Question

Choose the correct alternative answer for the questions given below.

(i) Which one of the following is an irrational number ?
(A) $\sqrt{\frac{16}{25}}$ (B) $\sqrt{5}$ (C) $\frac{3}{9}$ (D) $\sqrt{196}$

(ii) Which of the following is an irrational number?

(A) 0.17 (B) $1.\overline{513}$ (C) $0.27\overline{46}$ (D) 0.101001000.....

(iii) Decimal expansion of which of the following is non-terminating recurring ?

(A) $\frac{2}{5}$ (B) $\frac{3}{16}$ (C) $\frac{3}{11}$ (D) $\frac{137}{25}$

iv) Every point on the number line represent, which of the following numbers?

(A) Natural numbers (B) Irrational numbers (C) Rational numbers (D) Real numbers.

(v) The number $0.\stackrel{°}{4}$ in $\frac{p}{q}$form is .....
(A) $\frac{4}{9}$ (B) $\frac{40}{9}$ (C)$\frac{3.6}{9}$ (D) $\frac{36}{9}$

(vi) What is $\sqrt{n}$ , if n is not a perfect square number ?
(A) Natural number (B) Rational number
(C) Irrational number (D) Options A, B, C all are correct.

(vii) Which of the following is not a surd ?
(A) $\sqrt{7}$ (B) $\sqrt[3]{17}$ (C) $\sqrt[3]{64}$ (D) $\sqrt{193}$

(viii) What is the order of the surd $\sqrt[3]{\sqrt{5}}$ ?
(A) 3 (B) 2 (C) 6 (D) 5

(ix) Which one is the conjugate pair of $2\sqrt{5}+\sqrt{3}$ ?
(A) $-$$2\sqrt{5}+\sqrt{3}$ (B) $-2\sqrt{5}-\sqrt{3}$ (C) $2\sqrt{3}-\sqrt{5}$ (D) $\sqrt{3}+2\sqrt{5}$

(x) The value of $\left|12-\left(13+7\right)\times 4\right|$ is...............
(A) $-$68 (B) 68 (C) $-$32 (D) 32

Answer 1

(i) Since,

$\sqrt{\frac{16}{25}}$ = $\frac{4}{5}$ is a rational number; $\frac{3}{9}$ is a rational number; $\sqrt{196}$ = 14 is a rational number; and $\sqrt{5}$ is an irrational number.

Hence, the correct option is (B).

(ii) Since,

0.17 has a terminating decimal expansion, so, it is rational number;

$1.\overline{513}$ has non-terminating recurring decimal expansion, so, it is rational number;

$0.27\overline{46}$ has non-terminating recurring decimal expansion, so, it is rational number;

0.101001000..... has non-terminating non-recurring decimal expansion, so, it is irrational number;

Hence, the correct option is (D).

(iii)

(A) Since, 5 = $2^0\times 5^1$, which is in the form of $2^m\times 5^n$, where m and n are non-negative integers.

So, the decimal expansion of $\frac{2}{5}$ is terminating.

(B) Since, 16 = $2^4\times 5^0$, which is in the form of $2^m\times 5^n$, where m and n are non-negative integers.

So, the decimal expansion of $\frac{3}{16}$ is terminating.

(C) Since, 11 = $2^0\times 5^0\times {11}^1$, which is not in the form of $2^m\times 5^n$, where m and n are non-negative integers.

So, the decimal expansion of $\frac{3}{11}$​ is non-terminating recurring.

(D) Since, 25 = $2^0\times 5^2$, which is in the form of $2^m\times 5^n$, where m and n are non-negative integers.

So, the decimal expansion of $\frac{137}{25}$ is terminating.

Hence, the correct option is (C).

(iv) Since, every point on the number line represents a real number.

Hence, the correct option is (D).

(v)

$$\begin{gathered} \mathrm{Let}x=0.\overline{4}...\left(\mathrm{i}\right) \\ \mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}10,\mathrm{we}\mathrm{get} \\ 10x=4.\overline{4}...\left(\mathrm{ii}\right) \\ \mathrm{Subtracting}\left(\mathrm{i}\right)\mathrm{from}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get} \\ 9x=4 \\ \Rightarrow x=\frac{4}{9} \end{gathered}$$

Hence, the correct option is (A).

(vi) If n is not a perfect square number, then $\sqrt{n}$ is an irrational number.

Hence, the correct option is (C).

(vii) Since, $\sqrt[3]{64}$ = 4

Hence, the correct option is (C).

(viii) Since, $\sqrt[3]{\sqrt{5}}={\left(\sqrt{5}\right)}^{\frac{1}{3}}={\left[{\left(5\right)}^{\frac{1}{2}}\right]}^{\frac{1}{3}}=5^{\frac{1}{6}}=\sqrt[6]{5}$

So, the order of the surd $\sqrt[3]{\sqrt{5}}$ is 6.

Hence, the correct option is (C).

(ix) Since, the conjugate pair of ($2\sqrt{5}+\sqrt{3}$) is ($-$$2\sqrt{5}+\sqrt{3}$).

​Hence, the correct option is (A).

(x) Since, $\left|12-\left(13+7\right)\times 4\right|=\left|12-20\times 4\right|=\left|12-80\right|=\left|-68\right|=68$

So, the value of $\left|12-\left(13+7\right)\times 4\right|$ is 68 .

​Hence, the correct option is (B).