The equation $3^{sin 2 x + 2 {cos}^{2} x} + 3^{1 - sin 2 x + 2 {sin}^{2} x} = 28$ is satisfied for the values of $x$ given by

\frac{3 π}{8}

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\frac{3 π}{2}

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\frac{3 π}{4}

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\frac{π}{8}

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Solution

The correct option is B $\frac{3 π}{8}$

$3^{s i n 2 x + 2 c o s^{2} x} + 3^{1 - s i n 2 x + 2 (1 - c o s^{2} x)} = 28 3^{s i n 2 x + 2 c o s^{2} x} + (\frac{3^{3}}{3^{s i n 2 x + 2 c o s^{2} x}}) = 28 l e t 3^{s i n 2 x + 2 c o s^{2} x} = t ∴ t + (\frac{27}{t}) = 28 t^{2} + 27 = 28 t t^{2} - 28 t + 27 = 0 t^{2} - 27 t - t + 27 = 0 t (t - 27) - 1 (t - 27) = 0 (t - 27) (t - 1) = 0 ∴ t = 27 o r t = 1 i f t = 27, t h e n s i n 2 x + 2 c o s^{2} x = 3 w h i c h i s n o t p o s s i b l e i f t = 1, t h e n s i n 2 x + 2 c o s^{2} x = 1 a t x = (\frac{3 π}{8})$

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