# Frank solutions for Class 9 Maths ICSE chapter 1 - Irrational Numbers [Latest edition]

## Chapter 1: Irrational Numbers

Exercise 1.1Exercise 1.2Exercise 1.3
Exercise 1.1

### Frank solutions for Class 9 Maths ICSE Chapter 1 Irrational Numbers Exercise 1.1

Exercise 1.1 | Q 1.1

State if the following fraction has a terminating decimal

(3)/(5)

Exercise 1.1 | Q 1.2

State if the following fraction has a terminating decimal

(5)/(7)

Exercise 1.1 | Q 1.3

State if the following fraction has a terminating decimal

(25)/(49)

Exercise 1.1 | Q 1.4

State if the following fraction has a terminating decimal

(37)/(40)

Exercise 1.1 | Q 1.5

State if the following fraction has a terminating decimal

(57)/(64)

Exercise 1.1 | Q 1.6

State if the following fraction has a terminating decimal.

(59)/(75)

Exercise 1.1 | Q 1.7

State if the following fraction has a terminating decimal.

(89)/(125)

Exercise 1.1 | Q 1.8

State if the following fraction has a terminating decimal.

(125)/(213)

Exercise 1.1 | Q 1.9

State if the following fraction has a terminating decimal.

(147)/(160)

Exercise 1.1 | Q 2.1

Express the following decimal as a rational number.

0.93

Exercise 1.1 | Q 2.2

Express the following decimal as a rational number.

4.56

Exercise 1.1 | Q 2.3

Express the following decimal as a rational number.

0.614

Exercise 1.1 | Q 2.4

Express the following decimal as a rational number.

21.025

Exercise 1.1 | Q 3.1

Convert the following fraction into a decimal :

(3)/(5)

Exercise 1.1 | Q 3.2

Convert the following fraction into a decimal :

(8)/(11)

Exercise 1.1 | Q 3.3

Convert the following fraction into a decimal :

(-2)/(7)

Exercise 1.1 | Q 3.4

Convert the following fraction into a decimal :

(12)/(21)

Exercise 1.1 | Q 3.5

Convert the following fraction into a decimal :

(13)/(25)

Exercise 1.1 | Q 3.6

Convert the following fraction into a decimal :

(2)/(3)

Exercise 1.1 | Q 4.1

Express the following decimal as a rational number.

0.7

Exercise 1.1 | Q 4.2

Express the following decimal as a rational number.

0.35

Exercise 1.1 | Q 4.3

Express the following decimal as a rational number.

0.89

Exercise 1.1 | Q 4.4

Express the following decimal as a rational number.

0.057

Exercise 1.1 | Q 4.5

Express the following decimal as a rational number.

0.763

Exercise 1.1 | Q 4.6

Express the following decimal as a rational number.

2.67

Exercise 1.1 | Q 4.7

Express the following decimal as a rational number.

4.6724

Exercise 1.1 | Q 4.8

Express the following decimal as a rational number.

0.017

Exercise 1.1 | Q 4.9

Express the following decimal as a rational number.

17.027

Exercise 1.1 | Q 5.1

Insert a rational number between:

(2)/(5) and (3)/(4)

Exercise 1.1 | Q 5.2

Insert a rational number between:

(3)/(4) and (5)/(7)

Exercise 1.1 | Q 5.3

Insert a rational number between:

(4)/(3) and (7)/(5)

Exercise 1.1 | Q 5.4

Insert a rational number between:

(5)/(9) and (6)/(7)

Exercise 1.1 | Q 6.1

Insert a rational number between:

3 and 4

Exercise 1.1 | Q 6.2

Insert a rational number between:

7.6 and 7.7

Exercise 1.1 | Q 6.3

Insert a rational number between:

8 and 8.04

Exercise 1.1 | Q 6.4

Insert a rational number between:

101 and 102

Exercise 1.1 | Q 7.1

Insert three rational numbers between:

0 and 1

Exercise 1.1 | Q 7.2

Insert three rational number between:

6 and 7

Exercise 1.1 | Q 7.3

Insert three rational number between:

-3 and 3

Exercise 1.1 | Q 7.4

Insert three rational number between:

-5 and -4

Exercise 1.1 | Q 8.1

Insert five rational number between:

(2)/(5) and (2)/(3)

Exercise 1.1 | Q 8.2

Insert five rational number between:

-(3)/(4) and -(2)/(5)

Exercise 1.1 | Q 9.1

Find the greatest and the smallest rational number among the following.

