Page 1.24

### Exercise 1.1

Type 1

#### Question 1

**State whether the following statements are true or false ?**

**Also give reasons for your answer.**

**(i) Every whole number is a natural number**

Sol : False, since 0 (zero) is a whole number but not a natural number.

**(ii) Every integer is a rational number**

Sol : True, since every integer *m* may be written in the form so it is a rational number.

**(iii) Every rational number is an integer**

Sol : False, since is a rational number but it is not an integer

**(iv) If any rational number is an integer , then q=±1**

Sol : False, since (integer) but q=2

#### Question 2

**Write the following integers in the form of rational number **

**(i) 9**

Sol :

**(ii) -13**

Sol :

**(iii) 20**

Sol :

#### Question 3

**(i) Is a rational number, if p=0 ?**

Sol :

Yes , because by definition of rational number which can be expressed in term form of p/q when q is non zero integer and p is a integer .

**(ii) Is a rational number , if q=0 ?**

Sol :

No , According to definition of rational number which can be expressed in term form of p/q when q is non zero integer and p is a integer .

#### Question 4

**Fill up the blanks with the word terminating, non—terminating, repeating.**

[Note:Let x = p/q be a rational number, such that the prime factorization of q is of the form 2^{n} 5^{m}, where *n*, *m* are non-negative integers. Then x has a decimal expansion which terminates.]

**(i) On changing to a decimal , it will be __ decimal**

Sol:

Terminating

**(ii) On changing to a decimal, it will be __ decimal**

Sol:

Terminating

**(iii) On changing to a decimal, it will be __ decimal**

Sol:

Non-Terminating

**(iv) If denominator of a rational number has prime factors 2 and 5 only , the can be written in __ decimal form**

Sol:

Terminating

#### Question 5

**Without writing following rational numbers in decimal forms , state which will have terminating decimal expansion ?**

[Note:Let x = p/q be a rational number, such that the prime factorization of q is of the form 2^{n} 5^{m}, where *n*, *m* are non-negative integers. Then x has a decimal expansion which terminates.]

**(i) **

Sol:

Here q=11 and it is not in the form of 2^{n}5^{m}, it is not a terminating decimal.

**(ii) **

Sol:

Here q=9

Can be written as 3^{2 }and it is not in the form of 2^{n}5^{m}, it is not a terminating decimal.

**(iii) **

Sol:

Here q=16

Can be written as 2^{4 }and it is in the form of 2^{n}5^{m}, it is a terminating decimal.

**(iv) **

Sol :

Here q=30

Can be written as 2×3×5

Since 3 is also there and it is not in the form of 2^{n}5^{m}, it is not a terminating decimal.

#### Question 6

**Without writing following rational numbers in decimal forms , state which will have non-terminating decimal expansion ?**

[Note:Let x = p/q be a rational number, such that the prime factorization of q is of the form 2^{n} 5^{m}, where *n*, *m* are non-negative integers. Then x has a decimal expansion which terminates.]

**(i) **

Sol :

Here q=9

Can be written as 3^{2 }and it is not in the form of 2^{n}5^{m}, it is not a terminating decimal.

**(ii) **

Sol :

Here q=8

Can be written as 2^{3 }and it is in the form of 2^{n}5^{m}, it is a terminating decimal.

**(iii) **

Sol :

Here q=25

Can be written as 5^{2 }and it is in the form of 2^{n}5^{m}, it is a terminating decimal.

**(iv) **

Sol :

Here q=20

Can be written as 2^{2}×5 and it is in the form of 2^{n}5^{m}, it is a terminating decimal.

#### Question 7

**State which of the following rational numbers represent terminating decimal expansion ?**

^{n} 5^{m}, where *n*, *m* are non-negative integers. Then x has a decimal expansion which terminates.]

