Chapter 1 : Rational and Irrational Numbers - Selina Solutions for Class 9 Maths ICSE

Mathematics in ICSE Class 9 is one of the most challenging and trickiest subjects of all. It includes complex topics such as logarithms, expansions, indices and Pythagoras Theorem which are difficult to understand for an average student. TopperLearning provides study materials for ICSE Class 9 Mathematics to make the subject easy and help students to clear all their concepts. Our study materials comprise numerous video lessons, question banks, revision notes and sample papers which help achieve success in the examination.

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

Question 1

State, whether the following numbers are rational or not:

(i) (ii) (iii)

(iv) (v) (vi)

Solution 1

(i) $open parentheses 2 plus square root of 2 close parentheses squared equals 2 squared plus 2 open parentheses 2 close parentheses open parentheses square root of 2 close parentheses plus open parentheses square root of 2 close parentheses squared$

Irrational

(ii)

Irrational

(iii)

Rational

(iv)

Irrational

(v) Rational

(vi) Rational

Question 2

Find the square of:

$open parentheses i close parentheses fraction numerator 3 square root of 5 over denominator 5 end fraction space space space open parentheses i i close parentheses square root of 3 plus square root of 2 space space open parentheses i i i close parentheses square root of 5 minus 2 space space space open parentheses i v close parentheses space 3 plus 2 square root of 5$

Solution 2

(i)

$open parentheses fraction numerator 3 square root of 5 over denominator 5 end fraction close parentheses squared equals fraction numerator 3 squared open parentheses square root of 5 close parentheses squared over denominator 5 squared end fraction space space space space space space space space space space space space space space space space space equals fraction numerator 9 cross times 5 over denominator 25 end fraction space space space space space space space space space space space space space space space space space equals 9 over 5 space space space space space space space space space space space space space space space space space equals 1 4 over 5$

(ii)

(iii)

$open parentheses square root of 5 minus 2 close parentheses squared equals open parentheses square root of 5 close parentheses squared minus 2 open parentheses square root of 5 close parentheses open parentheses 2 close parentheses plus open parentheses 2 close parentheses squared space space space space space space space space space space space space space space space space space equals 5 minus 4 square root of 5 plus 4 space space space space space space space space space space space space space space space equals 9 minus 4 square root of 5$

(iv)

$open parentheses 3 plus 2 square root of 5 close parentheses squared equals 3 squared plus 2 open parentheses 3 close parentheses open parentheses 2 square root of 5 close parentheses plus open parentheses 2 square root of 5 close parentheses squared space space space space space space space space space space space space space space space space space space space equals 9 plus 12 square root of 5 plus 20 space space space space space space space space space space space space space space space space space space space equals 29 plus 12 square root of 5$

Question 3

State, in each case, whether true or false:

(i)

(ii)

(iii)

(iv) is an irrational number

(v) is 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) which is true

(iii) True.

(iv) False because

$2 over 7 equals 0. top enclose 285714$

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

(v) True because which is recurring and non-terminating

(vi) True

(vii) False

(viii) True.

Question 4

Given universal set =

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

(i)

(ii)

(iii)

(iv)

Question 5

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

Solution 5

Let us suppose that and are rational numbers

and (Where a, b 7 and b, y 0 x , y)

Squaring both sides

a² and x² are odd as 3b² and 5y² are odd .

a and x are odd....(1)

Let a = 3c, x = 5z

a² = 9c², x² = 25z²

3b² = 9c², 5y² = 25z²(From equation )

b² =3c², y² = 5z²

b² and y² are odd as 3c² and 5z² are odd .

b 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 are rational i.e, a, b and x, y do not have any common factors other than.

is not rational

and are irrational.

Question 6

Prove that each of the following numbers is irrational:

Solution 6

Let be a rational number.

⇒ = x

Squaring on both the sides, we get

Here, x is a rational number.

⇒ x²is a rational number.

⇒ x² - 5 is a rational number.

⇒ is also a rational number.

is a rational number.

But is an irrational number.

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 is a rational number is wrong.

∴ is an irrational number.