(6)/(7),(9)/(14) and (23)/(28)

Exercise 1.1 | Q 9.2

Find the greatest and the smallest rational number among the following.

(-2)/(3) , (-7)/(9) and (-5)/(6)

Exercise 1.1 | Q 10.1

Arrange the following rational numbers in ascending order.

(4)/(5),(6)/(7) and (7)/(10)

Exercise 1.1 | Q 10.2

Arrange the following rational numbers in ascending order.

(-7)/(12), (-3)/(10) and (-2)/(5)

Exercise 1.1 | Q 10.3

Arrange the following rational numbers in ascending order.

(10)/(9),(13)/(12) and (19)/(18)

Exercise 1.1 | Q 10.4

Arrange the following rational numbers in ascending order.

(7)/(4), (-6)/(5) and (-5)/(2)

Exercise 1.1 | Q 11.1

Arrange the following rational numbers in descending order.

(7)/(13),(8)/(15), and (3)/(5)

Exercise 1.1 | Q 11.2

Arrange the following rational numbers in descending order.

(4)/(3), (-14)/(5) and (17)/(15)

Exercise 1.1 | Q 11.3

Arrange the following rational numbers in descending order.

(-7)/(10), (-8)/(15) and (-11)/(30)

Exercise 1.1 | Q 11.4

Arrange the following rational numbers in descending order.

(-3)/(8),(2)/(5) and (-1)/(3)

Exercise 1.1 | Q 12.1

Find the value of:

2.65 + 1.25

Exercise 1.1 | Q 12.2

Find the value of:

1. 32 - 0.91

Exercise 1.1 | Q 12.3

Find the value of:

2.12 - 0.45

Exercise 1.1 | Q 12.4

Find the value of:

1.35 + 1.5

Exercise 1.2

### Frank solutions for Class 9 Maths ICSE Chapter 1 Irrational Numbers Exercise 1.2

Exercise 1.2 | Q 1.1

State whether the following number is rational or irrational

(3 + sqrt(3))^2

Exercise 1.2 | Q 1.2

State whether the following number is rational or irrational

(5 - sqrt(5))^2

Exercise 1.2 | Q 1.3

State whether the following number is rational or irrational

(2 + sqrt(2))(2 - sqrt(2))

Exercise 1.2 | Q 1.4

State whether the following number is rational or irrational

((sqrt5)/(3sqrt(2)))^2

Exercise 1.2 | Q 1.4

State if the following is a surd. Give reasons.

root(3)(-27)

Exercise 1.2 | Q 2.1

Check whether the square of the following is rational or irrational:

3sqrt(2)

Exercise 1.2 | Q 2.2

Check whether the square of the following is rational or irrational:

3 + sqrt(2)

Exercise 1.2 | Q 2.3

Check whether the square of the following is rational or irrational:

(3sqrt(2))/(2)

Exercise 1.2 | Q 2.4

Check whether the square of the following is rational or irrational:

sqrt(2) + sqrt(3)

Exercise 1.2 | Q 3

Show that sqrt(5) is an irrational numbers. [Use division method]

Exercise 1.2 | Q 4

Without using division method show that sqrt(7) is an irrational numbers.

Exercise 1.2 | Q 5.1

Write a pair of irrational numbers whose sum is irrational.

Exercise 1.2 | Q 5.2

Write a pair of irrational numbers whose sum is rational.

Exercise 1.2 | Q 5.3

Write a pair of irrational numbers whose difference is irrational.

Exercise 1.2 | Q 5.4

Write a pair of irrational numbers whose difference is rational.

Exercise 1.2 | Q 5.5

Write a pair of irrational numbers whose product is irrational.

Exercise 1.2 | Q 5.6

Write a pair of irrational numbers whose product is rational.

