇒ x2 is an irrational number.

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that √5 – 2 is a rational number is wrong.
∴ √5 – 2 is an irrational number.

Question 7

Write a pair of irrational numbers whose sum is irrational.

√3 + 5 and √5 – 3 are irrational numbers whose sum is irrational.
( √3 + 5  ) + ( √5 – 3 ) = √3 + √5 + 5 – 3 = √3 + √5 + 2 which is irrational.

#### Question 8

Write a pair of irrational numbers whose sum is rational.

√3 + 5 and  4 – √3 are two irrational numbers whose sum is rational.
( √3 + 5  ) + ( 4 – √3 ) = √3 + 5+ 4 – √3 = 9

#### Question 9

Write a pair of irrational numbers whose difference is irrational.

√3 + 2 and √2 – 3 are two irrational numbers whose difference is irrational.
( √3 + 2  ) – ( √2 – 3 ) = √3 – √2 + 2 + 3 = √3 – √2 + 5 which is irrational.

#### Question 10

Write a pair of irrational numbers whose difference is rational.

√5 – 3 and √5 + 3 are irrational numbers whose difference is rational.
( √5 – 3 ) – ( √5 + 3  ) = √5 – 3 – √5 – 3 = -6 which is rational.

#### Question 11

Write a pair of irrational numbers whose product is irrational.

Consider two irrational numbers ( 5 + √2 ) and ( √5 – 2 )
Thus, the product, ( 5 + √2 ) x ( √5 – 2 ) = 5√5 – 10 + √10 – 2√2 is irrational.

#### Question 12

Write a pair of irrational numbers whose product is rational.

Consider √2 as an irrational number.
√2×√2= √4= 2 which is a rational number.

#### Question 13.1

Write in ascending order: 3√5 and 4√3

#### Question 13.2

Write in ascending order :  253 and323

#### Question 13.3

Write in ascending order : 6√5, 7√3, and 8√2

6√5 = 62×5=180

7√3 = 72×3=147

8√2 = 82×2=128

and 128 < 147 < 180

∴ √128 < √147 < √180

⇒ 8√2 < 7√3 < 6√5

#### Question 13.4

Write in ascending order :  6√5, 7√3 and 8√2

65=62×5=180

73=72×3=147

82=82×2=128

and 128 < 147 < 180

∴ 128<147<180

⇒ 82<73<65

#### Question 14.1

Write in descending order:

#### Question 14.2

Write in descending order: 7√3 and 3√7

Compare :

#### Question 16.

Insert two irrational numbers between 5 and 6.

#### Question 17.

Insert five irrational numbers between 2√5 and 3√3.

We know that 2√5 = 4×5 = √20 and 3√3 = √20

Thus, We have, √20 < √21 < √22 < √23 < √24 < √25 < √26 < √27.
So any five irrational numbers between 2√5 and 3√3 are :
√21, √22, √23, √24, and √26.

#### Question 18.

Write two rational numbers between √2 and √3.

#### Question 19.

Write three rational numbers between √3 and √5.

#### Question 20.3

Simplify : ( 3 + √2 )( 4 + √7 )

( 3 + √2 )( 4 + √7 )
= 3 x 4 + 3 x √7 + 4 x √2 + √2  x √7
= 12 + 3√7 + 4√2 + √14

#### Question 20.4

Simplify : (√3 – √2 )2

(√3 – √2 )2

= ( √3 )2 + ( √2 )2 – 2 x √3 x √2

= 3 + 2 – 2√6

= 5 – 2√6

End of Exe-1 B Concise Rational and Irrational Numbers for ICSE Class-9

### Selina Solutions of Exercise – 1(C) Rational and Irrational Numbers Concise Maths for ICSE Class-9

#### Question 1.1

State, with reason, of the following is surd or not : √180

√180 = 2×2×5×3×3 = 6√5 which is irrational.
∴ √180 is a surds.

#### Question 1.2

State, with reason, of the following is surd or not :

#### Question 1.5

State, with reason, of the following is surd or not :

#### Question 1.6

State, with reason, of the following is surd or not :
…..

#### Question 1.7

State, with reason, of the following is surd or not : √π

√π not a surds as π is irrational.

