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If for all values of a one root of the equation $x^{2} - 3 a x + f (a) = 0$ is double of the other, $f (x)$ is equal to

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$2 x$

$x^{2}$

$2 x^{2}$

$2 \sqrt{x}$

answer is C.

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Detailed Solution

Let $α$ be one root of $x^{2} - 3 a x + f (a) = 0$
$\Rightarrow α + 2 α = 3 a \Rightarrow 3 α = 3 a$
and $α = a ..... (i)$
$\Rightarrow f (a) = 2 α^{2} = 2 a^{2}$ [using Eq.(i)]
$\Rightarrow f (x) = 2 x^{2}$

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