myhelper: RD Sharma 2020 solution class 9 chapter 1 Number System MCQS

RD Sharma 2020 solution class 9 chapter 1 Number System MCQS

MCQS

Page-1.40

Question 1:

Mark the correct alternative in each of the following:

1. Which one of the following is a correct statement?
(a) Decimal expansion of a rational number is terminating
(b) Decimal expansion of a rational number is non-terminating
(c) Decimal expansion of an irrational number is terminating
(d) Decimal expansion of an irrational number is non-terminating and non-repeating

Answer 1:

The decimal expansion of an irrational number is non-terminating and non- repeating. Thus, we can say that a number, whose decimal expansion is non-terminating and non- repeating, called irrational number. And the decimal expansion of rational number is either terminating or repeating. Thus, we can say that a number, whose decimal expansion is either terminating or repeating, is called a rational number.
Hence the correct option is

Question 2:

Which one of the following statements is true?
(a) The sum of two irrational numbers is always an irrational number
(b) The sum of two irrational numbers is always a rational number
(c) The sum of two irrational numbers may be a rational number or an irrational number
(d) The sum of two irrational numbers is always an integer

Answer 2:

Since,

and

are two irrational number and

Therefore, sum of two irrational numbers may be rational
Now, let

and

be two irrational numbers and

Therefore, sum of two irrational number may be irrational
Hence the correct option is

Question 3:

Which of the following is a correct statement?
(a) Sum of two irrational numbers is always irrational
(b) Sum of a rational and irrational number is always an irrational number
(c) Square of an irrational number is always a rational number
(d) Sum of two rational numbers can never be an integer

Answer 3:

The sum of irrational number and rational number is always irrational number.
Let a be a rational number and b be an irrational number.
Then,

As 2ab is irrational therefore

is irrational.
Hence

is irrational.
Therefore answer is

Question 4:

Which of the following statements is true?
(a) Product of two irrational numbers is always irrational
(b) Product of a rational and an irrational number is always irrational
(c) Sum of two irrational numbers can never be irrational
(d) Sum of an integer and a rational number can never be an integer

Answer 4:

Since we know that the product of rational and irrational number is always an irrational. For example: Let

are rational and irrational numbers respectively and their product is

Hence the correct option is

Question 5:

Which of the following is irrational?

(a)

\sqrt{\frac{4}{9}}

(b)

\sqrt{\frac{4}{5}}

(c)

\sqrt{7}

(d)

\sqrt{81}

Answer 5:

Given that

And 7 is not a perfect square.
Hence the correct option is

Question 6:

Which of the following is irrational?

(i) 0.14

(ii)

0.14 \bar{16}

(iii)

0 . \bar{1416}

(iv) 0.1014001400014...

Answer 6:

Given that

Here

is non-terminating or non-repeating. So it is an irrational number.
Hence the correct option is

Question 7:

Which of the following is rational?

(a)

\sqrt{3}

(b)

π

(c)

\frac{4}{0}

(d)

\frac{0}{4}

Answer 7:

Given that

Here,

, this is the form of

. So this is a rational number
Hence the correct option is

Question 8:

The number 0.318564318564318564 ........ is:
(a) a natural number
(b) an integer
(c) a rational number
(d) an irrational number

Answer 8:

Since the given number

is repeating, so it is rational number because rational number is always either terminating or repeating
Hence the correct option is

Question 9:

If n is a natural number, then

\sqrt{n}

is

(a) always a natural number
(b) always an irrational number
(c) always an irrational number
(d) sometimes a natural number and sometimes an irrational number

Answer 9:

The term “natural number” refers either to a member of the set of positive integer

.
And natural number starts from one of counting digit .Thus, if n is a natural number then sometimes n is a perfect square and sometimes it is not.
Therefore, sometimes

is a natural number and sometimes it is an irrational number
Hence the correct option is

Page-1.41

Question 10:

Which of the following numbers can be represented as non-terminating, repeating decimals?

(a) $\frac{39}{24}$

(b) $\frac{3}{16}$

(c) $\frac{3}{11}$

(d) $\frac{137}{25}$

Answer 10:

Given that

Here is repeating but non-terminating.

Hence the correct option is.