Let be a rational number.

⇒ = x

Squaring on both the sides, we get

Here, x is a rational number.

⇒ x²is a rational number.

⇒ 11 - x² is a rational number.

⇒ is also a rational number.

is a rational number.

But is an irrational number.

is 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 is a rational number is wrong.

∴ is an irrational number.

Let be a rational number.

⇒ = x

Squaring on both the sides, we get

Here, x is a rational number.

⇒ x²is a rational number.

⇒ 9 - x² is a rational number.

⇒ is also a rational number.

is a rational number.

But is an irrational number.

is 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 is a rational number is wrong.

∴ is an irrational number.

Question 7

Write a pair of irrational numbers whose sum is irrational.

Solution 7

are irrational numbers whose sum is irrational.

which is irrational.

Question 8

Write a pair of irrational numbers whose sum is rational.

Solution 8

and are two irrational numbers whose sum is rational.

Question 9

Write a pair of irrational numbers whose difference is irrational.

Solution 9

and are two irrational numbers whose difference is irrational.

which is irrational.

Question 10

Write a pair of irrational numbers whose difference is rational.

Solution 10

and are irrational numbers whose difference is rational.

which is rational.

Question 11

Write a pair of irrational numbers whose product is irrational.

Solution 11

$C o n s i d e r space t w o space i r r a t i o n a l space n u m b e r s space open parentheses 5 plus square root of 2 close parentheses space a n d space open parentheses square root of 5 minus 2 close parentheses T h u s comma space t h e space p r o d u c t comma space open parentheses 5 plus square root of 2 close parentheses space cross times space open parentheses square root of 5 minus 2 close parentheses equals 5 square root of 5 minus 10 plus square root of 10 minus 2 square root of 2 space i s space i r r a t i o n a l.$

Question 12

Write in ascending order:

(i)

(ii)

(iii)

Solution 12

(i)

and 45 < 48

(ii)

and40 < 54

(iii)

and 128 < 147 < 180

Question 13

Write in ascending order:

(i)

(ii)

(iii)

Solution 13

(i)

and 45 < 48

(ii)

and40 < 54

(iii)

and 128 < 147 < 180

Question 14

Write in descending order:

(i)

(ii)

Solution 14

(i)

Since 162 > 96

(ii)

141 > 63

Question 15

Compare.

$left parenthesis straight i right parenthesis space 6 root of 15 space and space 4 root of 12 left parenthesis ii right parenthesis space square root of 24 space and space 3 root of 35$

Solution 15

(i) and

Make powers and same

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

and

(ii) and

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

Question 16

Insert two irrational numbers between 5 and 6.

Solution 16

$W e space k n o w space t h a t space 5 equals square root of 25 space a n d space 6 equals square root of 36. T h u s space c o n s i d e r space t h e space n u m b e r s comma space square root of 25 less than square root of 26 less than square root of 27 less than square root of 28 less than square root of 29 less than square root of 30 less than square root of 31 less than square root of 32 less than square root of 33 less than square root of 34 less than square root of 35 less than square root of 36 T h e r e f o r e comma space a n y space t w o space i r r a t i o n a l space n u m b e r s space b e t w e e n space 5 space a n d space 6 space i s space square root of 27 space a n d space square root of 28$

Question 17

Insert five irrational numbers between and .

Solution 17

$W e space k n o w space t h a t space 2 square root of 5 equals square root of 4 cross times 5 end root equals square root of 20 space a n d space 3 square root of 3 equals square root of 27 T h u s comma space w e space h a v e comma space square root of 20 less than square root of 21 less than square root of 22 less than square root of 23 less than square root of 24 less than square root of 25 less than square root of 26 less than square root of 27 S o space a n y space f i v e space i r r a t i o n a l space n u m b e r s space b e t w e e n space 2 square root of 5 space a n d space 3 square root of 3 space a r e : square root of 21 comma square root of 22 comma square root of 23 comma square root of 24 space a n d space square root of 26$

