Question

The Number of solutions of the equation ${\sin }^{3}x\cos x+{\sin }^{2}x{\cos }^{2}x+\sin x{\cos }^{3}x=1$ in the interval $[0,2\pi ]$ is

• A. 0
• B. 2
• C. 3
• D. infinite

Answer 1

Answer: (A)

${\sin }^{3}x\cos x+{\sin }^{2}x{\cos }^{2}x+\sin x{\cos }^{3}x=1$
$\Rightarrow \sin x\cos x\left({\sin }^{2}x+\sin x\cos x+{\cos }^{2}x\right)=1$
$\Rightarrow \frac{\sin 2x}{2}\left(1+\frac{\sin 2x}{2}\right)=1$
$\Rightarrow \sin 2x(2+\sin 2x)=4$
$\Rightarrow {\sin }^{2}2x+2\sin 2x-4=0$
$\Rightarrow \sin 2x=\frac{-2\pm \sqrt{4+16}}{2}$
$\Rightarrow \sin 2x=-1\pm \sqrt{5}$
This is not possible. (since $-1\le \sin 2x\le 1$ )
Hence the given equation has no solution.