**(i) **

Sol :

Denominator = 8 = 2×2×2

Can be written as terminating expansion [is in form 2^{n}5^{m}]

**(ii) **

Sol :

Denominator = 7

Can not be written as terminating expansion [not in form 2^{n}5^{m}]

**(iii) **

Sol :

Denominator = 40 = 2×2×2×5

Can be written as terminating expansion [is in form 2^{n}5^{m}]

**(iv) **

Sol :

Denominator = 130 = 2×5×13

Can not be written as terminating expansion because it is not in the form of 2^{n}5^{m} [as 13 is present in prime factorization]

**(v) **

Sol :

Denominator = 35 = 7×5

Can not be written as terminating expansion because it is not in the form of 2^{n}5^{m} [as 7 is present in prime factorization]

**(vi) **

Sol :

Denominator = 128 = 2×2×2×2×2×2×2

Can be written as terminating expansion [ is in the form of 2^{n}5^{m}]

**(vii) **

Sol :

Denominator = 138 = 2×3×23

Can not be written as terminating expansion because it is not in the form of 2^{n}5^{m} [as 3 and 23 is present in prime factorization]

**(viii) **

Sol :

Denominator = 50 = 2×5×5

Can be written as terminating expansion because it is in the form of 2^{n}5^{m}

#### Question 8

**State which of the following rational numbers represent non-terminating decimal expansion ?**

^{n} 5^{m}, where *n*, *m* are non-negative integers. Then x has a decimal expansion which terminates.]

**(i) **

Sol :

Denominator = 7

Can not be written as terminating expansion [not in form 2^{n}5^{m}]

**(ii) **

Sol :

Denominator = 10 = 2×5

Can be written as terminating expansion [is in form 2^{n}5^{m}]

**(iii) **

Sol :

Denominator = 18 = 2×3×3

Can not be written as terminating expansion because it is not in the form of 2^{n}5^{m} [as 3 is present in prime factorization]

**(iv) **

Sol :

Denominator = 250 =2×5×5×5

Can be written as terminating expansion [is in form 2^{n}5^{m}]

**(v) **

Sol :

Denominator = 21 = 3×7

Can not be written as terminating expansion because it is not in the form of 2^{n}5^{m} [as 3 and 7 is present in prime factorization]

**(vi) **

Sol :

Denominator = 30 = 2×3×5

Can not be written as terminating expansion because it is not in the form of 2^{n}5^{m} [as 3 is present in prime factorization]

**(vii) **

Sol :

Denominator = 121 = 11×11

Can not be written as terminating expansion because it is not in the form of 2^{n}5^{m} [as 11 is present in prime factorization]

**(viii) **

Sol :

Denominator = 60 = 2×2×3×5

Can not be written as terminating expansion because it is not in the form of 2^{n}5^{m} [as 3 is present in prime factorization]

#### Question 9

**Write the following in decimal form and state, what kind of decimal expansion each has ?**

**(i)**

Sol :

Its denominator 100 = 2^{2}×5^{2}=(2×5)^{2}

Since denominator has prime factors 2 and 5 only , so its decimal expansion is terminating

Now ,

**(ii)**

Sol :

Its denominator 3

Since its denomination is not in the form of 2^{m}×5^{n} , therefore its decimal expansion is non-terminating recuring

**(iii) **

Its denominator 11

Since its denomination is not in the form of 2^{m}×5^{n} , therefore its decimal expansion is non-terminating recuring

**(iv) **

Its denominator 8 = 2^{3}

Since denominator has prime factors 2 only , so its decimal expansion is terminating

=0.875

**(v) **

Its denominator 10 = 2×5

Since denominator has prime factors 2 and 5 only , so its decimal expansion is terminating

=0.3

**(vi) **

Its denominator 3

Since its denomination is not in the form of 2^{m}×5^{n} , therefore its decimal expansion is non-terminating recuring

#### Question 10

**Write the following rational numbers in the decimal form:**

**(i) **

Sol : 0.3125

**(ii) **

Sol : 3.3

**(iii) **

Sol : 0.27

**(iv) **

Sol :

**(v) **

Sol :

**(vi) **

Sol : 0.654

**(vii) **

Sol : 0.83

**(viii) **

Sol : 1.2692307

Type 2

#### Question 11

**Show that the following numbers can be represented in the form where p and q are integers and q≠0:**

**(i) **

Sol :

Let

Then , x=1.2727.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 27 , in which number of digits is 2 .

