# R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 1 - Number Systems

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## Chapter 1 - Number Systems Exercise VSAQ

Question 1

What can you say about the sum of a rational number and an irrational number?

Solution 1

The sum of a rational number and an irrational number is irrational.

Example: 5 +  is irrational.

Question 2

Solve .

Solution 2

Question 3

The number  will terminate after how many decimal places?

Solution 3

Thus, the given number will terminate after 3 decimal places.

Question 4

Find the value of (1296)0.17× (1296)0.08.

Solution 4

(1296)0.17× (1296)0.08

Question 5

Simplify .

Solution 5

Question 6

Find an irrational number between 5 and 6.

Solution 6

An irrational number between 5 and 6 =

Question 7

Find the value of .

Solution 7

Question 8

Rationalise

Solution 8

Question 9

Solve for x: .

Solution 9

Question 10

Simplify (32)1/5 + (-7)0 + (64)1/2.

Solution 10

Question 11

Evaluate .

Solution 11

Question 12

Simplify .

Solution 12

Question 13

If a = 1, b = 2 then find the value of (ab + ba)-1.

Solution 13

Given, a = 1 and b = 2

Question 14

Simplify .

Solution 14

Question 15

Give an example of two irrational numbers whose sum as well as product is rational.

Solution 15

Question 16

Is the product of a rational and irrational numbers always irrational? Give an example.

Solution 16

Yes, the product of a rational and irrational numbers is always irrational.

For example,

Question 17

Give an example of a number x such that x2 is an irrational number and x3 is a rational number.

Solution 17

Question 18

Write the reciprocal of ().

Solution 18

The reciprocal of ()

Question 19

Solution 19

Question 20

Simplify

Solution 20

Question 21

If 10x = 64, find the value of .

Solution 21

Question 22

Evaluate

Solution 22

Question 23

Simplify .

Solution 23

## Chapter 1 - Number Systems Exercise MCQ

Question 1

Which of the following is a rational number?

(a)

(b) π

(c)

(d) 0

Solution 1

Correct option: (d)

0 can be written as  where p and q are integers and q ≠ 0.

Question 2

A rational number between -3 and 3 is

(a) 0

(b) -4.3

(c) -3.4

(d) 1.101100110001....

Solution 2

Correct option: (a)

On a number line, 0 is a rational number that lies between -3 and 3.

Question 3

Two rational numbers between  are

(a)

(b)

(c)

(d)

Solution 3

Correct option: (c)

Two rational numbers between

Question 4

Every point on number line represents

(a) a rational number

(b) a natural number

(c) an irrational number

(d) a unique number

Solution 4

Correct option: (d)

Every point on number line represents a unique number.

Question 5

Which of the following is a rational number?

Solution 5

Question 6

Every rational number is

(a) a natural number

(b) a whole number

(c) an integer

(d)a real number

Solution 6

Question 7

Between any two rational numbers there

(a) is no rational number

(b) is exactly one rational number

(c) are infinitely many rational numbers

(d)is no irrational number

Solution 7

Question 8

The decimal representation of a rational number is

(a) always terminating

(b) either terminating or repeating

(c) either terminating or non-repeating

(d)neither terminating nor repeating

Solution 8

Question 9

The decimal representation of an irrational number is

(a) always terminating

(b) either terminating or repeating

(c) either terminating or non-repeating

(d)neither terminating nor repeating

Solution 9

Question 10

The decimal expansion that a rational number cannot have is

(a) 0.25

(b)

(c)

(d) 0.5030030003....

Solution 10

Correct option: (d)

The decimal expansion of a rational number is either terminating or non-terminating recurring.

The decimal expansion of 0.5030030003…. is non-terminating non-recurring, which is not a property of a rational number.

Question 11

Which of the following is an irrational number?

(a) 3.14

(b) 3.141414....

(c) 3.14444.....

(d) 3.141141114....

Solution 11

Correct option: (d)

The decimal expansion of an irrational number is non-terminating non-recurring.

Hence, 3.141141114….. is an irrational number.

Question 12

A rational number equivalent to  is

(a)

(b)

(c)

(d)

Solution 12

Correct option: (d)

Question 13

Choose the rational number which does not lie between

(a)

(b)

(c)

(d)

Solution 13

Correct option: (b)

Given two rational numbers are negative and  is a positive rational number.

So, it does not lie between

Question 14

Π is

(a) a rational number

(b) an integer

(c) an irrational number

(d) a whole number

Solution 14

Correct option: (c)

Π = 3.14159265359…….., which is non-terminating non-recurring.

Hence, it is an irrational number.

Question 15

The decimal expansion of  is

(a) finite decimal

(b) 1.4121

(c) nonterminating recurring

(d) nonterminating, nonrecurring

Solution 15

Correct option: (d)

The decimal expansion of , which is non-terminating, non-recurring.

Question 16

Which of the following is an irrational number?

(a)

(b)

(c) 0.3799

(d)

Solution 16

Correct option: (a)

The decimal expansion of , which is non-terminating, non-recurring.

Hence, it is an irrational number.

Question 17

Hoe many digits are there in the repeating block of digits in the decimal expansion of

(a) 16

(b) 6

(c) 26

(d) 7

Solution 17

Correct option: (b)

Question 18

Which of the following numbers is irrational?

(a)

(b)

(c)

(d)

Solution 18

Correct option: (c)

The decimal expansion of , which is non-terminating, non-recurring.

Hence, it is an irrational number.

