# Rational or Irrational Numbers MCQ Quiz - Objective Question with Answer for Rational or Irrational Numbers - Download Free PDF

Last updated on Dec 1, 2022

In this article, a mock test on Rational or Irrational Numbers is formatted by Testbook. It’s a very common yet requisite part of Mathematics and Quantitative Aptitude in many famed national recruitment examinations. Rational or Irrational Numbers MCQs Quiz helps you in analysing your level of preparation for this particular section and helps in evaluating your skills. To ace up your preparation in this section, solve Rational or Irrational Numbers and check out some helpful tricks to solve these questions easily.

## Latest Rational or Irrational Numbers MCQ Objective Questions

#### Rational or Irrational Numbers Question 1:

Identify the rational number which is different from the other three : $$\frac{2}{3}, \frac{-4}{5}, \frac{1}{2}, \frac{1}{3}$$

1. -4/5
2. 2/3
3. 1/3
4. 1/2

Option 1 : -4/5

#### Rational or Irrational Numbers Question 1 Detailed Solution

Calculation:

$$\frac{-4}{5}$$ is the rational number which is different from the other three, as it lies on the left side of zero while others lie on the right side of zero on the number line.

#### Rational or Irrational Numbers Question 2:

What is the value of $$\frac{1}{1~3~5}+\frac{1}{3~5~7}+\ldots+\frac{1}{13~15~17}$$?

1. $$\dfrac{9}{85}$$
2. $$\dfrac {5} {85}$$
3. $$\dfrac{7}{85}$$
4. $$\dfrac{35}{153}$$

Option 3 : $$\dfrac{7}{85}$$

#### Rational or Irrational Numbers Question 2 Detailed Solution

Given:

$$\frac{1}{1~3~5}+\frac{1}{3~5~7}+\ldots+\frac{1}{13~15~17}$$

Calculations:

Tn = $$\dfrac{1}{(2n-1)(2n+1)(2n-3)}$$

Tn = $$\dfrac{1}{8}$$($$\dfrac{1}{2n-1}$$ - $$\dfrac{2}{2n+1}$$ + $$\dfrac{1}{2n+3}$$)

T1 = $$\dfrac{1}{8}$$($$\dfrac{1}{1}$$ - $$\dfrac{2}{3}$$ + $$\dfrac{1}{5}$$)

T2 = $$\dfrac{1}{8}$$($$\dfrac{1}{3}$$ - $$\dfrac{2}{5}$$ + $$\dfrac{1}{7}$$)

T3 = $$\dfrac{1}{8}$$($$\dfrac{1}{5}$$ - $$\dfrac{2}{7}$$ + $$\dfrac{1}{9}$$)

T4 = $$\dfrac{1}{8}$$($$\dfrac{1}{7}$$ - $$\dfrac{2}{9}$$ + $$\dfrac{1}{11}$$)

T5 = $$\dfrac{1}{8}$$($$\dfrac{1}{9}$$ - $$\dfrac{2}{11}$$ + $$\dfrac{1}{13}$$)

T6 = $$\dfrac{1}{8}$$($$\dfrac{1}{11}$$ - $$\dfrac{2}{13}$$ + $$\dfrac{1}{15}$$)

T7 = $$\dfrac{1}{8}$$($$\dfrac{1}{13}$$ - $$\dfrac{2}{15}$$ + $$\dfrac{1}{17}$$)

Now, Sn = $$\dfrac{1}{8}$$($$\dfrac{2}{3}$$ - $$\dfrac{1}{15}$$ + $$\dfrac{1}{17}$$)

= $$\dfrac{1}{8}$$ × $$\dfrac{170 -17 +15}{85}$$

= $$\dfrac{7}{85}$$

∴ The answer is $$\dfrac{7}{85}$$

#### Rational or Irrational Numbers Question 3:

What is the value of $$\frac{1}{3 \times 7}+\frac{1}{7 \times 11}+\frac{1}{11 \times 15} \ldots \ldots . .+\frac{1}{899 \times 903}$$

1. $$\frac{21}{509}$$
2. $$\frac{18}{403}$$
3. $$\frac{25}{301}$$
4. $$\frac{29}{31}$$

