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GATE CS 2011 · General Aptitude

Question 57

MCQPrinted as GA Q.571 markWrong answer: −1/3Easy (estimate)

Question

If log⁡(P)=12log⁡(Q)=13log⁡(R)\log(P) = \frac{1}{2}\log(Q) = \frac{1}{3}\log(R), then which of the following options is TRUE?

Choose one option
Official answer and solution
Official answer
B
From
Official GATE 2011 answer key

Solution

Express the logarithms in terms of a common parameter k to determine the algebraic relation between P, Q, and R.

  1. Let log⁡P=12log⁡Q=13log⁡R=k\log P = \frac{1}{2} \log Q = \frac{1}{3} \log R = k.
  1. Convert each logarithmic expression to exponential form (assuming base 10):
  • log⁡P=k  ⟹  P=10k\log P = k \implies P = 10^k
  • log⁡Q=2k  ⟹  Q=102k\log Q = 2k \implies Q = 10^{2k}
  • log⁡R=3k  ⟹  R=103k\log R = 3k \implies R = 10^{3k}
  1. Test the relation in Option B:
Q2=(102k)2=104kQ^2 = (10^{2k})^2 = 10^{4k}
P⋅R=10k⋅103k=10k+3k=104kP \cdot R = 10^k \cdot 10^{3k} = 10^{k + 3k} = 10^{4k}

Since Q2=PRQ^2 = P R, Option B is strictly TRUE.

Final answer

B

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper
GATE 2011 Computer Science and Information Technology (CS)
Question
57 of 65, printed as GA Q.57
Section
General Aptitude

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