Ncert exemplar solutions mathematics class 9 Solutions for Chapter 1 number systems

Exercises in Chapter 1 number systems Grade 9

Number Systems - Exercise 1.1

Number Systems - Exercise 1.2

Number Systems - Exercise 1.3

Number Systems - Exercise 1.4

Questions in Number Systems - Exercise 1.1

Value of $(256)^{0.16}\times(256)^{0.09}$ is ______.

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

The number obtained on rationalizing the denominator of $\frac{1}{\sqrt{7}-2}$ is _________.

The product $\sqrt[3]{2}\cdot\sqrt[4]{2}\cdot\sqrt[12]{32}$ equals _________.

After rationalizing the denominator of $\frac{7}{3\sqrt{3}-2\sqrt{2}}$ , we get the denominator as _______.

The value of $\frac{\sqrt{32}+\sqrt{48}}{\sqrt{8}+\sqrt{12}}$ is equal to ________.

If $\sqrt{2}=1.4142$ , then $\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$ is equal to _________.

$\sqrt[4]{\sqrt[3]{2^2}}$ is equal to _______

Value of $\sqrt[4]{(81)^{-2}}$ is ________.

Which of the following is equal to x?

The decimal expansion of the number √2 is __________.

Every rational number is _______

Between two rational numbers _____

Decimal representation of a rational number cannot be ___________

The product of any two irrational numbers is _____.

Which of the following is irrational?

Which of the following is irrational?

$\frac{1}{\sqrt{9}-\sqrt{8}}$ is equal to ________.

A rational number between √2 and √3 is ___________>

√10 x √15 is equal to _____

2√3 + √3 is equal to ______________

Every rational number is

(A) a natural number

(B) an integer

(D) a whole number

Between two rational numbers

(A) there is no rational number

(B) there is exactly one rational number

(D) there are only rational numbers and no irrational numbers

Decimal representation of a rational number cannot be

(A) terminating

(B) non-terminating

(D) non-terminating non-repeating

The product of any two irrational numbers is

(A) always an irrational number

(B) always a rational number

(D) sometimes rational, sometimes irrational

The decimal expansion of the number √2 is

(A) a finite decimal

(B) 1.41421

(D) non-terminating non-recurring

Which of the following is irrational?

(A) √4/√9

(B) √12/√3

(D) √81

Which of the following is irrational?

A rational number between 2 and 3 is

(A) (√2+√3)/2

(B) (√2. √3)/2

(D) 1.8

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

(A) 19/10

(B) 1999/1000

(D) 1/9

2√3 + √3 is equal to

(A) 2√6

(B) 6

(D) 4√6

Questions in Number Systems - Exercise 1.2

$0.5918$ is rational number.

$\frac{\sqrt{12}}{\sqrt{75}}$ is rational number.

$(1+\sqrt{5})-(4+\sqrt{5})$ is a rational number.

Let x and y be rational and irrational numbers, respectively. x + y is an irrational number.

Let x be rational and y be irrational. xy is irrational number only.

$\frac{\sqrt{2}}{3}$ is a rational number.

Number of rational numbers between 15 and 18 is finite.

The square of an irrational number is always rational.

There are infinitely many integers between any two integers.

$1.010010001\dots$ is a rational number.

There are numbers that can be written in the form p/q, q≠0, and p, q both are not integers.

$-\sqrt{0.4}$ is rational number.

$\frac{\sqrt{12}}{\sqrt{3}}$ is not a rational number as √12 and √3 are not integers.

$10.124124\dots$ is an irrational number.

$\frac{\sqrt{15}}{\sqrt{3}}$ is written in the form p/q , where q ≠ 0, so it is a rational number.

$\sqrt{196}$ is rational number.

$\sqrt{\frac{9}{27}}$ is a rational number.

$3\sqrt{18}$ is a rational number.

$\frac{\sqrt{28}}{\sqrt{343}}$ is rational number.

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.

Questions in Number Systems - Exercise 1.3

Which is the rational number between 0 and 0.1?

Rationalize the denominator in $\frac{4}{\sqrt{3}}$ . Evaluate by taking $\sqrt{3}=1.732$ up to three decimal places and select the correct option.