Question 18

Write two rational numbers between

Solution 18

We want rational numbers a/b and c/d such that: < a/b < c/d <

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

Let us take 2.25 and 2.56 as $square root of 2.25 end root equals 1.5 space a n d space square root of 2.56 end root equals 1.6$

Thus we have,

$square root of 2 less than square root of 2.25 end root less than square root of 2.56 end root less than square root of 3 rightwards double arrow square root of 2 less than 1.5 less than 1.6 less than square root of 3 rightwards double arrow square root of 2 less than 15 over 10 less than 16 over 10 less than square root of 3 rightwards double arrow square root of 2 less than 3 over 2 less than 8 over 5 less than square root of 3 T h e r e f o r e space a n y space t w o space r a t i o n a l space n u m b e r s space b e t w e e n space square root of 2 space a n d space square root of 3 space a r e : space 3 over 2 space a n d space 8 over 5$

Question 19

Write three rational numbers between

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

$square root of 3.24 end root equals 1.8 comma space square root of 3.61 end root equals 1.9 comma space square root of 4 equals 2 comma space square root of 4.41 end root equals 2.1 space a n d space square root of 4.84 end root equals 2.2$

$T h u s space w e space space h a v e comma square root of 3 less than square root of 3.24 end root less than square root of 3.61 end root less than square root of 4 less than square root of 4.41 end root less than square root of 4.84 end root less than square root of 5 rightwards double arrow square root of 3 less than 1.8 less than 1.9 less than 2 less than 2.1 less than 2.2 less than square root of 5 rightwards double arrow square root of 3 less than 18 over 10 less than 19 over 10 less than 2 less than 21 over 10 less than 22 over 10 less than square root of 5 rightwards double arrow square root of 3 less than 9 over 5 less than 19 over 10 less than 2 less than 21 over 10 less than 11 over 5 less than square root of 5 T h e r e f o r e comma space a n y space t h r e e space r a t i o n a l space n u m b e r s space b e t w e e n space square root of 3 space a n d space square root of 5 space a r e : 9 over 5 comma 19 over 10 space a n d space 21 over 10$

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

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

Question 1

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

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Solution 1

(i) Which is irrational

is a surds

(ii) Which is irrational

is a surds

(iii)

is a surds

(iv) which is rational

is not a surds

(v)

is not a surds

(vi) = -5

is is not a surds

(vii) not a surds as is irrational

(viii) is not a surds because is irrational.

Question 2

Write the lowest rationalising factor of:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

Solution 2

(i) which is rational

lowest rationalizing factor is

(ii)

lowest rationalizing factor is

(iii)

lowest rationalizing factor is

(iv)

$open parentheses 7 minus square root of 7 close parentheses open parentheses 7 plus square root of 7 close parentheses equals 49 minus 7 equals 42$

Therefore, lowest rationalizing factor is

(v) $square root of 18 minus square root of 50$

lowest rationalizing factor is

(vi)

$open parentheses square root of 5 minus square root of 2 close parentheses open parentheses square root of 5 plus square root of 2 close parentheses equals open parentheses square root of 5 close parentheses squared minus open parentheses square root of 2 close parentheses squared equals 3$

Therefore lowest rationalizing factor is

(vii)

$open parentheses square root of 13 plus 3 close parentheses open parentheses square root of 13 minus 3 close parentheses equals open parentheses square root of 13 close parentheses squared minus 3 squared equals 13 minus 9 equals 4$

Its lowest rationalizing factor is

(viii)

$15 minus 3 square root of 2 equals 3 open parentheses 5 minus square root of 2 close parentheses space space space space space space space space space space space space space space space space space equals 3 open parentheses 5 minus square root of 2 close parentheses open parentheses 5 plus square root of 2 close parentheses space space space space space space space space space space space space space space space space space equals 3 cross times open square brackets 5 squared minus open parentheses square root of 2 close parentheses squared close square brackets space space space space space space space space space space space space space space space space space equals 3 cross times open square brackets 25 minus 2 close square brackets space space space space space space space space space space space space space space space space equals 3 cross times 23 space space space space space space space space space space space space space space space space equals 69$