Exercise 1.2 | Q 6.1

Compare the following:

root(4)(12) and root(3)(15)

Exercise 1.2 | Q 6.2

Compare the following:

root(3)(48) and sqrt(36)

Exercise 1.2 | Q 7.1

Write the following in ascending order:

2sqrt(5), sqrt(3) and 5sqrt(2)

Exercise 1.2 | Q 7.2

Write the following in ascending order:

2root(3)(3), 4root(3)(3) and 3root(3)(3)

Exercise 1.2 | Q 7.3

Write the following in ascending order:

5sqrt(7), 7sqrt(5) and 6sqrt(2)

Exercise 1.2 | Q 7.4

Write the following in ascending order:

7root(3)(5), 6root(3)(4) and 5root(3)(6)

Exercise 1.2 | Q 8.1

Write the following in descending order:

sqrt(2), root(3)(5) and root(4)(10)

Exercise 1.2 | Q 8.2

Write the following in descending order:

5sqrt(3), sqrt(15) and 3sqrt(5)

Exercise 1.2 | Q 8.3

Write the following in descending order:

sqrt(6), root(3)(8) and root(4)(3)

Exercise 1.2 | Q 9

Insert two irrational numbers between 3 and 4.

Exercise 1.2 | Q 10

Insert five irrational number's between 2sqrt(3) and 3sqrt(5).

Exercise 1.2 | Q 11

Write two rational numbers between sqrt(3) and sqrt(7)

Exercise 1.2 | Q 12

Write four rational numbers between sqrt(2) and sqrt(3)

Exercise 1.2 | Q 13.1

State if the following is a surd. Give reasons.

sqrt(150)

Exercise 1.2 | Q 13.2

State if the following is a surd. Give reasons.

root(3)(4)

Exercise 1.2 | Q 13.3

State if the following is a surd. Give reasons.

root(3)(50). root(3)(20)

Exercise 1.2 | Q 13.5

State if the following is a surd. Give reasons.

sqrt(2 + sqrt(3)

Exercise 1.2 | Q 13.6

State if the following is a surd. Give reasons.

root(12)(8). ÷ root(6)(6)

Exercise 1.2 | Q 14

Represent the number sqrt(7) on the number line.

Exercise 1.3

### Frank solutions for Class 9 Maths ICSE Chapter 1 Irrational Numbers Exercise 1.3

Exercise 1.3 | Q 1.1

Simplify by rationalising the denominator in the following.

(3sqrt(2))/sqrt(5)

Exercise 1.3 | Q 1.2

Simplify by rationalising the denominator in the following.

(1)/(5 + sqrt(2))

Exercise 1.3 | Q 1.3

Simplify by rationalising the denominator in the following.

(1)/(sqrt(3) + sqrt(2))

Exercise 1.3 | Q 1.4

Simplify by rationalising the denominator in the following.

(2)/(3 + sqrt(7)

Exercise 1.3 | Q 1.5

Simplify by rationalising the denominator in the following.

(5)/(sqrt(7) - sqrt(2))

Exercise 1.3 | Q 1.6

Simplify by rationalising the denominator in the following.

(42)/(2sqrt(3) + 3sqrt(2)

Exercise 1.3 | Q 1.7

Simplify by rationalising the denominator in the following.

(sqrt(3) + 1)/(sqrt(3) - 1)

Exercise 1.3 | Q 1.8

Simplify by rationalising the denominator in the following.

(sqrt(5) - sqrt(7))/sqrt(3)

Exercise 1.3 | Q 1.9

Simplify by rationalising the denominator in the following.

(3 - sqrt(3))/(2 + sqrt(2)

Exercise 1.3 | Q 2.01

Simplify by rationalising the denominator in the following.

(5 + sqrt(6))/(5 - sqrt(6)

Exercise 1.3 | Q 2.02

Simplify by rationalising the denominator in the following.

(4 + sqrt(8))/(4 - sqrt(8)

Exercise 1.3 | Q 2.03

Simplify by rationalising the denominator in the following.

(sqrt(15) + 3)/(sqrt(15) - 3)

Exercise 1.3 | Q 2.04

Simplify by rationalising the denominator in the following.

(sqrt(7) - sqrt(5))/(sqrt(7) + sqrt(5)

Exercise 1.3 | Q 2.05

Simplify by rationalising the denominator in the following.

