This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Irrational and Rational Numbers”.

1. If x is a number whose simplest form is \(\frac {p}{q}\), where p and q are integers and q≠0, then x is a terminating decimal only when q is of the form _________

a) 3^{m}×5^{n}

b) 2^{m}×6^{n}

c) 2^{m}×5^{n}

d) 7^{m}×5^{n}

View Answer

Explanation: Let’s, take a number where q is of the form 2

^{m}×5

^{n}, say 2

^{50}×5

^{10}and p can be any integer

\(\frac {p}{2^{50}\times 5^{10}} = \frac {p \times 5^{40}}{10^{50}}\)

The number \(\frac {p \times 5^{40}}{10^{50}}\) will terminate after 50 decimal places.

Hence, if q is of the form 2

^{m}×5

^{n}, it will terminate after some decimal places.

2. If x is a number whose simplest form is \(\frac {p}{q}\), where p and q are integers and q ≠ 0, then x is a non-terminating repeating decimal only when q is not of the form ________

a) 2^{m}×2^{n}

b) 5^{m}×5^{n}

c) 2^{m}×5^{n}

d) 3^{m}×4^{n}

View Answer

Explanation: Let’s, take a number where q is of the form 2

^{m}×5

^{n}, say 2

^{50}×5

^{10}and p can be any integer

\(\frac {p}{2^{50}\times 5^{10}} = \frac {p \times 5^{40}}{10^{50}}\)

The number \(\frac {p \times 5^{40}}{10^{50}}\) will terminate after 50 decimal places.

Hence, if q is of the form 2

^{m}×5

^{n}, it will terminate after some decimal places.

3. Which of the following rational is non-terminating repeating decimal?

a) 0.25

b) \(\frac {4}{5}\)

c) \(\frac {4}{55}\)

d) \(\frac {2}{5}\)

View Answer

Explanation: The value of \(\frac {4}{55}\) is 0.07272727272…., which is non-terminating repeating decimal.

The other numbers terminate after few places of decimal.

4. The terminating rational number from the following numbers is _________

a) \(\frac {4}{9}\)

b) \(\frac {4}{3}\)

c) 0.146

d) \(\frac {4}{5}\)

View Answer

Explanation: The value of \(\frac {4}{5}\) is 0.8, which is terminating decimal.

5. The simplest form of the rational number 0.196 is ________

a) \(\frac {1}{6}\)

b) \(\frac {3}{6}\)

c) \(\frac {13}{66}\)

d) \(\frac {2}{5}\)

View Answer

Explanation:

10x = 1.969696…..(1)

1000x = 196.9696…(2)

Subtracting (1) from (2)

We get,

990x=195

x = \(\frac {195}{990} = \frac {13}{66}\)

6. The numbers of the form \(\frac {p}{q}\) are integers, and q≠0 are called irrational number.

a) True

b) False

View Answer

Explanation:

Irrational numbers cannot be written in the form of \(\frac {p}{q}\).

For example, ∛4 cannot be written in a fraction form as it has non-terminating and non-repeating decimals.

7. After how many places of decimal, will the decimal expansion of the rational number \(\frac {57}{2^4 5^6}\) terminate?

a) 4

b) 6

c) 7

d) 8

View Answer

Explanation:

We have,

\(\frac {57}{2^4 5^6} = \frac {57 \times 2^2}{2^6 5^6} = \frac {228}{10^6}\) = 0.000228

The number \(\frac {57}{2^4 5^6}\) will terminate after 6 decimal places.

8. From the following numbers, which number is not a rational number?

a) π

b) \(\frac {22}{7}\)

c) \(\frac {3}{4}\)

d) 0.666666…..

View Answer

Explanation: A rational number has terminating or non-terminating but repeating decimals.

In case of π, it has a non-terminating as well as non-repeating decimal.

The other three numbers have terminating or non-terminating but repeating decimal, therefore, they are rational numbers.

Hence, it is an irrational number.

9. An irrational number has ________

a) Non-terminating decimal

b) Non-repeating decimal

c) Non-terminating and non-repeating decimal

d) Terminating decimal

View Answer

Explanation: An irrational number has both non-terminating as well as non-repeating decimals.

For example, the number 1.353353335… has non-terminating as well as non-repeating decimals.

10. Which of the following numbers is not an irrational number?

a) π

b) \(\frac {22}{7}\)

c) 1.5353353335….

d) 2.7878878887….

View Answer

Explanation:

An irrational number is expressible in the decimal form as non-terminating and non-repeating decimals.

From the given options,

π, 1.5353353335…, 2.7878878887… are non-terminating and non-repeating decimal.

Whereas, \(\frac {22}{7}\) is non-terminating but repeating decimal.

11. The product of \(\frac {33}{2}\) and \(\frac {5}{4}\) is an irrational number.

a) True

b) False

View Answer

Explanation:

\(\frac {33}{2} \times \frac {5}{4} = \frac {165}{8}\)

\(\frac {165}{8}\) is a rational number

12. The product of a rational and an irrational number is rational number.

a) True

b) False

View Answer

Explanation: Take a rational and an irrational number, say 2 and 3√3

Product of 2 × 3√3 = 6√3.

6√3 is an irrational number

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

13. The product of two irrational numbers is an irrational number.

a) True

b) False

View Answer

Explanation: Consider an irrational number, say √10

√10 × √10=10

10 is a rational number. Hence, the product of two irrational numbers is not always irrational.

14. The sum of two rational numbers is a rational number.

a) False

b) True

View Answer

Explanation: Consider two rational numbers, say \(\frac {8}{9}, \frac {3}{5}\)

Sum of these number = \(\frac {8}{9} + \frac {3}{5} = \frac {67}{45}\), which is rational number.

Hence, the sum of two rational numbers is a rational number.

15. The sum of two irrational numbers is a rational number.

a) False

b) True

View Answer

Explanation: Consider two irrational numbers, say, √2 and √5

Sum of these number = √2 + √3 = 3.14626… which is an irrational number.

Hence, the sum of two irrational numbers is an irrational number.

**Sanfoundry Global Education & Learning Series – Mathematics – Class 10**.

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