Question 11:

Every point on a number line represents
(a) a unique real number
(b) a natural number
(c) a rational number
(d) an irrational number

Answer 11:

In basic mathematics, number line is a picture of straight line on which every point is assumed to correspond to real number.

Hence the correct option is.

Question 12:

Which of the following is irrational?

(a) 0.15

(b) 0.01516

(c) $0 . \bar{1516}$

(d) 0.5015001500015.

Answer 12:

Given decimal numbers are

Here the number is non terminating or non-repeating.

Hence the correct option is.

Question 13:

The number $1. top enclose 27$ in the form $p over q$ , where p and q are integers and q ≠ 0, is
(a) $14 over 9$

(b) $14 over 11$

(c) $14 over 13$

(d) $14 over 15$

Answer 13:

Given that

The correct option is.

Question 14:

The number $0 . \bar{3}$ in the form $\frac{p}{q}$ , where p and q are integers and q ≠ 0, is
(a) $\frac{33}{100}$

(b) $\frac{3}{10}$

(c) $\frac{1}{3}$

(d) $\frac{3}{100}$

Answer 14:

Given number is

The correct option is

Question 15:

$0.3 \bar{2}$ when expressed in the form $\frac{p}{q}$ (p, q are integers q ≠ 0), is

(a) $\frac{8}{25}$

(b) $\frac{29}{90}$

(c) $\frac{32}{99}$

(d) $\frac{32}{199}$

Answer 15:

Given that

Now we have to express this number into form

Let X = $0.3 \bar{2}$

The correct option is

Question 16:

$23 . \bar{43}$ when expressed in the form $\frac{p}{q}$ (p, q are integers q ≠ 0), is

(a) $\frac{2320}{99}$

(b) $\frac{2343}{100}$

(c) $\frac{2343}{999}$

(d) $\frac{2320}{199}$

Answer 16:

Given that

Now we have to express this number into the form of
Let

$x = 23.43 x = 23 + 0.4343 . . . x = 23 + \frac{43}{99} x = \frac{2277 + 43}{99} = \frac{2320}{99}$

The correct option is

Question 17:

$0 . \bar{001}$ when expressed in the form $\frac{p}{q}$ (p, q are integers, q ≠ 0), is

(a) $\frac{1}{1000}$

(b) $\frac{1}{100}$

(c) $\frac{1}{1999}$

(d) $\frac{1}{999}$

Answer 17:

Given that

Now we have to express this number into form

The correct option is

Question 18:

The value of $0 . \bar{23}$ + $0 . \bar{22}$ is

(a) $0 . \bar{45}$

(b) $0 . \bar{43}$

(c) $0 . \bar{45}$

(d) $0.45$

Answer 18:

Given that

Let

Now we have to find the value of

The correct option is

Question 19:

An irrational number between 2 and 2.5 is

(a) $\sqrt{11}$

(b) $\sqrt{5}$

(c) $\sqrt{22.5}$

(d) $\sqrt{12.5}$

Answer 19:

Let

Here a and b are rational numbers. So we observe that in first decimal place a and b have distinct. According to question a < b.so an irrational number between 2 and 2.5 is OR

Hence the correct answer is.

Question 20:

The number of consecutive zeros in $2^{3} \times 3^{4} \times 5^{4} \times 7$ , is

(a) 3

(b) 2

(c) 4

(d) 5

Answer 20:

We are given the following expression and asked to find out the number of consecutive zeros

We basically, will focus on the powers of 2 and 5 because the multiplication of these two number gives one zero. So

$2^{3} \times 3^{4} \times 5^{4} \times 7 = 2^{3} \times 5^{4} \times 3^{4} \times 7 = 2^{3} \times 5^{3} \times 5 \times 3^{4} \times 7 = {(2 \times 5)}^{3} \times 5 \times 3^{4} \times 7 = 10^{3} \times 5 \times 3^{4} \times 7 = 5 \times 81 \times 7 \times 1000 = 2835000$

Therefore the consecutive zeros at the last is 3

So the option (a) is correct

Page-1.42

Question 21:

The smallest rational number by which $\frac{1}{3}$ should be multiplied so that its decimal expansion terminates after one place of decimal, is

(a) $\frac{1}{10}$

(b) $\frac{3}{10}$

(c) 3

(d) 30

Answer 21:

Give number is. Now multiplying by in the given number, we have

Hence the correct option is

Question 1:

The decimal expansion of a rational number is either ______ or _______.