Its lowest rationalizing factor is $5 plus square root of 2$

(ix)

$3 square root of 2 plus 2 square root of 3 equals open parentheses 3 square root of 2 plus 2 square root of 3 close parentheses open parentheses 3 square root of 2 minus 2 square root of 3 close parentheses space space space space space space space space space space space space space space space space space space space space space equals open parentheses 3 square root of 2 close parentheses squared minus open parentheses 2 square root of 3 close parentheses squared space space space space space space space space space space space space space space space space space space space space space space space equals 9 cross times 2 minus 4 cross times 3 space space space space space space space space space space space space space space space space space space space space space space space equals 18 minus 12 space space space space space space space space space space space space space space space space space space space space space space space space equals 6$

its lowest rationalizing factor is

Question 3

Rationalise the denominators of :

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

Solution 3

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

Question 4

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

(i)

(ii)

(iii)

(iv)

Solution 4

(i)

(ii)

(iii)

(iv)

Question 5

Simplify:

(i)

(ii)

Solution 5

(i)

(ii)

Question 6

If= and y = ; find:

(i) x²(ii) y²

(iii) xy(iv) x² + y² + xy.

Solution 6

(i)

(ii)

(iii) xy =

(iv) x² + y² + xy = 161 -

= 322 + 1 = 323

Question 7

$begin mathsize 11px style If space straight m equals fraction numerator 1 over denominator 3 minus 2 square root of 2 end fraction space and space straight n equals fraction numerator 1 over denominator 3 plus 2 square root of 2 end fraction comma space find colon end style$

(i) m²

(ii) n²

(iii) mn

Solution 7

$left parenthesis straight i right parenthesis space straight m equals fraction numerator 1 over denominator 3 minus 2 square root of 2 end fraction space space space space equals fraction numerator 1 over denominator 3 minus 2 square root of 2 end fraction cross times fraction numerator 3 plus 2 square root of 2 over denominator 3 plus 2 square root of 2 end fraction space space space space equals fraction numerator 3 plus 2 square root of 2 over denominator open parentheses 3 close parentheses squared minus open parentheses 2 square root of 2 close parentheses squared end fraction space space space space equals fraction numerator 3 plus 2 square root of 2 over denominator 9 minus 8 end fraction space space space space equals 3 plus 2 square root of 2 rightwards double arrow straight m squared equals open parentheses 3 plus 2 square root of 2 close parentheses squared space space space space space space space space space space equals open parentheses 3 close parentheses squared plus 2 cross times 3 cross times 2 square root of 2 plus open parentheses 2 square root of 2 close parentheses squared space space space space space space space space space space equals 9 plus 12 square root of 2 plus 8 space space space space space space space space space space equals 17 plus 12 square root of 2$

$left parenthesis ii right parenthesis space straight n equals fraction numerator 1 over denominator 3 plus 2 square root of 2 end fraction space space space space equals fraction numerator 1 over denominator 3 plus 2 square root of 2 end fraction cross times fraction numerator 3 minus 2 square root of 2 over denominator 3 minus 2 square root of 2 end fraction space space space space equals fraction numerator 3 minus 2 square root of 2 over denominator open parentheses 3 close parentheses squared minus open parentheses 2 square root of 2 close parentheses squared end fraction space space space space equals fraction numerator 3 plus 2 square root of 2 over denominator 9 minus 8 end fraction space space space space equals 3 minus 2 square root of 2 rightwards double arrow straight n squared equals open parentheses 3 minus 2 square root of 2 close parentheses squared space space space space space space space space space space equals open parentheses 3 close parentheses squared minus 2 cross times 3 cross times 2 square root of 2 plus open parentheses 2 square root of 2 close parentheses squared space space space space space space space space space space equals 9 minus 12 square root of 2 plus 8 space space space space space space space space space space equals 17 minus 12 square root of 2$

$left parenthesis iii right parenthesis space space mn equals open parentheses 3 plus 2 square root of 2 close parentheses open parentheses 3 minus 2 square root of 2 close parentheses equals open parentheses 3 close parentheses squared minus open parentheses 2 square root of 2 close parentheses squared equals 9 minus 8 equals 1$

