MCQ test, Set 8

Answer: (C) $49$

$\displaystyle \left(\sqrt[3]{7}\right)^{6} = \left(7^{\frac{1}{3}}\right)^{6} = 7^{2} = 49$

Answer: (D) $b^{2}$

$\displaystyle a^{m} = b^{n} \Rightarrow a = b^{\frac{n}{m}}$

$\displaystyle a = b^{\frac{6}{3}} = b^{2}$

Answer: (B) $32$

$= (2^{4})^{5} \times (2^{5})^{-3}$

$= 2^{20} \times 2^{-15}$

$= 2^{20 - 15} = 2^{5}$

$= 32$

Answer: (B) $\displaystyle \frac{25}{4}$

$\displaystyle (10^{3})^{\frac{2}{3}} \div (2^{3})^{\frac{4}{3}}$

$= 10^{2} \div 2^{4}$

$\displaystyle = 100 \div 16 = \frac{25}{4}$

Answer: (C) $-4$

$\displaystyle \left(-\frac{1}{64}\right)^{-\frac{1}{3}} = \left[\left(-\frac{1}{4}\right)^{3}\right]^{-\frac{1}{3}}$

$\displaystyle = \left(-\frac{1}{4}\right)^{-1} = \left(-4\right)^{1}$

$= -4$

Answer: (D) $\displaystyle \frac{12}{11}$

$\displaystyle \left(\frac{121}{144}\right)^{-\frac{1}{2}} = \left[\left(\frac{11}{12}\right)^{2}\right]^{-\frac{1}{2}}$

$\displaystyle = \left(\frac{11}{12}\right)^{-1} = \left(\frac{12}{11}\right)^{1}$

$\displaystyle = \frac{12}{11}$

Answer: (B) $4$

$\displaystyle = \frac{(2^{2})^{\frac{4}{3}} \times (5^{2})^{\frac{1}{3}}}{(2 \times 5)^{\frac{2}{3}}}$

$\displaystyle = 2^{\frac{8}{3} - \frac{2}{3}} \times 5^{\frac{2}{3} - \frac{2}{3}}$

$= 2^{2} \times 5^{0}$

$= 4$

Answer: (B) $\displaystyle x^{\frac{1}{c} - \frac{1}{a}}$

$\displaystyle = x^{\frac{a-b}{ab}} \cdot x^{\frac{b-c}{bc}} = x^{\frac{1}{b} - \frac{1}{a}} \cdot x^{\frac{1}{c} - \frac{1}{b}}$

$\displaystyle = x^{\frac{1}{c} - \frac{1}{a}}$

Answer: (D) $-6$

$p + q + r = 0 \Rightarrow p^{3} + q^{3} + r^{3} = 3pqr$

$= 3 \times (-2) = -6$

Answer: (A) $2$

$\displaystyle \frac{2^{5x}}{2^{2x}} \times (2^{3})^{2} \times 2^{2} = 2^{14}$

$\Rightarrow 2^{3x} \times 2^{6} \times 2^{2} = 2^{14}$

$\Rightarrow 2^{3x + 8} = 2^{14}$

$\Rightarrow 3x + 8 = 14 \Rightarrow 3x = 6 \Rightarrow x = 2$

Answer: (B) $3$

$\displaystyle 2\sqrt{2} = 2^{1} \times 2^{\frac{1}{2}} = 2^{\frac{3}{2}}$

$\displaystyle (2^{4})^{x} = \left(2^{\frac{3}{2}}\right)^{8}$

$\Rightarrow 2^{4x} = 2^{12}$

$\Rightarrow 4x = 12 \Rightarrow x = 3$

Answer: (D) $3$

$160 = 5^{1} \times 2^{5}$

$\Rightarrow 5^{2x - 5} \times 2^{x + 2} = 5^{1} \times 2^{5}$

$\Rightarrow 2x - 5 = 1$ and $x + 2 = 5$

$\Rightarrow x = 3$ and $x = 3$

$\therefore x = 3$

Answer: (A) $4$

$\displaystyle 3^{x-4} = \frac{3^{x}}{3^{4}}$

$\displaystyle = \frac{324}{81} = 4$

Answer: (D) $-4$

$\displaystyle \left[\left(\frac{5}{7}\right)^{\frac{1}{2}}\right]^{x} = \left(\frac{7}{5}\right)^{2}$