Multiply both sides of (i) by 10^{2 }or 100

100x=100×(1.2727..)

100x=127.2727.. (ii)

Subtracting (i) from (ii) , we get

100x-x=127.2727.. – 1.2727..

99x=126

**(ii) 0.3333…**

Sol :

Let x=0.3.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 3 , in which number of digits is 1 .

Multiply both sides of (i) by 10^{1 }or 10

10x=10×(0.3..)

10x=3.3.. (ii)

Subtracting (i) from (ii) , we get

10x-x=3.3.. – 0.3..

9x=3

**(iii) **

Sol :

Let x=0.6.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 6 , in which number of digits is 1 .

Multiply both sides of (i) by 10^{1 }or 10

10x=10×(0.6..)

10x=6.6.. (ii)

Subtracting (i) from (ii) , we get

10x-x=6.6.. – 0.6..

9x=6

**(iv) 0.2353532…**

Sol :

Let x=0.235353.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 235353 , in which number of digits is 6 .

Multiply both sides of (i) by 10^{6 }or 1000000

1000000x=1000000×(0.235353..)

1000000x=235353.235353.. (ii)

Subtracting (i) from (ii) , we get

1000000x-x=235353.235353.. – 0.235353..

99999x=235353

**(v)**

Sol :

Let x=3.142678.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 142678 , in which number of digits is 6 .

Multiply both sides of (i) by 10^{6 }or 1000000

1000000x=1000000×(3.142678..)

1000000x=3142678.142678.. (ii)

Subtracting (i) from (ii) , we get

1000000x-x=3142678.142678.. – 3.142678..

99999x=3142675

Type 3

#### Question 12

**Write the following in the form of , where p and q are integers and q≠0 :**

**(i) 0.25**

Sol :

**(ii) 0.54**

Sol :

**(iii) **

Sol :

Let x=6.4646.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 46 , in which number of digits is 2 .

Multiply both sides of (i) by 10^{2 }or 100

100x=100×(6.4646..)

100x=646.4646.. (ii)

Subtracting (i) from (ii) , we get

100x-x=646.4646.. – 6.4646..

99x=145

**(iv) **

Sol :

Let x=0.033.. (i)

Here, number of digits after decimal which are not in the repeating block, m=1 .

Now multiply both sides of (i) by 10^{m}=10^{1}=10 we get

10x=10×0.03..

10x=0.3.. (ii)

Again, number of digits in repeating block, n=1

Multiplying both sides of (ii) by 10^{n}=10^{1}=10 we get

100x=10×0.3..

100x=3.3.. (iii)

Subtracting (ii) from (iii) , we get

100x-10x=3.3.. – 0.3..

90x=3

**(v) **

Sol :

Let x=4.6732732.. (i)

Here, number of digits after decimal which are not in the repeating block, m=1 .

Now multiply both sides of (i) by 10^{m}=10^{1}=10 we get

10x=10×4.6732732..

10x=46.732732.. (ii)

Again, number of digits in repeating block, n=3

Multiplying both sides of (ii) by 10^{3}=10^{3}=1000 we get

10000x=1000×46.732732

10000x=46732.732.. (iii)

Subtracting (ii) from (iii) , we get

10000x-10x=46732.732.. – 46.732732..

9990x=46686

**(vi) **

Sol :

Let x=4.2727.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 27 , in which number of digits is 2 .

Multiply both sides of (i) by 10^{2 }or 100

100x=100×(4.2727..)

100x=427.27.. (ii)

Subtracting (i) from (ii) , we get

100x-x=427.27.. – 4.27..

99x=423

**(vii) **

Sol :

Let x=3.7.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 7 , in which number of digits is 1 .

Multiply both sides of (i) by 10^{1 }or 10

10x=10×(3.7..)

10x=37.7.. (ii)

Subtracting (i) from (ii) , we get

10x-x=37.7.. – 3.7..

9x=34

**(viii) **

Sol :

Let x=18.48.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 48 , in which number of digits is 2 .

