The number of all possible triplets $(a_{1}, a_{2}, a_{3})$ such that $a_{1} + a_{2} cos 2 x + a_{3} {sin}^{2} x = 0$ for all $x$ is/are

zero

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one

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three

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infinite

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none of these

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Solution

The correct option is E infinite
$a_{1} + a_{2} c o s 2 x + a_{3} s i n^{2} x = 0$
$a_{1} + a_{2} (1 - 2 s i n^{2} x) + a_{3} s i n^{2} x = 0$
$a_{1} + a_{2} + (a_{3} - 2 a_{2}) s i n^{2} x = 0$
$s i n^{2} x$ can take any value between $0$ to $1$
Also, there's only one relation in the three independents $a_{1}, a_{2}, a_{3}$ .
So, we can see that here infinite triplets are possible.

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The number of all the possible triplets $(a_{1}, a_{2}, a_{3})$ such that $a_{1} + a_{2} c o s (2 x) + a_{3} s i n^{2} (x) = 0$ for all x is

a_{1}, a_{2}, a_{3}

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