(3sqrt(5) + sqrt(7))/(3sqrt(5) - sqrt(7)

Exercise 1.3 | Q 2.06

Simplify by rationalising the denominator in the following.

(2sqrt(3) - sqrt(6))/(2sqrt(3) + sqrt(6)

Exercise 1.3 | Q 2.07

Simplify by rationalising the denominator in the following.

(5sqrt(3) - sqrt(15))/(5sqrt(3) + sqrt(15)

Exercise 1.3 | Q 2.08

Simplify by rationalising the denominator in the following.

(2sqrt(6) - sqrt(5))/(3sqrt(5) - 2sqrt(6)

Exercise 1.3 | Q 2.09

Simplify by rationalising the denominator in the following.

(7sqrt(3) -  5sqrt(2))/(sqrt(48) + sqrt(18)

Exercise 1.3 | Q 2.1

Simplify by rationalising the denominator in the following.

(sqrt(12) + sqrt(18))/(sqrt(75) - sqrt(50)

Exercise 1.3 | Q 3.1

Simplify the following

(3)/(5 - sqrt(3)) + (2)/(5 + sqrt(3)

Exercise 1.3 | Q 3.2

Simplify the following

(4 + sqrt(5))/(4 - sqrt(5)) + (4 - sqrt(5))/(4 + sqrt(5)

Exercise 1.3 | Q 3.3

Simplify the following

(sqrt(5) - 2)/(sqrt(5) + 2) - (sqrt(5) + 2)/(sqrt(5) - 2)

Exercise 1.3 | Q 3.4

Simplify the following

(sqrt(7) - sqrt(3))/(sqrt(7) + sqrt(3)) - (sqrt(7) + sqrt(3))/(sqrt(7) - sqrt(3)

Exercise 1.3 | Q 3.5

Simplify the following

(sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3)) + (sqrt(5) - sqrt(3))/(sqrt(5) + sqrt(3)

Exercise 1.3 | Q 4.1

Simplify the following :

sqrt(6)/(sqrt(2) + sqrt(3)) + (3sqrt(2))/(sqrt(6) + sqrt(3)) - (4sqrt(3))/(sqrt(6) + sqrt(2)

Exercise 1.3 | Q 4.2

Simplify the following :

(3sqrt(2))/(sqrt(6) - sqrt(3)) - (4sqrt(3))/(sqrt(6) - sqrt(2)) + (2sqrt(3))/(sqrt(6) + 2)

Exercise 1.3 | Q 4.3

Simplify the following :

(6)/(2sqrt(3) - sqrt(6)) + sqrt(6)/(sqrt(3) + sqrt(2)) - (4sqrt(3))/(sqrt(6) - sqrt(2)

Exercise 1.3 | Q 4.4

Simplify the following :

(7sqrt(3))/(sqrt(10) + sqrt(3)) - (2sqrt(5))/(sqrt(6) + sqrt(5)) - (3sqrt(2))/(sqrt(15) + 3sqrt(2)

Exercise 1.3 | Q 4.5

Simplify the following :

(4sqrt(3))/((2 - sqrt(2))) - (30)/((4sqrt(3) - 3sqrt(2))) - (3sqrt(2))/((3 + 2sqrt(3))

Exercise 1.3 | Q 5

If (sqrt(2.5) - sqrt(0.75))/(sqrt(2.5) + sqrt(0.75)) = "p" + "q"sqrt(30), find the values of p and q.

Exercise 1.3 | Q 6.01

In the following, find the values of a and b.

(sqrt(3) - 1)/(sqrt(3) + 1) = "a" + "b"sqrt(3)

Exercise 1.3 | Q 6.02

In the following, find the values of a and b:

(3 + sqrt(7))/(3 - sqrt(7)) = "a" + "b"sqrt(7)

Exercise 1.3 | Q 6.03

In the following, find the values of a and b:

(5 + 2sqrt(3))/(7 + 4sqrt(3)) = "a" + "b"sqrt(3)

Exercise 1.3 | Q 6.04

In the following, find the values of a and b:

(1)/(sqrt(5) - sqrt(3)) = "a"sqrt(5) - "b"sqrt(3)

Exercise 1.3 | Q 6.05

In the following, find the values of a and b:

