# NCERT solutions for Mathematics Exemplar Class 9 chapter 1 - Number Systems [Latest edition]

## Chapter 1: Number Systems

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4
Exercise 1.1 [Pages 2 - 5]

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 1 Number Systems Exercise 1.1 [Pages 2 - 5]

#### Write the correct answer in the following:

Exercise 1.1 | Q 1 | Page 2

Every rational number is ______.

• A natural number

• An integer

• A real number

• A whole number

Exercise 1.1 | Q 2 | Page 3

Between two rational numbers ______.

• There is no rational number

• There is exactly one rational number

• There are infinitely many rational numbers

• There are only rational numbers and no irrational numbers

Exercise 1.1 | Q 3 | Page 3

Decimal representation of a rational number cannot be ______.

• Terminating

• Non-terminating

• Non-terminating repeating

• Non-terminating non-repeating

Exercise 1.1 | Q 4 | Page 3

The product of any two irrational numbers is ______.

• Always an irrational number

• Always a rational number

• Always an integer

• Sometimes rational, sometimes irrational

Exercise 1.1 | Q 5 | Page 3

The decimal expansion of the number sqrt(2) is ______.

• A finite decimal

• 1.41421

• Non-terminating recurring

• Non-terminating non-recurring

Exercise 1.1 | Q 6 | Page 3

Which of the following is irrational?

• sqrt(4/9)

• sqrt(12)/sqrt(3)

• sqrt(7)

• sqrt(81)

Exercise 1.1 | Q 7 | Page 3

Which of the following is irrational?

• 0.14

• 0.14bar16

• 0.bar1416

• 0.4014001400014...

Exercise 1.1 | Q 8 | Page 3

A rational number between sqrt(2) and sqrt(3) is ______.

• (sqrt(2) + sqrt(3))/2

• (sqrt(2) * sqrt(3))/2

• 1.5

• 1.8

Exercise 1.1 | Q 9 | Page 4

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

• 19/10

• 1999/1000

• 2

• 1/9

Exercise 1.1 | Q 10 | Page 4

2sqrt(3) + sqrt(3) is equal to ______.

• 2sqrt(6)

• 6

• 3sqrt(3)

• 4sqrt(6)

Exercise 1.1 | Q 11 | Page 4

sqrt(10) xx sqrt(15) is equal to ______.

• 6sqrt(5)

• 5sqrt(6)

• sqrt(25)

• 10sqrt(5)

Exercise 1.1 | Q 12 | Page 4

The number obtained on rationalising the denominator of 1/(sqrt(7) - 2) is ______.

• (sqrt(7) + 2)/3

• (sqrt(7) - 2)/3

• (sqrt(7) + 2)/5

• (sqrt(7) + 2)/45

Exercise 1.1 | Q 13 | Page 4

1/(sqrt(9) - sqrt(8)) is equal to ______.

• 1/2(3 - 2sqrt(2))

• 1/(3 + 2sqrt(2)

• 3 - 2sqrt(2)

• 3 + 2sqrt(2)

Exercise 1.1 | Q 14 | Page 4

After rationalising the denominator of 7/(3sqrt(3) - 2sqrt(2)), we get the denominator as ______.

• 13

• 19

• 5

• 35

Exercise 1.1 | Q 15 | Page 4

The value of (sqrt(32) + sqrt(48))/(sqrt(8) + sqrt(12)) is equal to ______.

• sqrt(2)

• 2

• 4

• 8

Exercise 1.1 | Q 16 | Page 4

If sqrt(2) = 1.4142, then sqrt((sqrt(2) - 1)/(sqrt(2) + 1)) is equal to ______.

• 2.4142

• 5.8282

• 0.4142

• 0.1718

Exercise 1.1 | Q 17 | Page 5

root(4)root(3)(2^2) equals to ______.

• 2^(-1/6)

• 2^-6

• 2^(1/6)

• 2^6

Exercise 1.1 | Q 18 | Page 5

The product root(3)(2) * root(4)(2) * root(12)(32) equals to ______.

• sqrt(2)

• 2

• root(12)(2)

• root(12)(32)

Exercise 1.1 | Q 19 | Page 5

Value of root(4)((81)^-2) is ______.

• 1/9

• 1/3

• 9

• 1/81

Exercise 1.1 | Q 20 | Page 5

Value of (256)^0.16 xx (256)^0.09 is ______.

• 4

• 16

• 64

• 256.25

Exercise 1.1 | Q 21 | Page 5

Which of the following is equal to x?