Question 19

The product of two irrational numbers is

(a) always irrational

(b) always rational

(c) always an integer

(d)sometimes rational and sometimes irrational

Solution 19

Question 20

Which of the following is a true statement?

(a) The sum of two irrational numbers is an irrational number

(b) The product of two irrational numbers is an irrational number

(c) Every real number is always rational

(d) Every real number is either rational or irrational

Solution 20

Question 21

Which of the following is a true statement?

(a)

(b)

(c)

(d)

Solution 21

Question 22

A rational number lying between  is

(a)

(b)

(c) 1.6

(d) 1.9

Solution 22

Correct option: (c)

Question 23

Which of the following is a rational number?

(a)

(b) 0.101001000100001...

(c) π

(d) 0.853853853...

Solution 23

Correct option: (d)

The decimal expansion of a rational number is either terminating or non-terminating recurring.

Hence, 0.853853853... is a rational number.

Question 24

The product of a nonzero rational number with an irrational number is always a/an

(a) irrational number

(b) rational number

(c) whole number

(d) natural number

Solution 24

Correct option: (a)

The product of a non-zero rational number with an irrational number is always an irrational number.

Question 25

The value of , where p and q are integers and q 0, is

(a)

(b)

(c)

(d)

Solution 25

Correct option: (b)

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

An irrational number between 5 and 6 is

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

The sum of

(a)

(b)

(c)

(d)

Solution 33

Correct option: (b)

Let x =

i.e. x = 0.3333…. ….(i)

10x = 3.3333…. ….(ii)

On subtracting (i) from (ii), we get

9x = 3

Let y =

i.e. y = 0.4444…. ….(i)

10y = 4.4444…. ….(ii)

On subtracting (i) from (ii), we get

9y = 4

Question 34

The value of

(a)

(b)

(c)

(d)

Solution 34

Correct option: (c)

Let x =

i.e. x = 2.4545….  ….(i)

100x = 245.4545…….  ….(ii)

On subtracting (i) from (ii), we get

99x = 243

Let y =

i.e. y = 0.3636….  ….(iii)

100y = 36.3636….  ….(iv)

On subtracting (iii) from (iv), we get

99y = 36

Question 35

Which of the following is the value of ?

(a) -4

(b) 4

(c)

(d)

Solution 35

Correct option: (b)

Question 36

when simplified is

(a) positive and irrational

(b) positive and rational

(c) negative and irrational

(d) negative and rational

Solution 36

Correct option: (b)

Which is positive and rational number.

Question 37

when simplified is

(a) positive and irrational

(b) positive and rational

(c) negative and irrational

(d) negative and rational

Solution 37

Correct option: (b)

Which is positive and rational number.

Question 38

When  is divided by , the quotient is

(a)

(b)

(c)

(d)

Solution 38

Correct option: (c)

Question 39

The value of  is

(a) 10

(b)

(c)

(d)

Solution 39

Correct option: (a)

Question 40

The value of  is

(a)

(b)

(c)

(d)

Solution 40

Correct option: (b)

Question 41

= ?

(a)

(b)

(c)

(d) None of these

Solution 41

Correct option: (b)

Question 42

=?

(a)

(b) 2

(c) 4

(d) 8

Solution 42

Correct option: (b)

Question 43

(125)-1/3 = ?

(a) 5

(b) -5

(c)

(d)

Solution 43

Correct option: (c)

Question 44

The value of 71/2 81/2 is

(a) (28)1/2

(b) (56)1/2

(c) (14)1/2

(d) (42)1/2

Solution 44

Correct option: (b)

Question 45

After simplification,  is

(a) 132/15

(b) 138/15

(c) 131/3

(d) 13-2/15

Solution 45

Correct option: (d)

Question 46

The value of is

(a)

(b)

(c) 8

(d)

Solution 46

Correct option: (a)

Question 47

The value of is

(a) 0

(b) 2

(c)

(d)

Solution 47

Correct option: (b)

Question 48

The value of (243)1/5 is

(a) 3

(b) -3

(c) 5

(d)

Solution 48

Correct option: (a)

Question 49

93 + (-3)3 - 63 = ?

(a) 432

(b) 270

(c) 486

(d) 540

Solution 49

Correct option: (c)

93 + (-3)3 - 63 = 729 - 27 - 216 = 486

Question 50

Simplified value of  is

(a) 0

(b) 1

(c) 4

(d) 16

Solution 50

Correct option: (b)

Question 51

The value of is

(a) 2-1/6

(b) 2-6

(c) 21/6

(d) 26

Solution 51

Correct option: (c)

Question 52

Simplified value of (25)1/3× 51/3 is

(a) 25

(b) 3

(c) 1

(d) 5

Solution 52

Correct option: (d)

Question 53

The value of is

(a) 3

(b) -3

(c) 9

(d)

Solution 53

Correct option: (a)

Question 54

There is a number x such that x2 is irrational but x4 is rational. Then, x can be

(a)

(b)

(c)

(d)

Solution 54

Correct option: (d)

Question 55

If  then value of p is

(a)

(b)

(c)

(d)

Solution 55

Correct option: (b)

Question 56

The value of is

(a)

(b)

(c)

(d)

Solution 56

Correct option: (b)

Question 57

The value of xp-q xq - r xr - p is equal to

(a) 0

(b) 1

(c) x

(d) xpqr

Solution 57

Correct option: (b)

xp-q xq - r xr - p

= xp - q + q - r + r - p

= x0

= 1

Question 58

The value of  is

(a) -1

(b) 0

(c) 1

(d) 2

Solution 58

Correct option: (c)

Question 59

= ?

