Class 9 Maths Chapter 1 Practice Question Answers - Number System

Ans: (b)
We know that π is irrational number and Irrational numbers have decimal expansions that neither terminate nor become periodic.
So, correct answer is option B.

Ans: (c)
According to definition of irrational number, If written in decimal notation, an irrational number would have an infinite number of digits to the right of the decimal point, without recurring digits.
Hence, a number having non terminating and non repeating decimal representation is an irrational number.
So, option C is correct.

Ans: (c)
Recall that to find a rational number between r and s, you can add r and s and divide the sum by 2, that is (r+s)/2 lies between r and s.
For example, 5/2 is a number between 2 and 3.
We can proceed in this manner to find many more rational numbers between 2 and 3.
Hence, we can conclude that there are infinitely many rational numbers between any two given rational numbers.    

Ans: (c)
Every rational number is a number that can be expressed in the form $\frac{p}{q}\backslash frac\{p\}\{q\}$, where $p$ and $qq$ are integers and $q\ne 0q\; \backslash neq\; 0$. Rational numbers are a subset of real numbers, but not all rational numbers are natural, integers, or whole numbers.

Ans: (a)
The product of a non-zero rational number with an irrational number is always irrational. This is because multiplying a rational number with an irrational number does not eliminate the non-repeating, non-terminating nature of the irrational number. For example, 2 x √2 = 2√2 which is irrational.

Ans: (d)
⇒ A rational number is a type of real numbers which can be expressed in the form of p/q , where q ≠ 0.
⇒ All the numbers are rational as they are in the form of p/q , where q ≠ 0.

Ans:

Therefore, 10/3 = 3.3333…

7/8 = 0.875

1/7 = 0.1428571…

Ans: Construct BD of unit length perpendicular to OB (here, OA = AB = 1 unit) as shown in the figure.

By Pythagoras theorem,

OD = √(2 + 1) = √3

Taking O as the centre and OD as radius, draw an arc which intersects the number line at the point Q using a compass.

Therefore, Q corresponds to the value of √3 on the number line.

Ans:

Given,

√12√75

= √(2 × 2 × 3)√(3 × 5 × 5)

= 2√35√3

= 25

Here, both 2 and 5 are integer.
Therefore, √12√75  is a rational number.

Ans: (√3 + √7)(√3 – √7)

Using the identity (a + b)(a – b) = a² – b²,

(√3 + √7)(√3 – √7) = (√3)² – (√7)²

= 3 – 7

= -4

Ans: We need to rationalise

5 + 2√37 + 4√3

= (5 + 2√3)(7 + 4√3) × (7 - 4√3)(7 - 4√3)

= (5 + 2√3)(7 - 4√3)49 - 48

= (35 - 20√3 + 14√3 - 24)

= 11 - 6√3

Now comparing this with a−6√3 , we get
a = 11

Ans: (c)
All numbers that can be written in the form of p/q , where p and q are integers are rational numbers.

Option A:

√ 818 = √ 49 = 23

Hence, it is rational.

Option B:

√ 123 = √4 = 2

Hence, it is a rational number.

Option C:

√ 288 = √ 72

This cannot be simplified further.
This is an irrational number.
Option D:
√81 = 9
This is a rational number.
Hence, option C is correct.

Ans: (i) 7^2/3.7^1/5

Bases are equal, so add the powers.

7^{(2/3 + 1/5)}

= 7^{(10 + 3)/15}

= 7^13/15

(ii) 10^1/2/10^1/4

Bases are equal, so subtract the powers.

= 10 ^{(1/2 – 1/4)}

= 10^1/4

Ans:  (a)

(256)^0.16 x (256)^0.09 = (256)^{(0.16 + 0.09)}

= (256)^0.25

= (256)^(25/100)

= (256)^(1/4)

= (4⁴)^(1/4)

= 4^4(1/4)

= 4

Ans: (d)

Consider x = 1.999 …. (1)

Let us multiply equation (1) with 10

10x = 19.99 …. (2)

Subtracting equation (1) from (2)

9x = 18

Dividing both sides by 9

x = 2

Therefore, the value of 1.999... is 2.
Hence, option D is correct answer.

Ans: (b)
Since the denominator of both rational numbers are same. So, for getting the rational numbers between the given rational numbers, we only have to consider the numerators of the rational numbers.
Two numbers between 1 & 4 are 2 and 3.
So, two rational numbers between the given rational numbers will be 2/5 and 3/5
So, correct answer is option B.

Ans: (c)
The value of π is equal to 3.14159265358… which is a non-terminating and non-repeating decimal hence, π is an irrational number.

Ans: Since in 0.111222333444.... each number appears 3 times in a row is a non- terminating and non- recurring decimal expansion.
Hence, 0.111222333444.... is irrational.

Ans:
Given: x = √2 + 1

∴ 1x = 1√2 + 1

= √2 - 1(√2)² - 1² {Using method of Rationalisation}

= √2 - 12 - 1 = √2 - 1

∴ x + 1x = (√2 + 1 + √2 - 1) = 2√2

Sol: We have to find five rational numbers between 3/5 and 4/5.

So, let us write the given numbers by multiplying with 6/6, (here 6 = 5 + 1)
Now,

3/5 = (3/5) × (6/6) = 18/30
4/5 = (4/5) × (6/6) = 24/30

Thus, the required five rational numbers will be: 19/30, 20/30, 21/30, 22/30, 23/30