What is the value of a and b in $\frac{7+\sqrt{5}}{7-\sqrt{5}}-\frac{7-\sqrt{5}}{7+\sqrt{5}}=a+\frac{7}{11} \sqrt{5} b$ ?

If $a=2+\sqrt{3}$ , then what is the value of $a-\frac{1}{a}$ ?

Rationalize the denominator in $\frac{6}{\sqrt{6}}$ , take the value up to three decimal places and select the correct option.

Rationalize the denominator in $\frac{\sqrt{10}-\sqrt{5}}{2}$ and hence evaluate by taking $\sqrt{2}=1.414, \text { and } \sqrt{5}=2.236$ , upto three places of decimal.

Select the correct option.

Rationalize the denominator in $\frac{\sqrt{2}}{2+\sqrt{2}}$ and hence evaluate by taking $\sqrt{2}=1.414$ , upto three places of decimal.

Select the correct option.

Rationalize the denominator in $\frac{1}{\sqrt{3}+\sqrt{2}}$ and hence evaluate by taking $\sqrt{2}=1.414, \sqrt{3}=1.732$ , upto three places of decimal.

Select the correct option.

Rationalize the denominator of $\frac{2}{3\sqrt{3}}$ and select the correct option.

Simplify $4 \sqrt{28} \div 3 \sqrt{7}$ and select the correct option.

Which is the rational number between √2 and √3?

Which is the rational number between $-\frac{2}{5}\ and\ \frac{1}{2}$ ?

Which is the rational number between –1 and –2?

Which is the rational number between 0.15 and 0.16?

Simplify

𝑢² = $\frac{17}{4}$

The valus of u is a rational number.

Which is the rational number between 2.357 and 3.121?

Which is the rational number between 6.375289 and 6.375738?

Which is the rational number between 2 and 3?

Simplify y² = 9

The valus of y is a rational number.

Simplify x² = 5

The valus of x is a rational number.

Simplify:

z² = 0.04

The valus of z is an irrational number.

Which is the rational number between $\frac{1}{3}\ and\ \frac{1}{2}$ ?

Express 0.888… in the form p/q, where p and q are integers and q ≠ 0 and select the correct option.

Which is the rational number between $\frac{5}{7}\ and\ \frac{6}{7}$ ?

Which is the rational number between $\frac{1}{4}\ and\ \frac{1}{5}$ ?

Which is the rational number between 0.1 and 0.11?

Which is the rational number between 0.00010 and 0.0012?

Express 0.2 in the form p/q, where p and q are integers and q ≠ 0 and select the correct option.

Express $0.\overline{001}$ in the form p/q, where p and q are integers and q ≠ 0 and select the correct option.

Express 5.2.... in the form p/q, where p and q are integers and q ≠ 0 and select the correct option.

Express 0.2555… in the form p/q, where p and q are integers and q ≠ 0 and select the correct option.

Express $0.1\overline{34}$ in the form p/q, where p and q are integers and q ≠ 0 and select the correct option.

Express 0.00323232… in the form p/q, where p and q are integers and q ≠ 0 and select the correct option.

Express 0.404040… in the form p/q, where p and q are integers and q ≠ 0 and select the correct option.

Simplify $\sqrt{45}-3\sqrt{20}+4\sqrt{5}$ and select the correct option.

$0.142857142857\ldots=\frac{1}{7}$

Simplify $3 \sqrt{3}+2 \sqrt{27}+\frac{7}{\sqrt{3}}$ and select the correct option.

Rationalize the denominator of $\frac{3+\sqrt{2}}{4 \sqrt{2}}$ and select the correct option.

Rationalize the denominator of $\frac{16}{\sqrt{41}-5}$ and select the correct option.

Rationalize the denominator of $\frac{3 \sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and select the correct option.

Simplify $\frac{3}{\sqrt{8}}+\frac{1}{\sqrt{2}}$ and select the correct option.

Simplify $(\sqrt{3}-\sqrt{2})^{2}$ and select the correct option.

Simplify $\frac{2\sqrt{3}}{3}-\frac{\sqrt{3}}{6}$ and select the correct option.