Multiply both sides of (i) by 10^{2 }or 100

100x=100×(18.48..)

100x=1848.48.. (ii)

Subtracting (i) from (ii) , we get

100x-x=1848.48.. – 18.48..

99x=1830

Type 4

#### Question 13

**Find five rational numbers between 1 and 2.**

Sol :

Here a=1 and b=2 and n=5

Now,

The required rational numbers will be

⇒ , , , ,

⇒ , , , ,

⇒ , , , ,

⇒ , , , ,

⇒ , , , ,

#### Question 14

**Find five rational numbers between and **

[Hint: ] Now write five rational numbers between ]

Sol :

Let and

Five rational numbers are

#### Question 15

**Write one rational number between **

Sol :

Here , and n=1

The required rational numbers will be

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

#### Question 16

**Find three rational numbers between **

Sol :

Here , and n=3

Now,

The required rational numbers will be

⇒ , ,

⇒ , ,

⇒ , ,

⇒ , ,

⇒ , ,

⇒ , ,

⇒ , ,

#### Question 17

**Find three rational numbers between 0 and 0.2**

Sol :

Average of 0 and 0.2

Average of 0.1 and 0

Average of 0.05 and 0.2

The required rational numbers are 0.1 , 0.05 , 0.12

**Alternate method**

Here a=0 , b=0.2 and n=3

Now,

The required rational numbers will be

⇒ , ,

⇒ , ,

⇒ , ,

⇒0.05 , 0.1 , 0.15

#### Question 18

**Find two rational numbers between **

Sol :

Here , and n=2

Now,

The required rational numbers will be

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒ ,

#### Question 19

**Find one rational numbers between **

Sol :

Here ,

The required rational numbers will be

Average

⇒

#### Question 20

**Find two rational numbers between **

Sol :

Here , and n=2

Now,

The required rational numbers will be

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒

#### Question 21

**Find two rational numbers between **

Sol :

Here ,

Average

and another is

Average

The required rational numbers will be

⇒ ,

**Alternate Method**

Here , and n=2

Now,

The required rational numbers will be

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒ ,

⇒ ,

#### Question 22

**Find two rational numbers between 0 and 0.1**

Sol :

Here a=0 and b=0.1

The required rational numbers will be

Average of 0 and 0.1

Average of 0.05 and 0.1

Two rational numbers are

⇒0.05 , 0.07

#### Question 23

**How many rational numbers can be written between 2.5 and 2.6 ? Write ten of these numbers**

Sol :

⇒Infinitely many rational numbers can be written between 2.5 and 2.6

⇒ Ten rational numbers between 2.5 and 2.6 are 2.51 , 2.52 , 2.53 , 2.54 , 2.55 , 2.56 , 2.57 , 2.58 , 2.59 , 2.511 .

Type 5

#### Question 24

**If x and y are rational numbers then, show that the following are also rational numbers :**

Properties of rational numbers

⇒Sum of two rational numbers is rational number ..(i)

⇒Difference of two rational numbers is rational number ..(ii)

⇒Product of two rational numbers is rational number..(iii)

⇒Division of two rational numbers is rational number..(iv)

**(i) x ^{2}-y^{2}**

Sol :

x^{2}-y^{2 }

can be written as (x+b)(x-b)

x+b is a rational number by (i)

x-b is a rational number by (ii)

So , x^{2}-y^{2} is also a rational number

**(ii) x-y**

Sol :

x-y is a rational number by (ii)

**(iii) , where y≠0**

Sol :

, where y≠0 is a rational number by (iv)

**(iv) x+y**

Sol :

x+y is rational number by (i)

#### Question 25

**If a is a rational number , then prove that a ^{n }will be a rational number , where n is a rational number greater than 1 .**

Sol :

Rational Number is a number that can be expressed in the form p/q, where q not equal to zero.

We know that product of two rational number is always a rational number.

Hence if a is a rational number then

a^{2} = a × a is a rational number.

a^{3} = a^{2}× a is a rational number,

a^{4} = a^{3}×a is a rational number,

……

……

∴** **a^{n} = a^{n-1}×a is a rational number.