If a real variable x satisfies 3x2 = 27 9x, then the value of 2x2(2x)2 is:

GATE 2025 PI – Question 9

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

If a real variable $x$ satisfies $3^{x^2} = 27 \times 9^x$, then the value of $\frac{2^{x^2}}{(2^x)^2}$ is:

$2^{-1}$
$2^0$
$2^3$
$2^{15}$

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Show answer and explanation

Correct answer: (C) $2^3$

Explanation

Writing everything as a power of 3, $3^{x^2} = 3^3 \times 3^{2x} = 3^{3 + 2x}$, so $x^2 = 2x + 3$, that is $x^2 - 2x - 3 = 0$ and $x = 3$ or $x = -1$. The expression is $\frac{2^{x^2}}{2^{2x}} = 2^{x^2 - 2x} = 2^3$, since $x^2 - 2x = 3$ for both roots.

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