Solution of Chapter 1. Number Systems (NCERT Mathematics Exemplar Book)

Chapter Exercises

Exercise 1.1
Exercise 1.2
Exercise 1.3
Exercise 1.4

Exercise 1.1

1

Every rational number is
2

Between two rational numbers
3

Decimal representation of a rational number cannot be
4

The product of any two irrational numbers is
5

The decimal expansion of the number √2 is
6

Which of the following is irrational?
7

Which of the following is irrational?
8

A rational number between and is
9

The value of 1.999... in the form pq, where p and q are integers and q ≠ 0 , is
10

is equal to
11

is equal to
12

The number obtained on rationalising the denominator of is
13

is equal to
14

After rationalising the denominator of , we get the denominator as
15

The value of is equal to
16

If √2 = 1.4142, then is equal to
17

equals
18

The product equals
19

Value of is
20

Value of (256)0.16×(256)0.09 is
21

Which of the following is equal to x?

Exercise 1.2

1

Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.
2

Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example.
3

State whether the following statements are true or false? Justify your answer.

is a rational number.
3

State whether the following statements are true or false? Justify your answer.

There are infinitely many integers between any two integers.
3

State whether the following statements are true or false? Justify your answer.

Number of rational numbers between 15 and 18 is finite.
3

State whether the following statements are true or false? Justify your answer.

There are numbers which cannot be written in the form , ≠ 0, p, q both are integers.
3

State whether the following statements are true or false? Justify your answer.

The square of an irrational number is always rational.
3

State whether the following statements are true or false? Justify your answer.

is not a rational number as √12 and √3 are not integers.
3

State whether the following statements are true or false? Justify your answer.

is written in the form

4

Classify the following numbers as rational or irrational with justification:
4

Classify the following numbers as rational or irrational with justification:
4

Classify the following numbers as rational or irrational with justification:
4

Classify the following numbers as rational or irrational with justification:
4

Classify the following numbers as rational or irrational with justification:
4

Classify the following numbers as rational or irrational with justification:
4

Classify the following numbers as rational or irrational with justification:

0.5918
4

Classify the following numbers as rational or irrational with justification:

(1 + √5) − (4 + √5)
4

Classify the following numbers as rational or irrational with justification:

10.124124…
4

Classify the following numbers as rational or irrational with justification:

1.010010001…

Exercise 1.3

1

Find which of the variables x, y, z and u represent rational numbers and which irrational numbers:

(i) x2 = 5 (ii) y2 = 9


(iii) z2 = .04 (iv)

2

Find three rational numbers between

(i) –1 and –2 (ii) 0.1 and 0.11


(iii) (iv)

3

Insert a rational number and an irrational number between the following:

(i) 2 and 3 (ii) 0 and 0.1


(iii) (iv)


(v) 0.15 and 0.16 (vi) √2 and √3


(vii) 2.357 and 3.121


(viii) .0001 and .001


(ix) 3.623623 and 0.484848


(x) 6.375289 and 6.375738.
4

Represent the following numbers on the number line:

7, 7.2, −3/2 , −12/5
5

Locate and on the number line.
6

Represent geometrically the following numbers on the number line:
6

Represent geometrically the following numbers on the number line:
6

Represent geometrically the following numbers on the number line:
6

Represent geometrically the following numbers on the number line:
7

Express the following in the form , where p and q are integers and q ≠ 0:

0.2
7

Express the following in the form , where p and q are integers and q ≠ 0:

0.888...
7

Express the following in the form , where p and q are integers and q ≠ 0:
7

Express the following in the form , where p and q are integers and q ≠ 0:
7

Express the following in the form , where p and q are integers and q ≠ 0:

0.2555...
7

Express the following in the form , where p and q are integers and q ≠ 0:
7

Express the following in the form , where p and q are integers and q ≠ 0:

.00323232...
7

Express the following in the form , where p and q are integers and q ≠ 0:

.404040....
8

Show that 0.142857142857... =.
9

Simplify the following:
9

Simplify the following:
9

Simplify the following:
9

Simplify the following:
9

Simplify the following:
9

Simplify the following:
9

Simplify the following:
9

Simplify the following:
9

Simplify the following:
10

Rationalise the denominator
10

Rationalise the denominator
10

Rationalise the denominator
10

Rationalise the denominator
10

Rationalise the denominator
10

Rationalise the denominator
10

Rationalise the denominator
10

Rationalise the denominator
10

Rationalise the denominator
11

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

=

11

Find the values of a and b in each of the following:
11

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

=2−

11

Find the values of a and b in each of the following:
12

If

13

Rationalise the denominator in each of the following and hence evaluate by taking √2 =1.414, √3 =1.732 and √5 =2.236, upto three places of decimal.

(i) (ii)


(iii) (iv)


(v)

14

Simplify:
14

Simplify:
14

Simplify:
14

Simplify:
14

Simplify:
14

Simplify:
14

Simplify:

Exercise 1.4

1

Express in the form , where p and q are integers and q ≠ 0.
2

Simplify

3

If √2 =1.414, √3 =1.732, then find the value of .
4

If , then find the value of

5

If and , then find the value of x2 + y2.
6

Simplify:

7

Find the values of