(a) 2

(b)

(c)

(d)

Solution 59

Correct option: (a)

Question 60

If  then x = ?

(a) 1

(b) 2

(c) 3

(d) 4

Solution 60

Correct option: (d)

Question 61

If (33)2 = 9x then 5x = ?

(a) 1

(b) 5

(c) 25

(d) 125

Solution 61

Correct option: (d)

(33)2 = 9x

(32)3 = (32)x

x = 3

Then 5x = 53 = 125

Question 62

On simplification, the expression  equals

(a)

(b)

(c)

(d)

Solution 62

Correct option: (b)

Question 63

The simplest rationalisation factor of  is

(a)

(b)

(c)

(d)

Solution 63

Correct option: (d)

Thus, the simplest rationalisation factr of

Question 64

The simplest rationalisation factor of  is

(a)

(b)

(c)

(d)

Solution 64

Correct option: (b)

The simplest rationalisation factor of  is

Question 65

The rationalisation factor of  is

(a)

(b)

(c)

(d)

Solution 65

Correct option: (d)

Question 66

Rationalisation of the denominator of  gives

(a)

(b)

(c)

(d)

Solution 66

Correct option: (d)

Question 67

(a)

(b) 2

(c) 4

(d)

Solution 67

Correct option: (c)

Question 68

(a)

(b)

(c)

(d) None of these

Solution 68

Correct option: (c)

Question 69

(a)

(b) 14

(c) 49

(d) 48

Solution 69

Correct option: (b)

Question 70

(a) 0.075

(b) 0.75

(c) 0.705

(d) 7.05

Solution 70

Correct option: (c)

Question 71

(a) 0.375

(b) 0.378

(c) 0.441

(d) None of these

Solution 71

Correct option: (b)

Question 72

The value of  is

(a)

(b)

(c)

(d)

Solution 72

Correct option: (d)

Question 73

The value of  is

(a)

(b)

(c)

(d)

Solution 73

Correct option: (c)

Question 74

(a) 0.207

(b) 2.414

(c) 0.414

(d) 0.621

Solution 74

Correct option: (c)

Question 75

= ?

(a) 34

(b) 56

(c) 28

(d) 63

Solution 75

Correct option: (a)

Question 76

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

 Assertion (A) Reason (R) A rational number between two rational numbers p and q is .

Solution 76

Question 77

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

 Assertion (A) Reason (R) Square root of a positive integer which is not a perfect square is an irrational number.

Solution 77

Question 78

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

 Assertion (A) Reason (R) e is an irrational number. Π is an irrational number.

Solution 78

Question 79

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

 Assertion (A) Reason (R) The sum of a rational number and an irrational number is an irrational number.

Solution 79

Question 80

Match the following columns:

 Column I Column II (p) 14 (q) 6 (r) a rational number (s) an irrational number

(a)-…….,

(b)-…….,

(c)-…….,

(d)-…….,

Solution 80

Question 81

Match the following columns:

 Column I Column II

(a)-…….,

(b)-…….,

(c)-…….,

(d)-…….,

Solution 81

## Chapter 1 - Number Systems Exercise Ex. 1B

Question 1(i)

Without actual division, find which of the following rationals are terminating decimals.

Solution 1(i)

If the prime factors of the denominator are 2 and/or 5 then the rational number is a terminating decimal.

Since, 80 has prime factors 2 and 5,  is a terminating decimal.

Question 1(ii)

Without actual division, find which of the following rationals are terminating decimals.

Solution 1(ii)

If the prime factors of the denominators of the fraction are other than 2 and 5, then the rational number is not a terminating decimal.

Since, 24 has prime factors 2 and 3 and 3 is different from 2 and 5,

is not a terminating decimal.

Question 1(iii)

Without actual division, find which of the following rationals are terminating decimals.

Solution 1(iii)

If the prime factors of the denominators of the fraction are other than 2 and 5, then the rational number is not a terminating decimal.

Since 12 has prime factors 2 and 3  and 3 is different from 2 and 5,

is not a terminating decimal.

Question 1(iv)

Without actual division, find which of the following rational numbers are terminating decimals.

Solution 1(iv)

Since the denominator of a given rational number is not of the form 2m × 2n, where m and n are whole numbers, it has non-terminating decimal.

Question 2(i)

Write each of the following in decimal form and say what kind of decimal expansion each has.

Solution 2(i)

Hence, it has terminating decimal expansion.

Question 2(ii)

Write each of the following in decimal form and say what kind of decimal expansion each has.

Solution 2(ii)

Hence, it has terminating decimal expansion.

Question 2(iii)

Write each of the following in decimal form and say what kind of decimal expansion each has.

Solution 2(iii)

Hence, it has non-terminating recurring decimal expansion.

Question 2(iv)

Write each of the following in decimal form and say what kind of decimal expansion each has.

Solution 2(iv)

Hence, it has non-terminating recurring decimal expansion.

Question 2(v)

Write each of the following in decimal form and say what kind of decimal expansion each has.

Solution 2(v)

Hence, it has non-terminating recurring decimal expansion.

Question 2(vi)

Write each of the following in decimal form and say what kind of decimal expansion each has.

Solution 2(vi)

Hence, it has terminating decimal expansion.

Question 2(vii)

Write each of the following in decimal form and say what kind of decimal expansion each has.

Solution 2(vii)

Hence, it has terminating decimal expansion.