Simplify $\sqrt[4]{81}-8 \sqrt[3]{216}+15 \sqrt[3]{32}+\sqrt{225}$ and select the correct option.

Rationalize the denominator of $\frac{\sqrt{40}}{\sqrt{3}}$ and select the correct option.

Which is the rational number between 3.623623 and 0.484848?

Simplify $\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}$ and select the correct option.

Rationalize the denominator of $\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}$ and select the correct option.

Rationalize the denominator of $\frac{2+\sqrt{3}}{2-\sqrt{3}}$ and select the correct option.

Rationalize the denominator of $\frac{\sqrt{6}}{\sqrt{2}+\sqrt{3}}$ and select the correct option.

Rationalize the denominator of $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$ and select the correct option.

Simplify $4\sqrt{12}\times7\sqrt{6}$ and select the correct option.

What is the value of 'a' in $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a-6\sqrt{3}$ ?

What is the value of b in $\frac{\sqrt{2}+\sqrt{3}}{3 \sqrt{2}-2 \sqrt{3}}=2-b \sqrt{6}$ ?

What is the value of 'a' in $\frac{3-\sqrt{5}}{3+2 \sqrt{5}}=a \sqrt{5}-\frac{19}{11}$ ?

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) 𝑢² = 17/4

Find three rational numbers between

(i) –1 and –2

(ii) 0.1 and 0.11

(iii) 5/7 and 6/7

(iv) 1/4 and 1/5

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

(i) 2 and 3

(ii) 0 and 0.1

(iii) 1/3 and 1/2

(iv) – 2/5 and 1/2

(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.

Represent the following numbers on the number line:

7, 7.2, −3/2 , −12/5

Locate √5, √10 and √17 on the number line.

Represent geometrically the following numbers on the number line:

(i) √4.5

(ii) √5.6

(iii) √8.1

(iv) √2.3

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

(I) 0.2

(ii) 0.888...

(iii)

(iv)

(v) 0.2555...

(vi)

(vii) .00323232...

(viii) .404040...]

Questions in Number Systems - Exercise 1.4

What is the value of $\frac{4}{(216)^{-\frac{2}{3}}}+\frac{1}{(256)^{-\frac{3}{4}}}+\frac{2}{(243)^{-\frac{1}{3}}}$ ?

Simplify $(256)^{-\left(4\frac{3}{2}\right)}$ and select the correct option

If √2 =1.414, √3 =1.732, then what is the value of $\frac{4}{3\sqrt{3}-2\sqrt{2}}+\frac{3}{3\sqrt{3}+2\sqrt{2}}$ ?

If $a=\frac{3+\sqrt{5}}{2}$ , then what is the value of $a^2+\frac{1}{a^2}$ ?

Express $0.6+0.\overline{7}+0.4\overline{7}$ in the form p/q, where p and q are integers and q ≠ 0 and select the correct option.

If $x=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}},\text{ and }y=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ , then what is the value of $x^2+y^2$ ?

Simplify $\frac{7\sqrt{3}}{\sqrt{10}+\sqrt{3}}-\frac{2\sqrt{5}}{\sqrt{6}+\sqrt{5}}-\frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{2}}$

Express $0.6+0 . \overline{7}+0.4 \overline{7}$ in the form p/q, where p and q are integers and q ≠ 0.

Simplify:

$\frac{7 \sqrt{3}}{\sqrt{10}+\sqrt{3}}-\frac{2 \sqrt{5}}{\sqrt{6}+5}-\frac{3 \sqrt{2}}{\sqrt{15}+3 \sqrt{2}}$

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

$\frac{4}{3 \sqrt{3}-2 \sqrt{2}}+\frac{3}{3 \sqrt{3}+2 \sqrt{2}}$

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Chapters in this book

Number Systems

Polynomials

Coordinate Geometry

Linear Equations in Two Variables

Introduction to Euclid's Geometry

Lines and Angles

Triangles

Quadrilaterals

Areas of Parallelograms and Triangles

Circles

Constructions

Heron's Formula

Surface Areas and Volumes

Statistics and Probability