GATE CS 2024 Set 1 Q.5: Quantitative Aptitude

For positive non-zero real variables $p$ and $q$ , if

\log(p^2 + q^2) = \log p + \log q + 2\log 3,

then, the value of $\frac{p^4 + q^4}{p^2 q^2}$ is

Official answer and solution

Solution

Combine the logarithmic terms to express $(p^2 + q^2)/(pq)$ , then square both sides.

Using logarithm properties $\log a + \log b = \log(ab)$ and $2\log 3 = \log(3^2) = \log 9$ , the given equation becomes:

\log(p^2 + q^2) = \log(9pq) \implies p^2 + q^2 = 9pq

Dividing both sides by $pq$ gives:

\frac{p^2 + q^2}{pq} = 9

Squaring both sides:

\left(\frac{p^2 + q^2}{pq}\right)^2 = \frac{p^4 + 2p^2q^2 + q^4}{p^2q^2} = 81 \implies \frac{p^4 + q^4}{p^2q^2} + 2 = 81 \implies \frac{p^4 + q^4}{p^2q^2} = 79

Final answer

Option (A)

Common trap

Forgetting to subtract 2 when squaring $(p^2 + q^2)/(pq) = 9$ , or adding 2 to get 83.

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