Question 2(viii)

Write each of the following in decimal form and say what kind of decimal expansion each has.

Solution 2(viii)

Hence, it has non-terminating recurring decimal expansion.

Question 3(i)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(i)

Let x =

i.e. x = 0.2222…. ….(i)

10x = 2.2222…. ….(ii)

On subtracting (i) from (ii), we get

9x = 2

Question 3(ii)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(ii)

Let x =

i.e. x = 0.5353….  ….(i)

100x = 53.535353…. ….(ii)

On subtracting (i) from (ii), we get

99x = 53

Question 3(iii)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(iii)

Let x =

i.e. x = 2.9393….  ….(i)

100x = 293.939……. ….(ii)

On subtracting (i) from (ii), we get

99x = 291

Question 3(iv)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(iv)

Let x =

i.e. x = 18.4848….  ….(i)

100x = 1848.4848……. ….(ii)

On subtracting (i) from (ii), we get

99x = 1830

Question 3(v)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(v)

Let x =

i.e. x = 0.235235..…   ….(i)

1000x = 235.235235……. ….(ii)

On subtracting (i) from (ii), we get

999x = 235

Question 3(vi)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(vi)

Let x =

i.e. x = 0.003232..…

100x = 0.323232……. ….(i)

10000x = 32.3232…. ….(ii)

On subtracting (i) from (ii), we get

9900x = 32

Question 3(vii)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(vii)

Let x =

i.e. x = 1.3232323..… ….(i)

100x = 132.323232……. ….(ii)

On subtracting (i) from (ii), we get

99x = 131

Question 3(viii)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(viii)

Let x =

i.e. x = 0.3178178..…

10x = 3.178178…… ….(i)

10000x = 3178.178……. ….(ii)

On subtracting (i) from (ii), we get

9990x = 3175

Question 3(ix)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(ix)

Let x =

i.e. x = 32.123535..…

100x = 3212.3535…… ….(i)

10000x = 321235.3535……. ….(ii)

On subtracting (i) from (ii), we get

9900x = 318023

Question 3(x)

Express each of the following decimals in the form , where p, q are integers and q 0.

Solution 3(x)

Let x =

i.e. x = 0.40777..…

100x = 40.777…… ….(i)

1000x = 407.777……. ….(ii)

On subtracting (i) from (ii), we get

900x = 367

Question 4

Express as a fraction in simplest form.

Solution 4

Let x =

i.e. x = 2.3636….  ….(i)

100x = 236.3636……. ….(ii)

On subtracting (i) from (ii), we get

99x = 234

Let y =

i.e. y = 0.2323….  ….(iii)

100y = 23.2323…. ….(iv)

On subtracting (iii) from (iv), we get

99y = 23

Question 5

Express in the form of

Solution 5

Let x =

i.e. x = 0.3838….  ….(i)

100x = 38.3838….  ….(ii)

On subtracting (i) from (ii), we get

99x = 38

Let y =

i.e. y = 1.2727….  ….(iii)

100y = 127.2727…….  ….(iv)

On subtracting (iii) from (iv), we get

99y = 126

Question 9(v)

Without actual division, find which of the following rationals are terminating decimals.

Solution 9(v)

If the prime factors of the denominator are 2 and/or 5 then the rational number is a terminating decimal.

Since 125 has prime factor 5 only

is a terminating decimal.

## Chapter 1 - Number Systems Exercise Ex. 1C

Question 1

What are irrational numbers? How do they differ from rational numbers? Give examples.

Solution 1

Irrational number: A number which cannot be expressed either as a terminating decimal or a repeating decimal is known as irrational number. Rather irrational numbers cannot be expressed in the fraction form,

For example, 0.101001000100001 is neither a terminating nor a repeating decimal and so is an irrational number.

Also, etc. are examples of irrational numbers.

Question 2(iii)

Classify the following numbers as rational or irrational. Give reasons to support you answer.

Solution 2(iii)

We know that, if n is a not a perfect square, then is an irrational number.

Here,  is a not a perfect square number.

So, is irrational.

Question 2(v)

Classify the following numbers as rational or irrational. Give reasons to support you answer.

Solution 2(v)

is the product of a rational number and an irrational number .

Theorem: The product of a non-zero rational number and an irrational number is an irrational number.

Thus, by the above theorem, is an irrational number.

So, is an irrational number.

Question 2(i)

Classify the following numbers as rational or irrational. Give reasons to support your answer.

Solution 2(i)

Since quotient of a rational and an irrational is irrational, the given number is irrational.

Question 2(ii)

Classify the following numbers as rational or irrational. Give reasons to support your answer.

Solution 2(ii)

Question 2(iv)

Classify the following numbers as rational or irrational. Give reasons to support your answer.

Solution 2(iv)

Question 2(vi)

Classify the following numbers as rational or irrational. Give reasons to support your answer.

4.1276

Solution 2(vi)

The given number 4.1276 has terminating decimal expansion.

Hence, it is a rational number.

Question 2(vii)

Classify the following numbers as rational or irrational. Give reasons to support your answer.

Solution 2(vii)

Since the given number has non-terminating recurring decimal expansion, it is a rational number.

Question 2(viii)

Classify the following numbers as rational or irrational. Give reasons to support your answer.

1.232332333....

Solution 2(viii)

The given number 1.232332333.... has non-terminating and non-recurring decimal expansion.

Hence, it is an irrational number.

Question 2(ix)

Classify the following numbers as rational or irrational. Give reasons to support your answer.

