SELINA Solutions for Class 9 Maths Chapter 1 - Rational and Irrational Numbers

Chapter 1 - Rational and Irrational Numbers Exercise Ex. 1(A)

Question 1

Is zero a rational number? Can it be written in the form Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers, where p and q are integers and q≠0?

Solution 1

Yes, zero is a rational number.

As it can be written in the form of Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers, where p and q are integers and q≠0?

0 = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 2

Are the following statements true or false? Give reasons for your answers.

i. Every whole number is a natural number.

ii. Every whole number is a rational number.

iii. Every integer is a rational number.

iv. Every rational number is a whole number.

Solution 2

i. False, zero is a whole number but not a natural number.

ii. True, Every whole can be written in the form of Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers, where p and q are integers and q≠0.

iii. True, Every integer can be written in the form of Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers, where p and q are integers and q≠0.

iv. False.

Example: Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is a rational number, but not a whole number.

Question 3

Arrange Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers  in ascending order of their magnitudes.

Also, find the difference between the largest and smallest of these fractions. Express this difference as a decimal fraction correct to one decimal place.

Solution 3

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 4

Arrange Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers  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.

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 5(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 5(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 5(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 5(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 5(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 5(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 5(iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 5(iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 5(v)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 5(v)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 5(vi)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 5(vi)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 5(vii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 5(vii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 5(viii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 5(viii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Chapter 1 - Rational and Irrational Numbers Exercise Ex. 1(B)

Question 1

State, whether the following numbers are rational or not:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers(v) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers(vi) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 1

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

                      Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 Irrational

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

                      Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Irrational

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Rational

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersIrrational

(v) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersRational

(vi) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersRational

Question 2

Find the square of:

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 2

 (i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii)

 Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

 

Question 3

State, in each case, whether true or false:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis an irrational number

(v) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis a rational number.

(vi) All rational numbers are real numbers.

(vii) All real numbers are rational numbers.

(viii) Some real numbers are rational numbers.

Solution 3

(i) False

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberswhich is true

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersTrue.

(iv) False because 

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

which is recurring and non-terminating and hence it is rational

(v) True because Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberswhich is recurring and non-terminating


(vi) True

(vii) False

(viii) True.

Question 4

Given universal set =

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

From the given set, find :

(i) set of rational numbers

(ii) set of irrational numbers

(iii) set of integers

(iv) set of non-negative integers

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

 

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) 

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

(iii) 

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

(iv)  

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 5

Use method of contradiction to show that Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersare irrational numbers.

Solution 5

Let us suppose that Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersare rational numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers(Where a, b Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers7 and b, y Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers0 x , y)

Squaring both sides

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersa2 and x2 are odd as 3b2 and 5y2 are odd .

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersa and x are odd....(1)

Let a = 3c, x = 5z

a2 = 9c2, x2 = 25z2

3b2 = 9c2, 5y2 = 25z2(From equation Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersb2 =3c2, y2 = 5z2

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersb2 and y2 are odd as 3c2 and 5z2 are odd .

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersb and y are odd...(2)

From equation (1) and (2) we get a, b, x, y are odd integers.

i.e., a, b, and x, y have common factors 3 and 5 this contradicts our assumption that Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersare rational i.e, a, b and x, y do not have any common factors other than.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis not rational

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersare irrational.

Question 6

Prove that each of the following numbers is irrational:

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 6

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Let Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersbe a rational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers = x

Squaring on both the sides, we get

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Here, x is a rational number.

x2 is a rational number. 

x2 - 5 is a rational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis also a rational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is a rational number.

But Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is an irrational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is an irrational number.

x2- 5 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 Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is a rational number is wrong.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is an irrational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Let Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersbe a rational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers = x

Squaring on both the sides, we get

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Here, x is a rational number.

x2 is a rational number. 

11 - x2 is a rational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis also a rational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis a rational number.

But Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is an irrational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis an irrational number.

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 Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is a rational number is wrong.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is an irrational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Let Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers be a rational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers = x

Squaring on both the sides, we get

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Here, x is a rational number.

x2 is a rational number. 

9 - x2 is a rational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis also a rational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis a rational number.

But Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is an irrational number.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis an irrational 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 Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is a rational number is wrong.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is an irrational number.

Question 7

Write a pair of irrational numbers whose sum is irrational.

Solution 7

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersare irrational numbers whose sum is irrational.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberswhich is irrational.

Question 8

Write a pair of irrational numbers whose sum is rational.

Solution 8

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersare two irrational numbers whose sum is rational.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 9

Write a pair of irrational numbers whose difference is irrational.

Solution 9

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersare two irrational numbers whose difference is irrational.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberswhich is irrational.

Question 10

Write a pair of irrational numbers whose difference is rational.

Solution 10

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersare irrational numbers whose difference is rational.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberswhich is rational.

Question 11

Write a pair of irrational numbers whose product is irrational.

Solution 11

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 13

Write in ascending order:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 13

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

and 45 < 48 Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

and40 < 54 Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

and 128 < 147 < 180

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 14

Write in descending order:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 14

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Since 162 > 96

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

141 > 63 Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 15

Compare.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 15

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Make powers Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberssame

L.C.M. of 6,4 is 12

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

and Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

L.C.M. of 2 and 3 is 6.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers, Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

Question 16

Insert two irrational numbers between 5 and 6.

Solution 16

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

Question 17

Insert five irrational numbers between Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers.

Solution 17

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 18

Write two rational numbers between Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 18

We want rational numbers a/b and c/d such that: Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers< a/b < c/d < Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Consider any two rational numbers between 2 and 3 such that they are perfect squares.

Let us take 2.25 and 2.56 as Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Thus we have,

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers



Question 19

Write three rational numbers between Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 19

Consider some rational numbers between 3 and 5 such that they are perfect squares.

Let us take, 3.24, 3.61, 4, 4.41 and 4.84 as

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 20(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 20(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 20(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 20(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 20(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 20(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 20(iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 20(iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Chapter 1 - Rational and Irrational Numbers Exercise Ex. 1(C)

Question 1

State, with reasons, which of the following are surds and which are not:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(v) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(vi) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(vii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(viii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 1

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersWhich is irrational

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis a surd

 

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersWhich is irrational

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is a surd

 

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis a surd

 

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberswhich is rational

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis not a surd

 

(v) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis not a surd

 

(vi) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers= -5

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis is not a surd

(vii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersnot a surd as Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is irrational

 

(viii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis not a surd because Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is irrational.

Question 2

Write the lowest rationalising factor of:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(v) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(vi) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(vii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(viii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ix) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers


 

Solution 2

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberswhich is rational

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberslowest rationalizing factor is Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberslowest rationalizing factor is Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

(iii)  Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberslowest rationalizing factor is Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Therefore, lowest rationalizing factor is Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

(v) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numberslowest rationalizing factor is Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

  (vi) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Therefore lowest rationalizing factor is Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 
 (vii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Its lowest rationalizing factor is Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(viii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Its lowest rationalizing factor is Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 (ix) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

its lowest rationalizing factor is Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

Question 3

Rationalise the denominators of :

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(v) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(vi) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(vii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(viii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ix) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

Solution 3

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 (iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 (v)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 (vi)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 (vii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(viii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 (ix)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

Question 4

Find the values of 'a' and 'b' in each of the following:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 4

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 (ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 (iv) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

Question 5

Simplify:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 5

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers


(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 6

If x = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand y = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers; find:

(i) x2     (ii) y2

(iii) xy   (iv) x2 + y2 + xy.

 

Solution 6

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

 

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

 

(iii) xy = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

(iv) x2 + y2 + xy = 161 - Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

= 322 + 1 = 323

 

 

 

Question 7

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

(i) m

(ii) n2

(iii) mn

Solution 7

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 8

If x = 2Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers+ 2Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers, find:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 8

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

Question 9

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

 

 

 

Solution 9

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

 

 

 

 

 

 

 

 

 

 

 

 

Question 10

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

 

 

 

Solution 10

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Question 11

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 11

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 12

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 12

 Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 13(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 13(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 13(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 13(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 13(iii)

If Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers = 1.4 and Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers = 1.7, find the value of each of the following, correct to one decimal place:

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Solution 13(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 14

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 14

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 15

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 15

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Chapter 1 - Rational and Irrational Numbers Exercise Ex. 1(D)

Question 1

Simplify: Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 1

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 2

Simplify: Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 2

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 3

Evaluate, correct to one place of decimal, the expressionSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers, if Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers= 2.2 and Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers = 3.2.