• x^(12/7) - x^(5/7)

• root(12)((x^4)^(1/3)

• (sqrt(x^3))^(2/3)

• x^(12/7) xx x^(7/12)

Exercise 1.2 [Page 6]

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 1 Number Systems Exercise 1.2 [Page 6]

Exercise 1.2 | Q 1 | Page 6

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.

Exercise 1.2 | Q 2 | Page 6

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

#### State whether the following statement is True or False:

Exercise 1.2 | Q 3.(i) | Page 6

sqrt(2)/3 is a rational number.

• True

• False

Exercise 1.2 | Q 3.(ii) | Page 6

There are infinitely many integers between any two integers.

• True

• False

Exercise 1.2 | Q 3.(iii) | Page 6

Number of rational numbers between 15 and 18 is finite.

• True

• False

Exercise 1.2 | Q 3.(iv) | Page 6

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

• True

• False

Exercise 1.2 | Q 3.(v) | Page 6

The square of an irrational number is always rational.

• True

• False

Exercise 1.2 | Q 3.(vi) | Page 6

sqrt(12)/sqrt(3) is not a rational number as sqrt(12) and sqrt(3) are not integers.

• True

• False

Exercise 1.2 | Q 3.(vii) | Page 6

sqrt(15)/sqrt(3) is written in the form p/q, q ≠ 0 and so it is a rational number.

• True

• False

Exercise 1.2 | Q 4.(i) | Page 6

Classify the following numbers as rational or irrational with justification:

sqrt(196)

Exercise 1.2 | Q 4.(ii) | Page 6

Classify the following numbers as rational or irrational with justification:

3sqrt(18)

Exercise 1.2 | Q 4.(iii) | Page 6

Classify the following numbers as rational or irrational with justification:

sqrt(9/27)

Exercise 1.2 | Q 4.(iv) | Page 6

Classify the following numbers as rational or irrational with justification:

sqrt(28)/sqrt(343)

Exercise 1.2 | Q 4.(v) | Page 6

Classify the following numbers as rational or irrational with justification:

- sqrt(0.4)

Exercise 1.2 | Q 4.(vi) | Page 6

Classify the following numbers as rational or irrational with justification:

sqrt(12)/sqrt(75)

Exercise 1.2 | Q 4.(vii) | Page 6

Classify the following numbers as rational or irrational with justification:

0.5918

Exercise 1.2 | Q 4.(viii) | Page 6

Classify the following numbers as rational or irrational with justification:

(1 + sqrt(5)) - (4 + sqrt(5))

Exercise 1.2 | Q 4.(ix) | Page 6

Classify the following numbers as rational or irrational with justification:

10.124124...

Exercise 1.2 | Q 4.(x) | Page 6

Classify the following numbers as rational or irrational with justification:

1.010010001...

Exercise 1.3 [Pages 9 - 11]

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 1 Number Systems Exercise 1.3 [Pages 9 - 11]

Exercise 1.3 | Q 1.(i) | Page 9

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

x2 = 5

Exercise 1.3 | Q 1.(ii) | Page 9

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

y2 = 9

Exercise 1.3 | Q 1.(iii) | Page 9

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

z2 = 0.04

Exercise 1.3 | Q 1.(iv) | Page 9

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

u^2 = 17/4

Exercise 1.3 | Q 2.(i) | Page 9

Find three rational numbers between –1 and –2

Exercise 1.3 | Q 2.(ii) | Page 9

Find three rational numbers between 0.1 and 0.11

Exercise 1.3 | Q 2.(iii) | Page 9

Find three rational numbers between 5/7 and 6/7

Exercise 1.3 | Q 2.(iv) | Page 9

Find three rational numbers between 1/4 and 1/5

Exercise 1.3 | Q 3.(i) | Page 9

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

2 and 3

Exercise 1.3 | Q 3.(ii) | Page 9

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

0 and 0.1

Exercise 1.3 | Q 3.(iii) | Page 9

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

1/3 and 1/2

Exercise 1.3 | Q 3.(iv) | Page 9

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

(-2)/5 and 1/2

Exercise 1.3 | Q 3.(v) | Page 9

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

0.15 and 0.16

Exercise 1.3 | Q 3.(vi) | Page 9

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

sqrt(2) and sqrt(3)

Exercise 1.3 | Q 3.(vii) | Page 9

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

2.357 and 3.121

Exercise 1.3 | Q 3.(viii) | Page 9

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

0.0001 and 0.001

Exercise 1.3 | Q 3.(ix) | Page 9

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

3.623623 and 0.484848

Exercise 1.3 | Q 3.(x) | Page 9

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

6.375289 and 6.375738

Exercise 1.3 | Q 4.(i) | Page 9

Represent the following numbers on the number line:

7

Exercise 1.3 | Q 4.(ii) | Page 9

Represent the following numbers on the number line:

7.2

Exercise 1.3 | Q 4.(iii) | Page 9

Represent the following numbers on the number line:

(-3)/2

Exercise 1.3 | Q 4.(iv) | Page 9

Represent the following numbers on the number line:

(-12)/5

Exercise 1.3 | Q 5 | Page 9

Locate sqrt(5), sqrt(10) and sqrt(17) on the number line.

