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Surds and Indices Questions Answers

Surds and Indices Questions Answers (MCQ) listings with explanations are important for BANK PO, Clerk, IBPS, SBI-PO, RBI, MBA, MAT, CAT, IIFT, IGNOU, SSC CGL, CBI, CPO, CLAT, CTET, NDA, CDS, Specialist Officers and other competitive exams.

1) \begin{align} \left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}.\\\text{ What is the value of x ?} \end{aligned}

  • 1) 1.5
  • 2) 4.5
  • 3) 7.5
  • 4) 9.5
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Ans.   B
Explanation :
\begin{align}&\left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}\\\\ &\Rightarrow \left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{a}{b}\right)^{-(x-7)}\\\\ &\Rightarrow x - 2 = -(x - 7)\\\\ &\Rightarrow x - 2 = -x + 7\\\\ &\Rightarrow x-2 = -x + 7\\\\ &\Rightarrow 2x = 9\\\\ &\Rightarrow x = \dfrac{9}{2} = 4.5 \end{align}

2) \begin{aligned} \text{If }x = \left(8 + 3\sqrt{7}\right),\text{ what is the value of }\\\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)? \end{aligned}

  • 1) \begin{aligned} \sqrt{13} \end{aligned}
  • 2) \begin{aligned} \sqrt{14} \end{aligned}
  • 3) \begin{aligned} \sqrt{15} \end{aligned}
  • 4) \begin{aligned} \sqrt{16} \end{aligned}
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Ans.   B
Explanation :
\begin{align}&\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2\\\\ &= x - 2 + \dfrac{1}{x}\\\\ &= x + \dfrac{1}{x} - 2 \\\\ &= \left(8 + 3\sqrt{7}\right) + \dfrac{1}{\left(8 + 3\sqrt{7}\right)} - 2 \\\\ &= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{\left(8 + 3\sqrt{7}\right)\left(8 - 3\sqrt{7}\right)} - 2 \\\\ &= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{8^2 - \left(3\sqrt{7}\right)^2} - 2 \\\\ &= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{64 - 63} - 2 \\\\ &= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{1} - 2 \\\\ &= 8 + 3\sqrt{7} + 8 - 3\sqrt{7} - 2 \\\\ &= 14 \\\&\text{as }\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2 = 14\\\\ &\text{so ,}\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right) = \sqrt{14}\end{align}

3) \begin{aligned} \text{If } 5^{(a + b)} = 5 \times 25 \times 125 ,\\ \text{what is }(a + b)^2 \end{aligned}

  • 1) 25
  • 2) 28
  • 3) 36
  • 4) 44
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Ans.   C
Explanation :

4) \begin{aligned} \text{If }2x = \sqrt[3]{32}, \text{ then x is equal to} \end{aligned}

  • 1) \begin{aligned} \frac{5}{2} \end{aligned}
  • 2) \begin{aligned} \frac{2}{5} \end{aligned}
  • 3) \begin{aligned} \frac{3}{5} \end{aligned}
  • 4) \begin{aligned} \frac{5}{3} \end{aligned}
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Ans.   D
Explanation :
\begin{aligned} = (32)^{\frac{1}{3}}\= (2^5)^{\frac{1}{3}}\= 2^{\frac{5}{3}}\=> x= \frac{5}{3} \end{aligned}

5) \begin{aligned} \frac{1}{1+a^{(n-m)}} + \frac{1}{1+a^{(m-n)}} = ? \end{aligned}

  • 1) 1
  • 2) 2
  • 3) 3
  • 4) 4
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Ans.   A
Explanation :
\begin{aligned} = \frac{1}{\left( 1 + \frac{a^n}{a^m} \right)} + \frac{1}{\left( 1 + \frac{a^m}{a^n} \right)} \ = \frac{a^m}{(a^m+a^n)} + \frac{a^n}{(a^m+a^n)} \ = \frac{(a^m+a^n)}{(a^m+a^n)} = 1 \end{aligned}

6) \begin{aligned} \text{If } 3^{x-y} = 27 \text{ and } 3^{x+y} = 243, \ \text{ then find the value of x } \end{aligned}

  • 1) 1
  • 2) 2
  • 3) 3
  • 4) 4
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Ans.   D
Explanation :
\begin{aligned}3^{x-y} = 27 = 3^3 <=> x-y = 3 \text{... (i)}\\ 3^{x+y} = 243 = 3^5 <=> x+y = 5 \text{... (ii)} \ \text{ adding (i) and (ii)} => 2x = 8 \=> x = 4 \end{aligned}

7) \begin{aligned} x = 3 + 2\sqrt{2}, \text{ then the value of }\ (\sqrt{x} - \frac{1}{\sqrt{x}}) \end{aligned}

  • 1) 1
  • 2) 2
  • 3) 3
  • 4) 4
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Ans.   B
Explanation :
Clue: \begin{aligned} (\sqrt{x} - \frac{1}{\sqrt{x}})^2 = x + \frac{1}{x} - 2 \end{aligned} Now put the value of x to calculate the answer :)

8) \begin{aligned} \text{if }6^m = 46656, \\\text{ What is the value of }6^{m-2} \end{aligned}

  • 1) 7776
  • 2) 7782
  • 3) 1296
  • 4) 1290
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Ans.   C
Explanation :
\begin{aligned} 6^{m-2}\\ = \dfrac{6^m}{6^2}\\ = \dfrac{46656}{6^2}\\ = \dfrac{46656}{36} = 1296 \end{aligned}

9) Value of \begin{aligned} (256)^{\frac{5}{4}} \end{aligned}

  • 1) 1012
  • 2) 1024
  • 3) 1048
  • 4) 525
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Ans.   B
Explanation :
\begin{aligned} = (256)^{\frac{5}{4}} = (4^4)^{\frac{5}{4}} = 4^5 = 1024 \end{aligned}

10) \begin{aligned} \sqrt{8}^\frac{1}{3} \end{aligned}

  • 1) 2
  • 2) 4
  • 3) \begin{aligned} \sqrt{2} \end{aligned}
  • 4) 8
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Ans.   C
Explanation :
\begin{aligned} = ((8)^\frac{1}{2})^\frac{1}{3} = 8^{(\frac{1}{2} \times \frac{1}{3})} \end{aligned} \begin{aligned} = (8)^{\frac{1}{6}} \end{aligned} \begin{aligned} = (2)^{3(\frac{1}{6})} \end{aligned} \begin{aligned} = (2)^{\frac{1}{2}} \end{aligned}
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