3.040040004.....

Solution 2(ix)

The given number 3.040040004..... has non-terminating and non-recurring decimal expansion.

Hence, it is an irrational number.

Question 2(x)

Classify the following numbers as rational or irrational. Give reasons to support your answer.

2.356565656.....

Solution 2(x)

The given number 2.356565656..... has non-terminating recurring decimal expansion.

Hence, it is a rational number.

Question 2(xi)

Classify the following numbers as rational or irrational. Give reasons to support your answer.

6.834834....

Solution 2(xi)

The given number 6.834834.... has non-terminating recurring decimal expansion.

Hence, it is a rational number.

Question 3

Let x be a rational number and y be an irrational number. Is x + y necessarily an irrational number? Give an example in support of your answer.

Solution 3

We know that the sum of a rational and an irrational is irrational.

Hence, if x is rational and y is irrational, then x + y is necessarily an irrational number.

For example,

Question 4

Let a be a rational number and b be an irrational number. Is ab necessarily an irrational number? Justify your answer with an example.

Solution 4

We know that the product of a rational and an irrational is irrational.

Hence, if a is rational and b is irrational, then ab is necessarily an irrational number.

For example,

Question 5

Solution 5

No, the product of two irrationals need not be an irrational.

For example,

Question 6

Give an example of two irrational numbers whose

(i) difference is an irrational number.

(ii) difference is a rational number.

(iii) sum is an irrational number.

(iv) sum is an rational number.

(v) product is an irrational number.

(vi) product is a rational number.

(vii) quotient is an irrational number.

(viii) quotient is a rational number.

Solution 6

(i) Difference is an irrational number:

(ii) Difference is a rational number:

(iii) Sum is an irrational number:

(iv) Sum is an rational number:

(v) Product is an irrational number:

(vi) Product is a rational number:

(vii) Quotient is an irrational number:

(viii) Quotient is a rational number:

Question 7

Examine whether the following numbers are rational or irrational.

Solution 7

Question 8

Insert a rational and an irrational number between 2 and 2.5

Solution 8

Rational number between 2 and 2.5 =

Irrational number between 2 and 2.5 =

Question 9

How many irrational numbers lie between? Find any three irrational numbers lying between .

Solution 9

There are infinite irrational numbers between.

We have

Hence, three irrational numbers lying between  are as follows:

1.5010010001……., 1.6010010001…… and 1.7010010001…….

Question 10

Find two rational and two irrational numbers between 0.5 and 0.55.

Solution 10

Since 0.5 < 0.55

Let x = 0.5, y = 0.55 and y = 2

Two irrational numbers between 0.5 and 0.55 are 0.5151151115……. and 0.5353553555….

Question 11

Find three different irrational numbers between the rational numbers .

Solution 11

Thus, three different irrational numbers between the rational numbers  are as follows:

0.727227222….., 0.757557555….. and 0.808008000…..

Question 12

Find two rational numbers of the form between the numbers 0.2121121112... and  0.2020020002......

Solution 12

Let a and b be two rational numbers between the numbers 0.2121121112... and 0.2020020002......

Now, 0.2020020002...... <0.2121121112...

Then, 0.2020020002...... < a < b < 0.2121121112...

Question 13

Find two irrational numbers between 0.16 and 0.17.

Solution 13

Two irrational numbers between 0.16 and 0.17 are as follows:

0.1611161111611111611111…… and 0.169669666…….

Question 14(i)

State in each case, whether the given statement is true or false.

The sum of two rational numbers is rational.

Solution 14(i)

True

Question 14(ii)

State in each case, whether the given statement is true or false.

The sum of two irrational numbers is irrational.

Solution 14(ii)

False

Question 14(iii)

State in each case, whether the given statement is true or false.

The product of two rational numbers is rational.

Solution 14(iii)

True

Question 14(iv)

State in each case, whether the given statement is true or false.

The product of two irrational numbers is irrational.

Solution 14(iv)

False

Question 14(v)

State in each case, whether the given statement is true or false.

The sum of a rational number and an irrational number is irrational.

Solution 14(v)

True

Question 14(vi)

State in each case, whether the given statement is true or false.

The product of a nonzero rational number and an irrational number is a rational number.

Solution 14(vi)

False

Question 14(vii)

State in each case, whether the given statement is true or false.

Every real number is rational.

Solution 14(vii)

False

Question 14(viii)

State in each case, whether the given statement is true or false.

Every real number is either rational or irrational.

Solution 14(viii)

True

Question 14(ix)

State in each case, whether the given statement is true or false.

is irrational and is rational.

Solution 14(ix)

True

## Chapter 1 - Number Systems Exercise Ex. 1D

Question 1(i)

Solution 1(i)

We have:

Question 1(ii)

Solution 1(ii)

We have:

Question 1(iii)

Solution 1(iii)

Question 2(i)

Multiply:

Solution 2(i)

Question 2(ii)

Multiply:

Solution 2(ii)

Question 2(iii)

Multiply:

Solution 2(iii)

Question 2(iv)

Multiply:

Solution 2(iv)

Question 2(v)

Multiply:

Solution 2(v)

Question 2(vi)

Multiply:

Solution 2(vi)

Question 3(i)

Divide:

Solution 3(i)

Question 3(ii)

Divide:

Solution 3(ii)

Question 3(iii)

Divide:

Solution 3(iii)

Question 4(iii)

Simplify:

Solution 4(iii)

Question 4(iv)

Simplify:

Solution 4(iv)

Question 4(vi)

Simplify:

Solution 4(vi)

Question 4(i)

Simplify

Solution 4(i)

= 9 - 11

= -2

Question 4(ii)

Simplify

Solution 4(ii)

= 9 - 5

= 4

Question 4(v)

Simplify

Solution 4(v)

Question 5

Simplify

Solution 5

Question 6(i)

Examine whether the following numbers are rational or irrational:

Solution 6(i)

Thus, the given number is rational.