Question 8

If x = 2+ 2, find:

(i) (ii) (iii)

Solution 8

(i)

(ii)

(iii)

Question 9

Solution 9

Question 10

Solution 10

Question 11

$begin mathsize 11px style Show space that colon space fraction numerator 1 over denominator 3 minus 2 square root of 2 end fraction minus fraction numerator 1 over denominator 2 square root of 2 minus square root of 7 end fraction plus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 6 minus square root of 5 end fraction plus fraction numerator 1 over denominator square root of 5 minus 2 end fraction equals 5. end style$

Solution 11

$straight L. straight H. straight S. equals fraction numerator 1 over denominator 3 minus 2 square root of 2 end fraction minus fraction numerator 1 over denominator 2 square root of 2 minus square root of 7 end fraction plus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 6 minus square root of 5 end fraction plus fraction numerator 1 over denominator square root of 5 minus 2 end fraction space space space space space space space space space space space space equals fraction numerator 1 over denominator 3 minus square root of 8 end fraction minus fraction numerator 1 over denominator square root of 8 minus square root of 7 end fraction plus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 6 minus square root of 5 end fraction plus fraction numerator 1 over denominator square root of 5 minus 2 end fraction space space space space space space space space space space space space equals fraction numerator 1 over denominator 3 minus square root of 8 end fraction cross times fraction numerator 3 plus square root of 8 over denominator 3 plus square root of 8 end fraction minus fraction numerator 1 over denominator square root of 8 minus square root of 7 end fraction cross times fraction numerator square root of 8 plus square root of 7 over denominator square root of 8 plus square root of 7 end fraction plus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction cross times fraction numerator square root of 7 plus square root of 6 over denominator square root of 7 plus square root of 6 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space minus fraction numerator 1 over denominator square root of 6 minus square root of 5 end fraction cross times fraction numerator square root of 6 plus square root of 5 over denominator square root of 6 plus square root of 5 end fraction plus fraction numerator 1 over denominator square root of 5 minus 2 end fraction cross times fraction numerator square root of 5 plus 2 over denominator square root of 5 plus 2 end fraction space space space space space space space space space space space space equals fraction numerator 3 plus square root of 8 over denominator open parentheses 3 close parentheses squared minus open parentheses square root of 8 close parentheses squared end fraction minus fraction numerator square root of 8 plus square root of 7 over denominator open parentheses square root of 8 close parentheses squared minus open parentheses square root of 7 close parentheses squared end fraction plus fraction numerator square root of 7 plus square root of 6 over denominator open parentheses square root of 7 close parentheses squared minus open parentheses square root of 6 close parentheses squared end fraction minus fraction numerator square root of 6 plus square root of 5 over denominator open parentheses square root of 6 close parentheses squared minus open parentheses square root of 5 close parentheses squared end fraction plus fraction numerator square root of 5 plus 2 over denominator open parentheses square root of 5 close parentheses squared minus open parentheses 2 close parentheses squared end fraction space space space space space space space space space space space space equals fraction numerator 3 plus square root of 8 over denominator 9 minus 8 end fraction minus fraction numerator square root of 8 plus square root of 7 over denominator 8 minus 7 end fraction plus fraction numerator square root of 7 plus square root of 6 over denominator 7 minus 6 end fraction minus fraction numerator square root of 6 plus square root of 5 over denominator 6 minus 5 end fraction plus fraction numerator square root of 5 plus 2 over denominator 5 minus 4 end fraction space space space space space space space space space space space space equals 3 plus square root of 8 minus square root of 8 minus square root of 7 plus square root of 7 plus square root of 6 minus square root of 6 minus square root of 5 plus square root of 5 plus 2 space space space space space space space space space space space space equals 3 plus 2 space space space space space space space space space space space space equals 5 space space space space space space space space space space space space equals straight R. straight H. straight S.$

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

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

Question 1

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

⇒ 0 =

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 , 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 fractions. Express this difference as a decimal fraction correct to one decimal place.

Solution 3