JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means of two distinct positive numbers. Then, $G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2$ is equal to:

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Explanation

Solution

Maana do numbers $a$ aur $b$ hain.

$A_1 + A_2 = a + b$ .
$G_1, G_2, G_3$ are GMs $\implies G_1 = a(b/a)^{1/4}, G_2 = \sqrt{ab}, G_3 = b(a/b)^{1/4}$ .
Property: $G_1 G_3 = G_2^2 = ab$ .
Expression: $G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2 = G_1^4 + (ab)^2 + G_3^4 + (ab)^2 = G_1^4 + G_3^4 + 2(ab)^2$ .
$G_1^4 + G_3^4 = a^3 b + b^3 a = ab(a^2 + b^2)$ .
Total $= ab(a^2 + b^2 + 2ab) = ab(a+b)^2$ .
LHS $= (G_1 G_3)(A_1 + A_2)^2$ .

Correct Option: (2)

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