#### Chapters

Chapter 2: Polynomials

Chapter 3: Coordinate Geometry

Chapter 4: Linear Equation In Two Variables

Chapter 5: Introduction To Euclid's Geometry

Chapter 6: Lines & Angles

Chapter 7: Triangles

Chapter 8: Quadrilaterals

Chapter 9: Areas of Parallelograms & Triangles

Chapter 10: Circles

Chapter 11: Construction

Chapter 12: Heron's Formula

Chapter 13: Surface Area & Volumes

Chapter 14: Statistics & Probability

## Chapter 1: Number Systems

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

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

Every rational number is ______.

A natural number

An integer

A real number

A whole number

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

Decimal representation of a rational number cannot be ______.

Terminating

Non-terminating

Non-terminating repeating

Non-terminating non-repeating

The product of any two irrational numbers is ______.

Always an irrational number

Always a rational number

Always an integer

Sometimes rational, sometimes irrational

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

A finite decimal

1.41421

Non-terminating recurring

Non-terminating non-recurring

Which of the following is irrational?

`sqrt(4/9)`

`sqrt(12)/sqrt(3)`

`sqrt(7)`

`sqrt(81)`

Which of the following is irrational?

0.14

`0.14bar16`

`0.bar1416`

0.4014001400014...

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

`(sqrt(2) + sqrt(3))/2`

`(sqrt(2) * sqrt(3))/2`

1.5

1.8

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`

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

`2sqrt(6)`

6

`3sqrt(3)`

`4sqrt(6)`

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

`6sqrt(5)`

`5sqrt(6)`

`sqrt(25)`

`10sqrt(5)`

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`

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

`1/2(3 - 2sqrt(2))`

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

`3 - 2sqrt(2)`

`3 + 2sqrt(2)`

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

13

19

5

35

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

`sqrt(2)`

2

4

8

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

2.4142

5.8282

0.4142

0.1718

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

`2^(-1/6)`

`2^-6`

`2^(1/6)`

`2^6`

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

`sqrt(2)`

2

`root(12)(2)`

`root(12)(32)`

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

`1/9`

`1/3`

9

`1/81`

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

4

16

64

256.25

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)`

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 1 Number Systems Exercise 1.2 [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.

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:

`sqrt(2)/3` is a rational number.

True

False

There are infinitely many integers between any two integers.

True

False

Number of rational numbers between 15 and 18 is finite.

True

False

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

True

False

The square of an irrational number is always rational.

True

False

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

True

False

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

True

False

Classify the following numbers as rational or irrational with justification:

`sqrt(196)`

Classify the following numbers as rational or irrational with justification:

`3sqrt(18)`

Classify the following numbers as rational or irrational with justification:

`sqrt(9/27)`

Classify the following numbers as rational or irrational with justification:

`sqrt(28)/sqrt(343)`

Classify the following numbers as rational or irrational with justification:

`- sqrt(0.4)`

Classify the following numbers as rational or irrational with justification:

`sqrt(12)/sqrt(75)`

Classify the following numbers as rational or irrational with justification:

0.5918

Classify the following numbers as rational or irrational with justification:

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

Classify the following numbers as rational or irrational with justification:

10.124124...

Classify the following numbers as rational or irrational with justification:

1.010010001...

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

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

x^{2} = 5

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

y^{2} = 9

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

z^{2} = 0.04

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

`u^2 = 17/4`

Find three rational numbers between –1 and –2

Find three rational numbers between 0.1 and 0.11

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

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

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

2 and 3

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

0 and 0.1

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

`1/3` and `1/2`

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

`(-2)/5` and `1/2`

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

0.15 and 0.16

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

`sqrt(2)` and `sqrt(3)`

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

2.357 and 3.121

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

0.0001 and 0.001

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

3.623623 and 0.484848

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

6.375289 and 6.375738

Represent the following numbers on the number line:

7

Represent the following numbers on the number line:

7.2

Represent the following numbers on the number line:

`(-3)/2`

Represent the following numbers on the number line:

`(-12)/5`

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

Represent geometrically the following numbers on the number line:

`sqrt(4.5)`

Represent geometrically the following numbers on the number line:

`sqrt(5.6)`

Represent geometrically the following numbers on the number line:

`sqrt(8.1)`

Represent geometrically the following numbers on the number line:

`sqrt(2.3)`

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

0.2

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

0.888...

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

`5.bar2`

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

`0.bar001`

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

0.2555...

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

`0.1bar34`

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

0.00323232...

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

0.404040...

Show that 0.142857142857... = `1/7`

Simplify the following:

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

Simplify the following:

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

Simplify the following:

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

Simplify the following:

`4sqrt(28) ÷ 3sqrt(7) ÷ root(3)(7)`

Simplify the following:

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

Simplify the following:

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

Simplify the following:

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

Simplify the following:

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

Simplify the following:

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

Rationalise the denominator of the following:

`2/(3sqrt(3)`

Rationalise the denominator of the following:

`sqrt(40)/sqrt(3)`

Rationalise the denominator of the following:

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

Rationalise the denominator of the following:

`16/(sqrt(41) - 5)`

Rationalise the denominator of the following:

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

Rationalise the denominator of the following:

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

Rationalise the denominator of the following:

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

Rationalise the denominator of the following:

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

Rationalise the denominator of the following:

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

Find the values of a and b in the following:

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

Find the values of a and b in the following:

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

Find the values of a and b in the following:

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

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`

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

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)`

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)`

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`

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

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

Simplify:

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

Simplify:

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

Simplify:

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

Simplify:

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

Simplify:

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

Simplify:

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

Simplify:

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

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

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

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

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

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

If `x = (sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2))` and`y = (sqrt(3) - sqrt(2))/(sqrt(3) + sqrt(2))`, then find the value of x^{2} + y^{2}.

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

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

## Chapter 1: Number Systems

## 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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