Computer Sciences > Gate 2024 Set-1 > Log
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
log (𝑝2 + 𝑞2) = log 𝑝 + log 𝑞 + 2 log 3 ,
then, the value of (𝑝4+𝑞4)/(𝑝2𝑞2) is
Correct : a
Simplifing the given equation using logarithm properties:
log(p² + q²) = log p + log q + 2 log 3
log(p² + q²) = log p + log q + log 3²
log(p² + q²) = log(pq) + log 9
log(p² + q²) = log(9pq)
Since log(p² + q²) = log(9pq), we have:
p² + q² = 9pq
We need to find (p⁴ + q⁴)/(p²q²)
Let's use the identity: p⁴ + q⁴ = (p² + q²)² - 2p²q²
(p² + q²)²:
From Step 2: p² + q² = 9pq
(p² + q²)² = (9pq)²
(p² + q²)² = 81p²q²
Calculating p⁴ + q⁴:
p⁴ + q⁴ = (p² + q²)² - 2p²q²
p⁴ + q⁴ = 81p²q² - 2p²q²
p⁴ + q⁴ = 79p²q²
The final value:
(p⁴ + q⁴)/(p²q²) = 79p²q² / p²q²
(p⁴ + q⁴)/(p²q²) = 79
Answer: A) 79
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