The number of integral triplets $(a, b, c)$ such that $a + b cos 2 x + c {sin}^{2} x = 0$ for all $x$ is

0

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1

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3

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Infinitely many

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Solution

The correct option is D Infinitely many
We have,
$a + b cos 2 x + c {sin}^{2} x = 0$ for all $x$
$\Rightarrow a + b (1 - 2 {sin}^{2} x) + c {sin}^{2} x = 0$
$\Rightarrow a + b + (c - 2 b) {sin}^{2} x = 0$ for all $x$
$\Rightarrow a + b = 0$ and $c - 2 b = 0$
$\Rightarrow a = - b$ and $c = 2 b$
Thus, the triplets are $(- b, b, 2 b)$ , where $b ϵ R$ .
Hence, there are infinitely many triplets.

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