Exercise 1.3 | Q 6.(i) | Page 9

Represent geometrically the following numbers on the number line:

sqrt(4.5)

Exercise 1.3 | Q 6.(ii) | Page 9

Represent geometrically the following numbers on the number line:

sqrt(5.6)

Exercise 1.3 | Q 6.(iii) | Page 9

Represent geometrically the following numbers on the number line:

sqrt(8.1)

Exercise 1.3 | Q 6.(iv) | Page 9

Represent geometrically the following numbers on the number line:

sqrt(2.3)

Exercise 1.3 | Q 7.(i) | Page 10

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

0.2

Exercise 1.3 | Q 7.(ii) | Page 10

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

0.888...

Exercise 1.3 | Q 7.(iii) | Page 10

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

5.bar2

Exercise 1.3 | Q 7.(iv) | Page 10

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

0.bar001

Exercise 1.3 | Q 7.(v) | Page 10

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

0.2555...

Exercise 1.3 | Q 7.(iv) | Page 10

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

0.1bar34

Exercise 1.3 | Q 7.(vii) | Page 10

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

0.00323232...

Exercise 1.3 | Q 7.(viii) | Page 10

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

0.404040...

Exercise 1.3 | Q 8 | Page 10

Show that 0.142857142857... = 1/7

Exercise 1.3 | Q 9.(i) | Page 10

Simplify the following:

sqrt(45) - root(3)(20) + 4sqrt(5)

Exercise 1.3 | Q 9.(ii) | Page 10

Simplify the following:

sqrt(24)/8 + sqrt(54)/9

Exercise 1.3 | Q 9.(iii) | Page 10

Simplify the following:

root(4)(12) xx root(7)(6)

Exercise 1.3 | Q 9.(iv) | Page 10

Simplify the following:

4sqrt(28) ÷ 3sqrt(7) ÷  root(3)(7)

Exercise 1.3 | Q 9.(v) | Page 10

Simplify the following:

3sqrt(3) + 2sqrt(27) + 7/sqrt(3)

Exercise 1.3 | Q 9.(vi) | Page 10

Simplify the following:

(sqrt(3) - sqrt(2))^2

Exercise 1.3 | Q 9.(vii) | Page 10

Simplify the following:

root(4)(81) - 8root(3)(216) + 15root(5)(32) + sqrt(225)

Exercise 1.3 | Q 9.(viii) | Page 10

Simplify the following:

3/sqrt(8) + 1/sqrt(2)

Exercise 1.3 | Q 9.(ix) | Page 10

Simplify the following:

(2sqrt(3))/3 - sqrt(3)/6

Exercise 1.3 | Q 10.(i) | Page 10

Rationalise the denominator of the following:

2/(3sqrt(3)

Exercise 1.3 | Q 10.(ii) | Page 10

Rationalise the denominator of the following:

sqrt(40)/sqrt(3)

Exercise 1.3 | Q 10.(iii) | Page 10

Rationalise the denominator of the following:

(3 + sqrt(2))/(4sqrt(2))

Exercise 1.3 | Q 10.(iv) | Page 10

Rationalise the denominator of the following:

16/(sqrt(41) - 5)

Exercise 1.3 | Q 10.(v) | Page 10

Rationalise the denominator of the following:

(2 + sqrt(3))/(2 - sqrt(3))

Exercise 1.3 | Q 10.(vi) | Page 10

Rationalise the denominator of the following:

sqrt(6)/(sqrt(2) + sqrt(3))

Exercise 1.3 | Q 10.(vii) | Page 10

Rationalise the denominator of the following:

(sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2))

Exercise 1.3 | Q 10.(viii) | Page 10

Rationalise the denominator of the following:

(3sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3))

Exercise 1.3 | Q 10.(ix) | Page 10

Rationalise the denominator of the following:

(4sqrt(3) + 5sqrt(2))/(sqrt(48) + sqrt(18))

