Power, Indices and Surds MCQ Question with Answer

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Power, Indices and Surds: Quantitative Aptitude MCQ Question with Answer

 

Q1. Evaluate $\frac{\sqrt{24}+\sqrt{6}}{\sqrt{24}-\sqrt{6}}$

(a) 5

(b) 4

(c) 3

(d) 2

Answer: (c)

Solution: $\frac{\sqrt{24}+\sqrt{6}}{\sqrt{24}-\sqrt{6}}=\frac{2\sqrt{6}+\sqrt{6}}{2\sqrt{6}-\sqrt{6}}=\frac{3\sqrt{6}}{\sqrt{6}}=3$

Q2. The smallest of is: $\sqrt{8}+\sqrt{5},\sqrt{7}+\sqrt{6},\sqrt{10}+\sqrt{3}\mathrm{and} \sqrt{11}+\sqrt{2}$

(a) $\sqrt{8}+\sqrt{5}$

(b) $\sqrt{7}+\sqrt{6}$

(c) $\sqrt{10}+\sqrt{3}$

(d) $\sqrt{11}+\sqrt{2}$

Answer: (d)

Solution: $\sqrt{\left( \sqrt{8}+\sqrt{5} \right) ^2}=\sqrt{13+2\sqrt{40}}$

$\sqrt{\left( \sqrt{7}+\sqrt{6} \right) ^2}=\sqrt{13+2\sqrt{42}}$

$\sqrt{\left( \sqrt{10}+\sqrt{3} \right) ^2}=\sqrt{13+2\sqrt{30}}$

$\sqrt{\left( \sqrt{11}+\sqrt{2} \right) ^2}=\sqrt{13+2\sqrt{22}}$

Since $2\sqrt{22}$ is smallest then $\sqrt{11}+\sqrt{2}$ is smallest

Q3. Find the value of $\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}$

(a) 2

(b) $2^{\frac{1}{2}}$

(c) $2^{\frac{11}{12}}$

(d) $2^{\frac{31}{32}}$

Answer: (d)

Solution: $\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}=2^{\frac{2^5-1}{2^5}}=2^{\frac{31}{32}}$

Rules: $\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}}}}}}=\mathrm{X}^{\frac{\mathrm{X}^{\mathrm{n}}-1}{\mathrm{X}^{\mathrm{n}}}}$

where n is no. of times X repeated.

Q4. Find the value of $\sqrt{5\sqrt{5\sqrt{5…\mathrm{\alpha}}}}$

(a) 5

(b) $5^{\frac{7}{8}}$

(c) $5^{\frac{1}{8}}$

(d) $5^{\frac{1}{3}}$

Answer: (a)

Solution: $\sqrt{5\sqrt{5\sqrt{5…\mathrm{\alpha}}}}=5$

Rules: $\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}}}}}}=\mathrm{X}$

Q5. Find the value of $\sqrt{12+\sqrt{12+\sqrt{12….}}}$

(a) 3

(b) –3

(c) 2

(d) –2

Answer: (b)

Solution:

$\sqrt{12+\sqrt{12+\sqrt{12….}}}=\mathrm{x}\left( \mathrm{let} \right)$
$\Rightarrow 12 +\mathrm{x} =\,\,\mathrm{x}^2$ ( square both side)
$\Rightarrow \mathrm{x}^2-\mathrm{x}-12=0$
$\Rightarrow \mathrm{x}^2-4\mathrm{x}+3\mathrm{x}-12=0$
$\Rightarrow \mathrm{x}\left( \mathrm{x}-4 \right) +3\left( \mathrm{x}-12 \right) =0$
$\Rightarrow \left( \mathrm{x}-4 \right) \left( \mathrm{x}+3 \right) =0$

∴ x = 4 or -3

Here –3 is in option.

