Question

Number of solution(s) of equation ${\sin }^{3}x\cos x+{\sin }^{2}x{\cos }^{2}x+\sin x{\cos }^{3}x=1\forall x\in [0,2\pi ]$ is :

• A. $4$
• B. $2$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

${\sin }^{3}x\cos x+{\sin }^{2}x{\cos }^{2}x+\sin x{\cos }^{3}x=1$
$\Rightarrow \sin x\cos x[{\sin }^{2}x+\sin x\cos x+{\cos }^{2}x]=1\Rightarrow \frac{1}{2}\sin 2x[1+\frac{1}{2}\sin 2x]=1$
Let $\sin 2x=t$
$\therefore t(2+t)=4\Rightarrow t^2+2t-4=0\Rightarrow t=\frac{-2\pm 2\sqrt{5}}{2}=-1\pm \sqrt{5}\therefore \sin 2x=-1-\sqrt{5}<-1\left(\text{not possible}\right)$
or $\sin 2x=\sqrt{5}-1>1\left(\text{not possible}\right)$
Hence no solution