The number of values of the triplet $(a, b, c)$ for which $a cos 2 x + b {sin}^{2} x + c = 0$ is satisfied by all real $x$ , is

0

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2

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3

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infinte

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Solution

The correct option is B infinte
Given, $a cos 2 x + b {sin}^{2} x + c = 0$

$\Rightarrow a - 2 a {sin}^{2} x + b {sin}^{2} x + c = 0$

$\Rightarrow {sin}^{2} x = \frac{a + c}{2 a - b}$
As $0 \leq {sin}^{2} x \leq 1 \Rightarrow 0 \leq \frac{a + c}{2 a - b} \leq 1$
Then this hold for infinite values of $(a, b, c)$

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