Rules: $\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) +\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) +\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) ….}}}=-\mathrm{n},\left( \mathrm{n}+1 \right) $

Q6. Find the value of $\sqrt{30-\sqrt{30-\sqrt{30….}}}$

(a) 5

(b) 6

(c) –5

(d) 4

Answer: (a)

Solution: $\sqrt{30-\sqrt{30-\sqrt{30….}}}=\mathrm{x}\left( \mathrm{let} \right)$
$\Rightarrow 30 -\mathrm{x} =\,\,\mathrm{x}^2$ ( square both side)
$\Rightarrow \mathrm{x}^2+\mathrm{x}-30=0$
$\Rightarrow \mathrm{x}^2+6\mathrm{x}-5\mathrm{x}-30=0$
$\Rightarrow \mathrm{x}\left( \mathrm{x}+6 \right) -5\left( \mathrm{x}+6 \right) =0$
$\Rightarrow \left( \mathrm{x}+6 \right) \left( \mathrm{x}-5 \right) =0$

∴ x = -6 or 5

Here 5 is in option.

Rules: $\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) -\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) -\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) ….}}}=\mathrm{n},-\left( \mathrm{n}+1 \right) $

Q7. Find the square root of $105\frac{4}{64}$

(a) $15\frac{1}{4}$

(b) $15\frac{3}{4}$

(c) $10\frac{1}{4}$

(d) $6\frac{1}{4}$

Answer: (c)

Solution: $\sqrt{105\frac{4}{64}\,\,}=\,\,\sqrt{\frac{6724}{64}}=\frac{82}{8}=10\frac{1}{4}$

Q8. Find the value of

$\frac{1}{\sqrt{9}-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-\sqrt{4}}$

(a) 0

(b) 1

(c) 1/3

(d) 5

(e) 1/5

Answer: (d)

Solution: $\frac{1}{\sqrt{9}-\sqrt{8}}=\frac{\sqrt{9}+\sqrt{8}}{\left( \sqrt{9}-\sqrt{8} \right) \left( \sqrt{9}+\sqrt{8} \right)}=\sqrt{9}+\sqrt{8}$

Similarly $\frac{1}{\sqrt{8}-\sqrt{7}}=\sqrt{8}+\sqrt{7}$

$\frac{1}{\sqrt{7}-\sqrt{6}}=\sqrt{7}+\sqrt{6}$

$\frac{1}{\sqrt{6}-\sqrt{5}}=\sqrt{6}+\sqrt{5}$

$\frac{1}{\sqrt{5}-\sqrt{4}}=\sqrt{5}+\sqrt{4}$

$\therefore \sqrt{9}+\sqrt{8}-\sqrt{8}-\sqrt{7}+\sqrt{7}-\sqrt{6}-\sqrt{5}+\sqrt{5}+\sqrt{4}$

= 3 + 2 = 5

Q9. The value of $\left( \frac{1024}{243} \right) ^{-\frac{4}{5}}$ is:

(a) $\frac{81}{16}$

(b) $\frac{81}{256}$

(c) $\frac{4}{9}$

(d) $\frac{9}{4}$

(e) None of these

Answer: (b)

Solution:

$\left( \frac{1024}{243} \right) ^{-\frac{4}{5}}\,\,=\,\,\left( \frac{\left( 2 \right) ^{10}}{\left( 3 \right) ^5} \right) ^{^{-\frac{4}{5}}}=\,\,\frac{2^{10\times \left( -\frac{4}{5} \right)}}{3^{5\times \left( -\frac{4}{5} \right)}}\,\,=\,\,\frac{2^{-8}}{3^{-4}}=\left( \frac{3}{4} \right) ^4=\frac{81}{256}$

Q10. The value of $\left( \sqrt{125} \right) ^{\frac{1}{3}}$is:

(a) 2

(b) 4

(c) 5

(d) 8

(e) None of these

Answer: (e)

Solution: $\left( \sqrt{125} \right) ^{\frac{1}{3}}\,\,=\,\,\left( \sqrt{5^3} \right) ^{\frac{1}{3}}\,\,=\,\,\left( 5^{\frac{3}{2}} \right) ^{\frac{1}{3}}=\sqrt{5}$

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