Option 3 : $$\frac{25}{301}$$

#### Rational or Irrational Numbers Question 3 Detailed Solution

$$\frac{1}{3 \times 7}+\frac{1}{7 \times 11}+\frac{1}{11 \times 15} \ldots \ldots \frac{1}{899 \times 903}$$

$$\rightarrow \frac{1}{4}\left[\frac{1}{3 \times 7}+\frac{4}{7 \times 11}+\frac{4}{11 \times 15} \ldots \ldots \ldots \frac{4}{899 \times 903}\right]$$

$$\rightarrow \frac{1}{4}\left[\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15} \ldots \ldots \cdot \frac{1}{899}-\frac{1}{903}\right]$$

$$\rightarrow \frac{1}{4}\left[\frac{1}{3}-\frac{1}{903}\right]$$

$$\rightarrow \frac{1}{4}\left[\frac{301-1}{903}\right]$$

$$\rightarrow \frac{1}{4}\left[\frac{300}{903}\right]$$

$$\rightarrow \frac{25}{301}$$

#### Rational or Irrational Numbers Question 4:

What is the value of $$(2+\sqrt{2})+\left(\frac{1}{2+\sqrt{2}}\right)+\left(\frac{1}{2-\sqrt{2}}\right)+(2-\sqrt{2})$$?

1. 2
2. 4
3. 8
4. 6

Option 4 : 6

#### Rational or Irrational Numbers Question 4 Detailed Solution

$$(2+\sqrt{2})+(2-\sqrt{2})+\frac{1}{2+\sqrt{2}}+\frac{1}{2-\sqrt{2}}=4+\frac{4}{2}=6$$

#### Rational or Irrational Numbers Question 5:

What will be the maximum number of digits in the repeating block of digits in the decimal expansion of $$\frac{1}{7}$$?

1. 5
2. 8
3. 6
4. 7

Option 3 : 6

#### Rational or Irrational Numbers Question 5 Detailed Solution

Given:

The repeating block of digits in the decimal expansion of $$\frac{1}{7}$$.

Calculation:

$$1\over7$$ = 0.142857142857142857........

⇒ $$1\over7$$ = $$0.\overline{142857}$$

So, The maximum number of digits in the repeating block of digits is 6.

∴ The correct answer is 6.

## Top Rational or Irrational Numbers MCQ Objective Questions

#### Rational or Irrational Numbers Question 6

Write 0.135135.... in the form of p/q.

1. 5/37
2. 17/37
3. 19/7
4. 19/111

Option 1 : 5/37

#### Rational or Irrational Numbers Question 6 Detailed Solution

Given:

0.135135....

Concept used:

The numbers of the form (p/q), where q ≠ 0 and p and q is integer is known as rational number.

Calculation:

Let x = 0.135135....      ----(1)

Multiply equation (1) by 1000, we have

1000x = 135.135....      ----(2)

Subtract equation (1) from equation (2), we have

1000x - x = (135.135...) - (0.135135....)

⇒ 999x = 135

⇒ x = 135/999

⇒ x = 45/333

⇒ x = 5/37

∴ The 0.135135.... can be written as 5/37 in the form of p/q.

#### Rational or Irrational Numbers Question 7

Which of the following numbers will have an irrational square root?

1. 11025
2. 6025
3. 9025
4. 3025

Option 2 : 6025

#### Rational or Irrational Numbers Question 7 Detailed Solution

⇒ 11025 = 52 × 212

⇒ 6025 = 52 × 241

⇒ 9025 = 52 × 192

⇒ 3025 = 52 × 112

∴ 6025 will have irrational square root.

#### Rational or Irrational Numbers Question 8

Which of the following rational numbers does not lie between 3/5 and 4/5?

1. 19/30
2. 2/3
3. 7/10
4. 16/30

Option 4 : 16/30

#### Rational or Irrational Numbers Question 8 Detailed Solution

Calculation:

Taking all the fractions,

3/5, 4/5, 19/30, 2/3, 7/10, 16/30

Taking LCM of the denominator

LCM of (5, 5, 30, 3, 10 and 30) = 30

Converting all the denominators to 30

3/5 can be written as 18/30

4/5 can be written as 24/30

19/30 can be written as 19/30

2/3 can be written as 20/30

7/10 can be written as 21/30

16/30 can be written as 16/30

Representing this on the number line

So,

∴ 16/30 does not lie between 3/5 and 4/5.

#### Rational or Irrational Numbers Question 9

What is the value of $$\sqrt {19 + 8\sqrt 3 } = ?$$

1. $$4 + \sqrt 3$$
2. $$4 - \sqrt 3$$
3. $$8 + \sqrt 3$$
4. $$8 - \sqrt 3$$

Option 1 : $$4 + \sqrt 3$$

#### Rational or Irrational Numbers Question 9 Detailed Solution

$$\begin{array}{l} \sqrt {19 + 8\sqrt 3 } \\ = \sqrt {16 + 3 + 8\sqrt 3 } \\ = \sqrt {{4^2} + {{\left( {\sqrt 3 } \right)}^2} + 2\; \times \;4\; \times \;\sqrt 3 } \end{array}$$

∵ a+ b2 + 2ab = (a + b)2

$$= \sqrt {{{\left( {4 + \sqrt 3 } \right)}^2}}$$

= 4 + √ 3

#### Rational or Irrational Numbers Question 10

How many factors of number 4000 are perfect squares?

