Sum of the roots of $3^{s i n 2 x + 2 c o s^{2} x} + 3^{1 - s i n 2 x + 2 s i n^{2} x} = 28$ in $[- 2 π, 2 π]$ is

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π

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2 π

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\frac{3 π}{2}

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Solution

The correct option is B $π$
$3^{s i n 2 x + 2 c o s^{2} x} + 3^{1 - s i n 2 x + 2 s i n^{2} x} = 28 3^{s i n 2 x + 2 c o s^{2} x} + 3^{1 - s i n 2 x + 2 (1 - c o s^{2} x)} = 28 3^{s i n 2 x + 2 c o s^{2} x} + 3^{3} {.3}^{- s i n 2 x - 2 c o s^{2} x} = 28$
Let $t = 3^{s i n 2 x + 2 c o s^{2} x} \Rightarrow t + \frac{3^{3}}{t} = 28 \Rightarrow t^{2} - 28 t + 27 = 0 t = 1, 27 s i n 2 x + 2 c o s^{2} x = 0 o r s i n 2 x + 2 c o s^{2} x = 3 2 s i n x c o s x + 2 c o s^{2} x = 0 i . e s i n 2 x = 1, 2 c o s^{2} x = 1$ [which is not possible. ]
$2 c o s x (s i n x + c o s x) = 0 c o s x = 0, s i n x + c o s x = 0$
$c o s x = 0 \Rightarrow x = \frac{π}{2}, \frac{- π}{2}, \frac{3 π}{2}, \frac{- 3 π}{2} . s i n x + c o s x = 0 \Rightarrow t a n x = - 1 \Rightarrow x = \frac{- π}{4}, \frac{3 π}{4}, \frac{7 π}{4}, \frac{- 5 π}{4} ∴ s u m = π$

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