If the equation $a_{1} + a_{2} cos 2 x + a_{3} {sin}^{2} x = 0$ is satisfied by every real value of x then the number of possible values of the triplet is $(a_{1}, a_{2}, a_{3})$ is

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Solution

The correct option is C infinite
$a_{1} + a_{2} cos 2 x + a_{3} {sin}^{2} x = 0$
$a_{1} + a_{2} cos 2 x + a_{3} (\frac{1 - cos 2 x}{2}) = 0$ which is satisfied for all real values of $x$
If $a_{1} = - \frac{a_{3}}{2} = - a_{2}$
$a_{1} = - \frac{a_{3}}{2} = - a_{2} = - \frac{k}{2}$ (say)
$\Rightarrow a_{1} = - \frac{k}{2}, a_{2} = \frac{k}{2}, a_{3} = k$ for any $k \in R$
Hence, the required number of triplets is infinite.

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