Exercise 1.3 | Q 11.(i) | Page 10

Find the values of a and b in the following:

(5 + 2sqrt(3))/(7 + 4sqrt(3)) = a - 6sqrt(3)

Exercise 1.3 | Q 11.(ii) | Page 10

Find the values of a and b in the following:

(3 - sqrt(5))/(3 + 2sqrt(5)) = asqrt(5) - 19/11

Exercise 1.3 | Q 11.(iii) | Page 10

Find the values of a and b in the following:

(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) = 2 - bsqrt(6)

Exercise 1.3 | Q 11.(iv) | Page 10

Find the values of a and b in the following:

(7 + sqrt(5))/(7 - sqrt(5)) - (7 - sqrt(5))/(7 + sqrt(5)) = a + 7/11 sqrt(5)b

Exercise 1.3 | Q 12 | Page 11

If  a = 2 + sqrt(3), then find the value of a - 1/a.

Exercise 1.3 | Q 13.(i) | Page 11

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

4/sqrt(3)

Exercise 1.3 | Q 13.(ii) | Page 11

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

6/sqrt(6)

Exercise 1.3 | Q 13.(iii) | Page 11

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

(sqrt(10) - sqrt(5))/2

Exercise 1.3 | Q 13.(iv) | Page 11

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

sqrt(2)/(2 + sqrt(2)

Exercise 1.3 | Q 13.(v) | Page 11

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

1/(sqrt(3) + sqrt(2))

Exercise 1.3 | Q 14.(i) | Page 11

Simplify:

(1^3 + 2^3 + 3^3)^(1/2)

Exercise 1.3 | Q 14.(ii) | Page 11

Simplify:

(3/5)^4 (8/5)^-12 (32/5)^6

Exercise 1.3 | Q 14.(iii) | Page 11

Simplify:

(1/27)^((-2)/3)

Exercise 1.3 | Q 14.(iv) | Page 11

Simplify:

[((625)^(-1/2))^((-1)/4)]^2

Exercise 1.3 | Q 14.(v) | Page 11

Simplify:

(9^(1/3) xx 27^(-1/2))/(3^(1/6) xx 3^(- 2/3))

Exercise 1.3 | Q 14.(vi) | Page 11

Simplify:

64^(-1/3)[64^(1/3) - 64^(2/3)]

Exercise 1.3 | Q 14.(vii) | Page 11

Simplify:

(8^(1/3) xx 16^(1/3))/(32^(- 1/3))

Exercise 1.4 [Page 12]

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 1 Number Systems Exercise 1.4 [Page 12]

Exercise 1.4 | Q 1 | Page 12

Express 0.6 + 0.bar7 + 0.4bar7 in the form p/q, where p and q are integers and q ≠ 0.

Exercise 1.4 | Q 2 | Page 12

Simplify: (7sqrt(3))/(sqrt(10) + sqrt(3)) - (2sqrt(5))/(sqrt(6) + sqrt(5)) - (3sqrt(2))/(sqrt(15) + 3sqrt(2))

Exercise 1.4 | Q 3 | Page 12

If sqrt(2) = 1.414, sqrt(3) = 1.732, then find the value of 4/(3sqrt(3) - 2sqrt(2)) + 3/(3sqrt(3) + 2sqrt(2))

Exercise 1.4 | Q 4 | Page 12

If a = (3 + sqrt(5))/2, then find the value of a^2 + 1/a^2.

Exercise 1.4 | Q 5 | Page 12

If x = (sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) andy = (sqrt(3) - sqrt(2))/(sqrt(3) + sqrt(2)), then find the value of x2 + y2.

Exercise 1.4 | Q 6 | Page 12

Simplify: (256)^(-((-3)/4^2))

Exercise 1.4 | Q 7 | Page 12

Find the value of 4/((216)^(-2/3)) + 1/((256)^(- 3/4)) + 2/((243)^(- 1/5))

## Chapter 1: Number Systems

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4

## NCERT solutions for Mathematics Exemplar Class 9 chapter 1 - Number Systems

NCERT solutions for Mathematics Exemplar Class 9 chapter 1 (Number Systems) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 9 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 9 chapter 1 Number Systems are Introduction of Real Number, Concept of Irrational Numbers, Real Numbers and Their Decimal Expansions, Representing Real Numbers on the Number Line, Operations on Real Numbers, Laws of Exponents for Real Numbers.

Using NCERT Class 9 solutions Number Systems exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 9 prefer NCERT Textbook Solutions to score more in exam.

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