Question 6(ii)

Examine whether the following numbers are rational or irrational:

Solution 6(ii)

Clearly, the given number is irrational.

Question 6(iii)

Examine whether the following numbers are rational or irrational:

Solution 6(iii)

Thus, the given number is rational.

Question 6(iv)

Examine whether the following numbers are rational or irrational:

Solution 6(iv)

Thus, the given number is irrational.

Question 7

On her birthday Reema distributed chocolates in an orphanage. The total number of chocolates she distributed is given by .

(i) Find the number of chocolates distributed by her.

(ii) Write the moral values depicted here by Reema.

Solution 7

(i) Number of chocolates distributed by Reema

(ii) Loving, helping and caring attitude towards poor and needy children.

Question 8(i)

Simplify

Solution 8(i)

Question 8(ii)

Simplify

Solution 8(ii)

Question 8(iii)

Simplify

Solution 8(iii)

## Chapter 1 - Number Systems Exercise Ex. 1G

Question 1(iii)

Simplify:

Solution 1(iii)

Question 1(i)

Simplify

Solution 1(i)

Question 1(ii)

Simplify

Solution 1(ii)

Question 1(iv)

Simplify

Solution 1(iv)

Question 2(i)

Simplify:

Solution 2(i)

Question 2(ii)

Simplify:

Solution 2(ii)

Question 2(iii)

Simplify:

Solution 2(iii)

Question 3(i)

Simplify:

Solution 3(i)

Question 3(ii)

Simplify:

Solution 3(ii)

Question 3(iii)

Simplify:

Solution 3(iii)

Question 4(i)

Simplify:

Solution 4(i)

Question 4(ii)

Simplify:

Solution 4(ii)

Question 4(iii)

Simplify:

Solution 4(iii)

Question 5(i)

Evaluate:

Solution 5(i)

Question 5(ii)

Evaluate:

Solution 5(ii)

Question 5(iii)

Evaluate:

Solution 5(iii)

Question 5(iv)

Evaluate:

Solution 5(iv)

Question 5(v)

Evaluate:

Solution 5(v)

Question 5(vi)

Evaluate:

Solution 5(vi)

Question 6(i)

If a = 2, b = 3, find the value of (ab + ba)-1

Solution 6(i)

Given, a = 2 and b = 3

Question 6(ii)

If a = 2, b = 3, find the value of (aa + bb)-1

Solution 6(ii)

Given, a = 2 and b = 3

Question 7(i)

Simplify

Solution 7(i)

Question 7(ii)

Simplify

(14641)0.25

Solution 7(ii)

(14641)0.25

Question 7(iii)

Simplify

Solution 7(iii)

Question 7(iv)

Simplify

Solution 7(iv)

Question 8(i)

Evaluate

Solution 8(i)

Question 8(ii)

Evaluate

Solution 8(ii)

Question 8(iii)

Evaluate

Solution 8(iii)

Question 8(iv)

Evaluate

Solution 8(iv)

Question 9(i)

Evaluate

Solution 9(i)

Question 9(ii)

Evaluate

Solution 9(ii)

Question 9(iii)

Evaluate

Solution 9(iii)

Question 9(iv)

Evaluate

Solution 9(iv)

Question 10(i)

Prove that

Solution 10(i)

Question 10(ii)

Prove that

Solution 10(ii)

Question 10(iii)

Prove that

Solution 10(iii)

Question 11

Simplify and express the result in the exponential form of x.

Solution 11

Question 12

Simplify the product

Solution 12

Question 13(i)

Simplify

Solution 13(i)

Question 13(ii)

Simplify

Solution 13(ii)

Question 13(iii)

Simplify

Solution 13(iii)

Question 14(i)

Find the value of x in each of the following.

Solution 14(i)

Question 14(ii)

Find the value of x in each of the following.

Solution 14(ii)

Question 14(iii)

Find the value of x in each of the following.

Solution 14(iii)

Question 14(iv)

Find the value of x in each of the following.

5x - 3× 32x - 8 = 225

Solution 14(iv)

5x - 3 × 32x - 8 = 225

5x - 3× 32x - 8 = 52 × 32

x - 3 = 2 and 2x - 8 = 2

x = 5 and 2x = 10

x = 5

Question 14(v)

Find the value of x in each of the following.

Solution 14(v)

Question 15(i)

Prove that

Solution 15(i)

Question 15(ii)

Prove that

Solution 15(ii)

Question 15(iii)

Prove that

Solution 15(iii)

Question 15(iv)

Prove that

Solution 15(iv)

Question 16

If x is a positive real number and exponents are rational numbers, simplify

Solution 16

Question 17

If prove that m - n = 1.

Solution 17

Question 18

Write the following in ascending order of magnitude.

Solution 18

## Chapter 1 - Number Systems Exercise Ex. 1A

Question 1

Is zero a rational number? Justify.

Solution 1

A number which can be expressed as , where 'a' and 'b' both are integers and b ≠ 0, is called a rational number.

Since, 0 can be expressed as , it is a rational number.

