The number of solutions of the equation $16 (s i n^{5} x + c o s^{5} x) = 11 (s i n x + c o s x)$ in the interval $[0, 2 π]$ is

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Solution

The correct option is A 6
$16 (s i n^{5} x + c o s^{5} x) = 11 (s i n x + c o s x) = 0 \Rightarrow (s i n x + c o s x) {16 (s i n^{4} x - s i n^{3} x c o s x + s i n^{2} x c o s^{2} x - s i n x c o s^{3} x + c o s^{4} x) - 11} = 0 \Rightarrow (s i n x + c o s x) {16 (1 - s i n^{2} x c o s^{2} x - s i n x c o s x)} - 11 = 0 \Rightarrow (s i n x + c o s x) (4 s i n x c o s x - 1) (4 s i n x c o s x + 5) = 0 A s 4 s i n x c o s x + 5 \neq 0, W e h a v e s i n x + c o s x = 0, 4 s i n x c o s x - 1 = 0 t h e r e q u i r e d v a l u e s a r e \frac{π}{12}, \frac{5 π}{12}, \frac{9 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}, \frac{21 π}{12}, - T H E Y A R E 6 S O L U T I O N S O N [0, 2 π]$

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