If $0 \leq x \leq 2 π$ , then the number of solutions of $3 (sin x + cos x) - 2 ({sin}^{3} x + {cos}^{3} x) = 8$ is

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Solution

The correct option is A 0
$3 (c o s x + s i n x) - 2 (c o s^{3} x + s i n^{3} x) = 8$
$\Rightarrow 3 c o s x + 3 s i n x - 2 c o s x (1 - s i n^{2} x) - 2 s i n x (1 - c o s^{2} x) = 8$
$\Rightarrow c o s x + s i n x + 2 c o s x . s i n x (s i n x + c o s x) = 8$
$\Rightarrow (c o s x + s i n x) (1 + 2 c o s x . s i n x) = 8$
$\Rightarrow (c o s x + s i n x) (s i n^{2} x + c o s^{2} x + 2 c o s x . s i n x) = 8$
$\Rightarrow (c o s x + s i n x)^{3} = 8 \Rightarrow c o s x + s i n x = 2$
but we know, $s i n x + c o s x \leq \sqrt{2}$
Therefore, no solution exits.
Hence, option 'A' is correct.

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