If the equation $a_{1}$ + $a_{2} c o s 2 x$ + $a_{3} s i n^{2} x$ = 1 is satisfied by every real values of x, then the number of possible values of the triplet $(a_{1}, a_{2}, a_{3})$

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infinite

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Solution

The correct option is D
infinite

$a_{1} + a_{2} (2 c o s^{2} x - 1) + a_{3} (1 - c o s^{2} x)$ = 1

Or $(2 a_{2} - a_{3})$ $c o s^{2}$ x + $(a_{1} - a_{2} + a_{3} - 1)$ = 0

This can hold for all x if $2 a_{2} - a_{3}$ = 0 and $a_{1} - a_{2} + a_{3}$ - 1 = 0

As there are three variable and two equations, the number of solution is infinite.

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