1. 4
2. 5
3. 7
4. 6

Option 4 : 6

#### Rational or Irrational Numbers Question 10 Detailed Solution

Factors of 4000 = 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 125, 160, 200, 250, 400, 500, 800, 1000, 2000, 4000

The factors which are perfect squares = 1, 4, 16, 25, 100, 400.

∴ the required number is 6

#### Rational or Irrational Numbers Question 11

Between two rational numbers

1. there is no rational number
2. there is exactly one rational number
3. there are infinitely many rational numbers
4. there are only rational numbers and no irrational number

Option 3 : there are infinitely many rational numbers

#### Rational or Irrational Numbers Question 11 Detailed Solution

Concept used:

rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero.

Calculation:

Let, here we take two rational numbers 1/5 and 1/2

Then, rational numbers between and 7/20, 8/20, 9/20, 21/100, 22/100, 23/100, 24/100,...

Between two rational numbers, there are infinitely many rational numbers.

∴ Between two rational numbers, there are infinitely many rational numbers.

#### Rational or Irrational Numbers Question 12

Which of the following is the rational form of $$1.02{\bar3}$$ ?

1. $$\frac{{307}}{{300}}$$
2. $$\frac{{1023}}{{900}}$$
3. $$\frac{{308}}{{300}}$$
4. $$\frac{{1021}}{{900}}$$

Option 1 : $$\frac{{307}}{{300}}$$

#### Rational or Irrational Numbers Question 12 Detailed Solution

Shortcut Trick

We can solve this question by short method:

Write the number and subtract the non-repeated part and divide the result by as many as 9 as repeat part followed by and as many as 0 as non-repeating decimal

⇒ $$1.02{\bar3}$$ = (1023 - 102)/900

⇒ $$1.02{\bar3}$$ = 921/900

⇒ $$1.02{\bar3}$$ = 307/300

∴ The rational form of $$1.02{\bar3}$$ is 307/300.

Given:

Given number is $$1.02{\bar3}$$

Calculation

Let x = 1.02333.... ----(1)

multiply equation (1) by 100, we get

⇒ 100x = 102.333.... ----(2)

multiply equation (2) by 10, we get

⇒ 1000x = 1023.333.... ----(3)

Now, Subtracting equation (2) from equation (3), we get

⇒ 900x = 921

Divide each term by 3, we get

⇒ 300x = 307

⇒ x = 307/300

∴ The rational form of $$1.02{\bar3}$$ is 307/300.

#### Rational or Irrational Numbers Question 13

Which of the following is not an irrational number?

1. √2
2. √3
3. √4
4. π

Option 3 : √4

#### Rational or Irrational Numbers Question 13 Detailed Solution

Concept used:

Irrational numbers are those numbers which have decimal expansion that neither terminating nor repeating.

Calculation:

Like π = 3.1415926…….

√2 = 1.41421….

√3 = 1.73205……

But √4 = 2

∴ √4 is not an irrational number

#### Rational or Irrational Numbers Question 14

Which of the following is equivalent to $$0.\overline {56}$$? (the bar indicates repeating decimal)

1. 56/100
2. 56/1000
3. 56/99
4. 56/90

Option 3 : 56/99

#### Rational or Irrational Numbers Question 14 Detailed Solution

Given:

$$0.\overline {56}$$ = ?

Calculations:

Let x = 0.56565656..... (i)

100x = 56.565656.... (ii)

Equation (ii) - Equation (i)

99x = 56

∴ x = 56/99

#### Rational or Irrational Numbers Question 15

A number $${2.\overline{35}}$$ is:

1. a rational number
2. an irrational number
3. a rational number and a prime number
4. a prime number

Option 1 : a rational number

#### Rational or Irrational Numbers Question 15 Detailed Solution

Given:

Number = $${2.\overline{35}}$$

Concept used:

A number is rational if and only if its decimal representation is repeating or terminating.

Detailed solution:

$${2.\overline{35}}$$ is a repeating decimal (∵  $${2.\overline{35}} = 2.3535353535...$$)

∴ The given number $${2.\overline{35}}$$ is a rational number

Additional InformationA number is irrational if and only if its decimal representation is non-terminating and non-repeating.

Any number which cannot be written in the form of p/q is termed an irrational number.
While $${2.\overline{35}}$$  is a rational number as it can be changed to p/q form,

$${2.\overline{35}}$$= 233/99
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}