Answer 1:

The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.

Hence, the decimal expansion of a rational number is either terminating or recurring.

Question 2:

The decimal expansion of an irrational number is non-terminating and ________.

Answer 2:

The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.

In case of irrational number, the decimal expansion neither terminates nor repeats after finitely many digits.

Hence, the decimal expansion of an irrational number is non-terminating and non-repeating.

Question 3:

The decimal expansion of $\sqrt{2}$ is _______ and _________.

Answer 3:

$\sqrt{2}$ is an irrational number.

The decimal expansion of an irrational number neither terminates nor repeats after finitely many digits.

Hence, the decimal expansion of $\sqrt{2}$ is non-terminating and non-repeating.

Question 4:

The value of 1.999. in the form of $\frac{m}{n}$ , where m and n are integers and n ≠ 0, is _______.

Answer 4:

$Let x = 1.999 . . . . . . . . (1) Multiply (1) by 10 on both sides, we get 10 x = 19.999 . . . . . . . . (2) Subtracting (1) from (2), we get 10 x - x = 19.999 . . . . - 1.999 . . . . \Rightarrow 9 x = 18 \Rightarrow x = \frac{18}{9} \Rightarrow x = \frac{2}{1}$

Hence, the value of 1.999... in the form of $\frac{m}{n}$ , where m and n are integers and n ≠ 0, is $\frac{2}{1}$ .

Question 5:

Every recurring decimal is a _________ number.

Answer 5:

The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.

Hence, every recurring decimal is a rational number.

Question 6:

π is an _______ number.

Answer 6:

The decimal expansion of π neither terminates nor repeats after finitely many digits.

Therefore, it is an irrational number.

Hence, π is an irrational number.

Question 7:

The product of a non-zero rational number with an irrational number is always an ________ number.

Answer 7:

The product of a non-zero rational number with an irrational number always results in an irrational number.

Hence, the product of a non-zero rational number with an irrational number is always an irrational number.

Question 8:

The simplest form of $1.6$ is _______.

Answer 8:

$Let x = 1.6666 . . . . . . . . (1) Multiply (1) by 10 on both sides, we get 10 x = 16.6666 . . . . . . . . (2) Subtracting (1) from (2), we get 10 x - x = 16.6666 . . . . - 1.6666 . . . . \Rightarrow 9 x = 15 \Rightarrow x = \frac{15}{9} \Rightarrow x = \frac{5}{3}$

Hence, the simplest form of $1.6$ is $\frac{5}{3}$ .

Question 9:

$0.3 + 0.4$ is equal to _________.

Answer 9:

$Let x = 0.3333 . . . . . . . . (1) Multiply (1) by 10 on both sides, we get 10 x = 3.3333 . . . . . . . . (2) Subtracting (1) from (2), we get 10 x - x = 3.333 . . . . - 0.333 . . . . \Rightarrow 9 x = 3 \Rightarrow x = \frac{3}{9} . . . (3) Let y = 0.4444 . . . . . . . . (4) Multiply (1) by 10 on both sides, we get 10 y = 4.4444 . . . . . . . . (5) Subtracting (4) from (5), we get 10 y - y = 4.4444 . . . . - 0.4444 . . . . \Rightarrow 9 y = 4 \Rightarrow y = \frac{4}{9} . . . (6) Now, 0.3 + 0.4 = x + y = \frac{3}{9} + \frac{4}{9} (From (3) and (6)) = \frac{7}{9}$

Hence, $0.3 + 0.4$ is equal to $\frac{7}{9}$ .

Question 10:

The sum of a rational number and an irrational number is ________ number.

Answer 10:

The sum of a rational number and an irrational number always results in an irrational number.

Hence, the sum of a rational number and an irrational number is an irrational number.

Question 11:

Every real number is either ________ or _______ number.

Answer 11:

The real number includes all the rational as well as irrational numbers.

Hence, every real number is either rational or irrational number.

RD Sharma 2020 solution class 9 chapter 1 Number System MCQS

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