Question 2(i)

Represent each of the following rational numbers on the number line:

(i)

Solution 2(i)

(i)

Question 2(ii)

Represent each of the following rational numbers on the number line:

(ii)

Solution 2(ii)

(ii)

Question 2(iii)

Represent each of the following rational numbers on the number line:

Solution 2(iii)

Question 2(iv)

Represent each of the following rational numbers on the number line:

(iv) 1.3

Solution 2(iv)

(iv) 1.3

Question 2(v)

Represent each of the following rational numbers on the number line:

(v) -2.4

Solution 2(v)

(v) -2.4

Question 3(i)

Find a rational number lying between

Solution 3(i)

Question 3(ii)

Find a rational number lying between

1.3 and 1.4

Solution 3(ii)

Question 3(iii)

Find a rational number lying between

-1 and

Solution 3(iii)

Question 3(iv)

Find a rational number lying between

Solution 3(iv)

Question 3(v)

Find a rational number between

Solution 3(v)

Question 4

Find three rational numbers lying between

How many rational numbers can be determined between these two numbers?

Solution 4

Infinite rational numbers can be determined between given two rational numbers.

Question 5

Find four rational numbers between

Solution 5

We have

We know that 9 < 10 < 11 < 12 < 13 < 14 < 15

Question 6

Find six rational numbers between 2 and 3.

Solution 6

2 and 3 can be represented asrespectively.

Now six rational numbers between 2 and 3 are

.

Question 7

Find five rational numbers between

Solution 7

Question 8

Insert 16 rational numbers between 2.1 and 2.2.

Solution 8

Let x = 2.1 and y = 2.2

Then, x < y because 2.1 < 2.2

Or we can say that,

Or,

That is, we have,

We know that,

Therefore, we can have,

Therefore, 16 rational numbers between, 2.1 and 2.2 are:

So, 16 rational numbers between 2.1 and 2.2 are:

2.105, 2.11, 2.115, 2.12, 2.125, 2.13, 2.135, 2.14, 2.145, 2.15, 2.155, 2.16, 2.165, 2.17, 2.175, 2.18

Question 9(i)

State whether the given statement is true or false. Give reasons. for your answer.

Every natural number is a whole number.

Solution 9(i)

True. Since the collection of natural number is a sub collection of whole numbers, and every element of natural numbers is an element of whole numbers

Question 9(ii)

Write, whether the given statement is true or false. Give reasons.

Every whole number is a natural number.

Solution 9(ii)

False. Since 0 is whole number but it is not a natural number.

Question 9(iii)

State whether the following statements are true or false. Give reasons for your answer.

Every integer is a whole number.

Solution 9(iii)

False, integers include negative of natural numbers as well, which are clearly not whole numbers. For example -1 is an integer but not a whole number.

Question 9(iv)

Write, whether the given statement is true or false. Give reasons.

Ever integer is a rational number.

Solution 9(iv)

True. Every integer can be represented in a fraction form with denominator 1.

Question 9(v)

State whether the following statements are true or false. Give reasons for your answer.

Every rational number is an integer.

Solution 9(v)

False, integers are counting numbers on both sides of the number line i.e. they are both positive and negative while rational numbers are of the form . Hence, Every rational number is not an integer but every integer is a rational number.

Question 9(vi)

Write, whether the given statement is true or false. Give reasons.

Every rational number is a whole number.

Solution 9(vi)

False. Since division of whole numbers is not closed under division, the value of , may not be a whole number.

## Chapter 1 - Number Systems Exercise Ex. 1E

Question 1

Represent on the number line.

Solution 1

Draw a number line as shown.

On the number line, take point O corresponding to zero.

Now take point A on number line such that OA = 2 units.

Draw perpendicular AZ at A on the number line and cut-off arc AB = 1 unit.

By Pythagoras Theorem,

OB2 = OA2 + AB2 = 22 + 12 = 4 + 1 = 5

OB =

Taking O as centre and OB =  as radius draw an arc cutting real line at C.

Clearly, OC = OB =

Hence, C represents  on the number line.

Question 2

Locate on the number line.

Solution 2

Draw a number line as shown.

On the number line, take point O corresponding to zero.

Now take point A on number line such that OA = 1 unit.

Draw perpendicular AZ at A on the number line and cut-off arc AB = 1 unit.

By Pythagoras Theorem,

OB2 = OA2 + AB2 = 12 + 12 = 1 + 1 = 2

OB =

Taking O as centre and OB =  as radius draw an arc cutting real line at C.

Clearly, OC = OB =

Thus, C represents  on the number line.

Now, draw perpendicular CY at C on the number line and cut-off arc CE = 1 unit.

By Pythagoras Theorem,

OE2 = OC2 + CE2 = 2 + 12 = 2 + 1 = 3

OE =

Taking O as centre and OE =  as radius draw an arc cutting real line at D.

Clearly, OD = OE =

Hence, D represents  on the number line.

Question 3

Locate on the number line.

Solution 3

Draw a number line as shown.

On the number line, take point O corresponding to zero.

Now take point A on number line such that OA = 3 units.

Draw perpendicular AZ at A on the number line and cut-off arc AB = 1 unit.

By Pythagoras Theorem,

OB2 = OA2 + AB2 = 32 + 12 = 9 + 1 = 10

OB =

Taking O as centre and OB =  as radius draw an arc cutting real line at C.

Clearly, OC = OB =

Hence, C represents  on the number line.

Question 4

Locate on the number line.

Solution 4

Draw a number line as shown.