#### Question 1.8

State, with reason, of the following is surd or not :

is not a surds because 3 + √2 is irrational.

#### Question 2.1

Write the lowest rationalising factor of : 5√2

5√2 x 5√2 = 5 x 2 = 10 which is rational.

∴ lowest rationalizing factor is √2

#### Question 2.2.

Write the lowest rationalising factor of : √24

#### Question 2.3

Write the lowest rationalising factor of : √5 – 3

( √5 – 3 )( √5 + 3 ) = ( √5 )2 – (3)2 = 5 – 9 = -4

∴ lowest rationalizing factor is ( √5 + 3 )

#### Question 2.4

Write the lowest rationalising factor of : 7 – √7

7 – √7
( 7 – √7 )( 7 + √7 ) = 49 – 7 = 42
Therefore, lowest rationalizing factor is ( 7 + √7 ).

#### Question 2.5

Write the lowest rationalising factor of : √18 – √50

∴ lowest rationalizing factor is √2

#### Question 2.6

Write the lowest rationalising factor of : √5 – √2

√5 – √2
( √5 – √2 )( √5 + √2 ) = ( √5 )2 – ( √2 )2 = 3

Therefore lowest rationalizing factor is √5 + √2

#### Question 2.7

Write the lowest rationalising factor of : √13 + 3

( √13 + 3 )( √13 – 3 ) = ( √13 )2 – 32 = 13 – 9 = 4

Its lowest rationalizing factor is √13 – 3.

#### Question 2.8

Write the lowest rationalising factor of : 15 – 3√2

#### Question 2.9

Write the lowest rationalising factor of : 3√2 + 2√3

3√2 + 2√3
= ( 3√2 + 2√3 )( 3√2 – 2√3 )
= ( 3√2)2 – (2√3)2
= 9 x 2 – 4 x 3
= 18 – 12
= 6
its lowest rationalizing factor is 3√2 – 2√3.

#### Question 3.1

Rationalise the denominators of :

35×55=35/5

#### Question 3.2

Rationalise the denominators of :

#### Question 3.3

Rationalise the denominators of :

#### Question 3.4

Rationalise the denominators of :

#### Question 3.5

Rationalise the denominators of :

#### Question 3.6

Rationalise the denominators of :

#### Question 3.7

Rationalise the denominators of :

#### Question 3.8

Rationalise the denominators of :

#### Question 3.9

Rationalise the denominators of : …………………

#### Question 4.1

Find the values of ‘a’ and ‘b’ in each of the following :
………

#### Question 4.2

Find the values of ‘a’ and ‘b’ in each of the following:…..

#### Question 4.3

Find the values of ‘a’ and ‘b’ in each of the following:

#### Question 4.4

Find the values of ‘a’ and ‘b’ in each of the following

Simplify :

Simplify :
…………….

#### Question 6.1

f x =…………………………find

#### Question 6.2

f x =…………………; find : y2

#### Question 6.3

If x =…………………; find :  xy

#### Question 6.4

If x =………………….; find :
x2 + y2 + xy.

x2 + y2 + xy
= 161 – 72√5 + 161 +72√5 + 1

= 322 + 1 = 323

#### Question 7.1

If m = ………… find m2

#### Question 7.2

If m = ……………….., find n2

#### Question 8.1

If x = 2√3 + 2√2 , find : …..

#### Question 8.2

If x = 2√3 + 2√2 , find : ……..

#### Question 8.3

If x = 2√3 + 2√2 , find :…..

#### Question 9

If x = 1 – √2, find the value of ……

#### Question 10

If x = 5 – 2√6, find x2+1×2

#### Question 11

Show that : ……………..

#### Question 12

Rationalise the denominator of :

#### Question 13.1

If √2 = 1.4 and √3 = 1.7, find the value of :…………..

#### Question 13.2

If √2 = 1.4 and √3 = 1.7, find the value of : ………

Simplify :

Evaluate :

#### Question 15

If ; find the value of x2 – y2.

#### Question 16

Simplify :

— End of Exe-1 C Concise Rational and Irrational Numbers Solutions :–

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