(sqrt(3) - 2)/(sqrt(3) + 2) = "a"sqrt(3) + "b"

Exercise 1.3 | Q 6.06

In the following, find the values of a and b:

(sqrt(11) - sqrt(7))/(sqrt(11) + sqrt(7)) = "a" - "b"sqrt(77)

Exercise 1.3 | Q 6.07

In the following, find the values of a and b:

(7sqrt(3) - 5sqrt(2))/(4sqrt(3) + 3sqrt(2)) = "a" - "b"sqrt(6)

Exercise 1.3 | Q 6.08

In the following, find the values of a and b:

(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) = "a" - "b"sqrt(6)

Exercise 1.3 | Q 6.09

In the following, find the value of a and b:

(7 + sqrt(5))/(7 - sqrt(5)) - (7 - sqrt(5))/(7 + sqrt(5)) = "a" + "b"sqrt(5)

Exercise 1.3 | Q 6.1

In the following, find the value of a and b:

(sqrt(3) - 1)/(sqrt(3) + 1) + (sqrt(3) + 1)/(sqrt(3) - 1) = "a" + "b"sqrt(3)

Exercise 1.3 | Q 7.1

If x = (7 + 4sqrt(3)), find the value of

sqrt(x) + (1)/(sqrt(x)

Exercise 1.3 | Q 7.2

If x = (7 + 4sqrt(3)), find the value of

x^2 + (1)/x^2

Exercise 1.3 | Q 7.3

If x = (7 + 4sqrt(3)), find the values of

x^3 + (1)/x^3

Exercise 1.3 | Q 7.4

If x = (7 + 4sqrt(3)), find the values of :

(x + (1)/x)^2

Exercise 1.3 | Q 8.1

If x = (4 - sqrt(15)), find the values of

(1)/x

Exercise 1.3 | Q 8.2

If x = (4 - sqrt(15)), find the values of

x + (1)/x

Exercise 1.3 | Q 8.3

If x = (4 - sqrt(15)), find the values of

x^2 + (1)/x^2

Exercise 1.3 | Q 8.4

If x = (4 - sqrt(15)), find the values of

x^3 + (1)/x^3

Exercise 1.3 | Q 8.5

If x = (4 - sqrt(15)), find the values of :

(x + (1)/x)^2

Exercise 1.3 | Q 9

If x = ((2 + sqrt(5)))/((2 - sqrt(5)) and y = ((2 - sqrt(5)))/((2 + sqrt(5)), show that (x2 - y2) = 144sqrt(5).

Exercise 1.3 | Q 10.1

If x = ((sqrt(3) + 1))/((sqrt(3) - 1) and y = ((sqrt(3) - 1))/((sqrt(3) + 1), find the values of

x2 + y2

Exercise 1.3 | Q 10.2

If x = ((sqrt(3) + 1))/((sqrt(3) - 1) and y = ((sqrt(3) - 1))/((sqrt(3) + 1), find the values of

x3 + y3

Exercise 1.3 | Q 10.3

If x = ((sqrt(3) + 1))/((sqrt(3) - 1) and y = ((sqrt(3) - 1))/((sqrt(3) - 1), find the values of

x2 - y2 + xy

Exercise 1.3 | Q 11.1

If x = (1)/((3 - 2sqrt(2)) and y = (1)/((3 + 2sqrt(2)), find the values of

x2 + y2

Exercise 1.3 | Q 11.2

If x = (1)/((3 - 2sqrt(2)) and y = (1)/((3 + 2sqrt(2)), find the values of

x3 + y3

## Chapter 1: Irrational Numbers

Exercise 1.1Exercise 1.2Exercise 1.3

## Frank solutions for Class 9 Maths ICSE chapter 1 - Irrational Numbers

Frank solutions for Class 9 Maths ICSE chapter 1 (Irrational Numbers) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CISCE Class 9 Maths ICSE solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 9 Maths ICSE chapter 1 Irrational Numbers are Concept of Rational Numbers, Properties of Rational Numbers, Decimal Representation of Rational Numbers, Concept of Irrational Numbers, Concept of Real Numbers, Concept of Surds, Rationalisation of Surds, Simplifying an expression by rationalization of the Denominator.

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