GATE AG 2024 • Q5
MCQ • 1 Mark
Section 8: General Aptitude
Quantitative Aptitude
Subtopic: Data Interpretation
Question Statement:
For positive non-zero real variables p and q, if log(p² + q²) = log p + log q + 2 log 3, then the value of (p⁴ + q⁴)/(p²q²) is
Options:
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(A) 79
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(B) 81
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(C) 9
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(D) 83
Step-by-Step Solution & Explanation
Step 1 — remove the logarithms.
log(p² + q²) = log p + log q + 2 log 3 = log(p · q · 3²) = log(9pq)
⇒ p² + q² = 9pq
Step 2 — build the required expression. Square both sides:
(p² + q²)² = (9pq)²
p⁴ + 2p²q² + q⁴ = 81p²q²
p⁴ + q⁴ = 79p²q²
Step 3 — divide:
(p⁴ + q⁴)/(p²q²) = 79
Trap: forgetting to subtract the cross term 2p²q² gives 81 — option (B).
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