$\displaystyle \Rightarrow \left(\frac{5}{7}\right)^{\frac{x}{2}} = \left(\frac{5}{7}\right)^{-2}$

$\displaystyle \Rightarrow \frac{x}{2} = -2$

$\Rightarrow x = -4$

$\Rightarrow x = -4$

Answer: (A) $4$

$\displaystyle 2 \times 2^{x} + \frac{2^{x}}{2} = 40$

$\displaystyle \Rightarrow 2^{x}\left(2 + \frac{1}{2}\right) = 40$

$\displaystyle \Rightarrow 2^{x} \times \frac{5}{2} = 40$

$\displaystyle \Rightarrow 2^{x} = 40 \times \frac{2}{5} = 16 = 2^{4}$

$\Rightarrow x = 4$

Answer: (A) $9$

$\displaystyle 2^{\frac{n}{2}} = 512 = 2^{9} \Rightarrow \frac{n}{2} = 9 \Rightarrow n = 18$

$\displaystyle 3^{\frac{18}{9}} = 3^{2} = 9$

Answer: (C) $1$

$\displaystyle = \frac{5^{b}}{5^{b} + 5^{a}} + \frac{5^{a}}{5^{a} + 5^{b}} = 1$

Answer: (A) $x^{2b^{3}}$

$= x^{a^{3}+b^{3}} \div x^{a^{3}-b^{3}}$

$= x^{a^{3}+b^{3}-a^{3}+b^{3}} = x^{2b^{3}}$

Answer: (C) $3$

$5 = 2^{y} = (3^{x})^{y} = 3^{xy}$

$27 = 5^{z} = (3^{xy})^{z} = 3^{xyz}$

$3^{xyz} = 27 = 3^{3}$

$\Rightarrow xyz = 3$

Answer: (C) $\displaystyle \frac{2}{x} + \frac{2}{y}$

Let each $= k$: $\displaystyle 3 = k^{\frac{1}{x}}$, $\displaystyle 5 = k^{\frac{1}{y}}$, $\displaystyle 225 = k^{\frac{1}{z}}$

$225 = 3^{2} \times 5^{2}$, so $\displaystyle \frac{1}{z} = \frac{2}{x} + \frac{2}{y}$

Answer: (A) $\displaystyle \frac{1}{100}$

$\displaystyle \sqrt[n]{a^{m}} = a^{\frac{m}{n}}$

$\displaystyle \sqrt[5]{100000^{-2}} = 100000^{-\frac{2}{5}} = (10^{5})^{-\frac{2}{5}}$

$\displaystyle = 10^{-2} = \frac{1}{10^{2}} = \frac{1}{100}$

Answer: (B) $\displaystyle \frac{25}{6}$

$\displaystyle \frac{a^{m}}{a^{n}} = \frac{1}{a^{n-m}}$

$\displaystyle \left(\frac{\sqrt{6}}{5}\right)^{2} \div \left(\frac{\sqrt{6}}{5}\right)^{4} = \frac{1}{\left(\frac{\sqrt{6}}{5}\right)^{2}}$

$\displaystyle = \frac{25}{6}$

Answer: (D) $110$

$(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)$

$\displaystyle x^{3} = 10 + 10^{2} + 3 \cdot 10^{\frac{1}{3}} \cdot 10^{\frac{2}{3}} \cdot x = 110 + 30x$

$x^{3} - 30x = 110$

Answer: (C) $8$

$= (32)^{0.4 + 0.2} = (32)^{0.6}$

$\displaystyle 0.6 = \frac{3}{5}$

$\displaystyle = (2^{5})^{\frac{3}{5}} = 8$

Answer: (C) $8$

$\displaystyle = 2^{(-5) \times 3 \times (-\frac{1}{5})}$

$= 2^{3}$

$= 8$