Solution 3

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 4

 If x = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers, find the value of:

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 5

Show that:

(i) Negative of an irrational number is irrational number.

(ii) The product of a non - zero rational number and an irrational number is a rational number.

Solution 5

Proof:

(i)

The given statement is TRUE.

Let us assume that negative of an irrational number is a rational number.

Let p be an irrational number,

→ - p is a rational number.

→ - (-p) = p is a rational number.

But p is an irrational number.

Therefore our assumption was wrong.

So, Negative of an irrational number is irrational number.

(ii)

The given statement is FALSE.

We know that 3 is a non - zero rational number and Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersis an irrational number.

So, 3 ×Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers= 3Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers is an irrational.

Therefore, the product of a non - zero rational number and an irrational number is an irrational number.

Question 6

Draw a line segment of length Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers cm.

Solution 6

Since Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

So, first we need to find Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers and mark it on number line and then find Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

 

Steps to draw Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers on the number line are:

  1. Draw a number line and mark point O.
  2. Mark point A on it such that OA = 1 unit.
  3. Draw right triangle OAB such that A = 90° and AB = 1 unit.
  4. Join OB.
  5. By Pythagoras Theorem, Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers.
  6. Draw a line BC perpendicular to OB such that BC = 1.
  7. Join OC. Thus, OBC is a right triangle. Again by Pythagoras Theorem, we have Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 
  8. With centre as O and radius OC draw a circle which meets the number line at a point E.
  9. Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers units. Thus, OE represents Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers on the number line.

 

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

 

Question 7

Draw a line segment of length Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers cm.

Solution 7

Since, Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

We need to construct a right - angled triangle OAB, in which

A = 90°, OA = 2 units and AB = 2 units.

By Pythagoras theorem, we get

OB2 = OA2 + AB2

OB = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers units

STEPS:

  1. Draw a number line.
  2. Mark point A on it which is two points (units) from an initial point say O in the right/positive direction.
  3. Now, draw a line AB = 2 units which is perpendicular to A.
  4. Join OB to represent the hypotenuse of a triangle OAB, right angled at A.
  5. This hypotenuse of triangle OAB is showing a length of OB.
  6. With centre as O and radius as OB, draw an arc on the number line which cuts it at the point C.
  7. Here, C representsSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers.

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 8

Show that: Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 8

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational NumbersSelina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

Question 9

Show that:

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 9

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 10

Show that x is irrational, if:

(i) x2 = 6 (ii) x2 = 0.009 (iii) x2 = 27

Solution 10

(i) x2 = 6

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

(ii) x2 = 0.009

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers 

(iii) x2 = 27

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Question 11

Show that x is rational, if:

(i) x2 = 16 (ii) x2 = 0.0004 (iii) x2 = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Solution 11

(i) x2 = 16 

Taking square root on both the sides, we get

x = ± 4

As we know that 4 = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersand -4 =Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbersare rational numbers.

x is rational , if: x2 = 16 .

(ii) x2 = 0.0004 (iii) x2 = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

x2 = 0.0004 = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Taking square root on both the sides, we get

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers…. is a rational number.

x is rational , if: x2 =0.0004.

(iii) x2 = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Taking square root on both the sides, we get

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers…. is a rational number.

x is rational , if: x2 =Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers. 

Question 12

Using the following figure, show that BD =Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers.

Solution 12

From the image, it is cleared that

AB = x, BC = 1

AC = AB + BC = x + 1

O is a centre of the semi - circle with AC as diameter.

OA, OC and OD are the radii.

OA = OC = OD = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

OB = OC - BC = Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

mOBD = 90° … given in the image

By Pythagoras theorem, we get

OD2 = OB2 + BD2

⇒ BD2 = OD2 - OB2

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers

Selina Solutions Icse Class 9 Mathematics Chapter - Rational And Irrational Numbers