On the number line, take point O corresponding to zero.

Now take point A on number line such that OA = 2 units.

Draw perpendicular AZ at A on the number line and cut-off arc AB = 2 units.

By Pythagoras Theorem,

OB2 = OA2 + AB2 = 22 + 22 = 4 + 4 = 8

OB =

Taking O as centre and OB =  as radius draw an arc cutting real line at C.

Clearly, OC = OB =

Hence, C represents  on the number line.

Question 5

Represent  geometrically on the number line.

Solution 5

Draw a line segment AB = 4.7 units and extend it to C such that BC = 1 unit.

Find the midpoint O of AC.

With O as centre and OA as radius, draw a semicircle.

Now, draw BD AC, intersecting the semicircle at D.

Then, BD =  units.

With B as centre and BD as radius, draw an arc, meeting AC produced at E.

Then, BE = BD =  units.

Question 6

Represent on the number line.

Solution 6

Draw a line segment OB = 10.5 units and extend it to C such that BC = 1 unit.

Find the midpoint D of OC.

With D as centre and DO as radius, draw a semicircle.

Now, draw BE AC, intersecting the semicircle at E.

Then, BE =  units.

With B as centre and BE as radius, draw an arc, meeting AC produced at F.

Then, BF = BE =  units.

Question 7

Represent geometrically on the number line.

Solution 7

Draw a line segment AB = 7.28 units and extend it to C such that BC = 1 unit.

Find the midpoint O of AC.

With O as centre and OA as radius, draw a semicircle.

Now, draw BD AC, intersecting the semicircle at D.

Then, BD = units.

With D as centre and BD as radius, draw an arc, meeting AC produced at E.

Then, BE = BD = units.

Question 8

Represent  on the number line.

Solution 8

Draw a line segment OB = 9.5 units and extend it to C such that BC = 1 unit.

Find the midpoint D of OC.

With D as centre and DO as radius, draw a semicircle.

Now, draw BE AC, intersecting the semicircle at E.

Then, BE =  units.

With B as centre and BE as radius, draw an arc, meeting AC produced at F.

Then, BF = BE =  units.

Extend BF to G such that FG = 1 unit.

Then, BG =

Question 9

Visualize the representation of 3.765 on the number line using successive magnification.

Solution 9

Question 10

Visualize the representation of  on the number line up to 4 decimal places.

Solution 10

## Chapter 1 - Number Systems Exercise Ex. 1F

Question 1

Write the rationalising factor of the denominator in .

Solution 1

The rationalising factor of the denominator in  is

Question 2(i)

Rationalise the denominator of following:

Solution 2(i)

On multiplying the numerator and denominator of the given number by , we get

Question 2(ii)

Rationalise the denominator of following:

Solution 2(ii)

On multiplying the numerator and denominator of the given number by , we get

Question 2(iii)

Rationalise the denominator of following:

Solution 2(iii)

Question 2(iv)

Rationalise the denominator of following:

Solution 2(iv)

Question 2(v)

Rationalise the denominator of following:

Solution 2(v)

Question 2(vi)

Rationalise the denominator of each of the following.

Solution 2(vi)

Question 2(vii)

Rationalise the denominator of each of the following.

Solution 2(vii)

Question 2(viii)

Rationalise the denominator of each of the following.

Solution 2(viii)

Question 2(ix)

Rationalise the denominator of each of the following.

Solution 2(ix)

Question 3(i)

find the value to three places of decimals, of each of the following.

Solution 3(i)

Question 3(ii)

find the value to three places of decimals, of each of the following.

Solution 3(ii)

Question 3(iii)

find the value to three places of decimals, of each of the following.

Solution 3(iii)

Question 4(i)

Find rational numbers a and b such that

Solution 4(i)

Question 4(ii)

Find rational numbers a and b such that

Solution 4(ii)

Question 4(iii)

Find rational numbers a and b such that

Solution 4(iii)

Question 4(iv)

Find rational numbers a and b such that

Solution 4(iv)

Question 5(i)

find to three places of decimals, the value of each of the following.

Solution 5(i)

Question 5(ii)

find to three places of decimals, the value of each of the following.

Solution 5(ii)

Question 5(iii)

find to three places of decimals, the value of each of the following.

Solution 5(iii)

Question 5(iv)

find to three places of decimals, the value of each of the following.

Solution 5(iv)

Question 5(v)

find to three places of decimals, the value of each of the following.

Solution 5(v)

Question 5(vi)

find to three places of decimals, the value of each of the following.

Solution 5(vi)

Question 6(i)

Simplify by rationalising the denominator.

Solution 6(i)

Question 6(ii)

Simplify by rationalising the denominator.

Solution 6(ii)

Question 7(i)

Simplify:

Solution 7(i)

Question 7(ii)

Simplify

Solution 7(ii)

Question 7(iii)

Simplify

Solution 7(iii)

Question 7(iv)

Simplify

Solution 7(iv)

Question 8(i)

Prove that

Solution 8(i)

Question 8(ii)

Prove that

Solution 8(ii)

Question 9

Find the values of a and b if

Solution 9

Question 10

Simplify

Solution 10

Question 11

Solution 11

Thus, the given number is rational.

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

*Question modified

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

.

Solution 21

Question 22(i)

Rationalise the denominator of each of the following.

Solution 22(i)

Question 22(ii)

Rationalise the denominator of each of the following.

Solution 22(ii)

Question 22(iii)

Rationalise the denominator of each of the following.

Solution 22(iii)

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

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