Question

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

• A. 7
• B. 6
• C. 5
• D. 4

Answer 1

Answer: (C)

The number of solutions in the interval $[0, \pi]$ of the equation
$\sin ^{3} x \cos 3 x+\sin 3 x \cos ^{3} x=0$ is
equal toThe given equation can be written as
$\left(\frac{3 \sin x-\sin 3 x}{4}\right) \cos 3 x+\sin 3 x\left(\frac{3 \cos x+\cos 3 x}{4}\right)$
$\sin x \cos 3 x+\cos x \sin 3 x=0$
So, $\sin (x+3 x)=0 \Rightarrow \sin 4 x=0$
$\Rightarrow x=\frac{n \pi}{4}, n \in Z . $
$\Rightarrow x=0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \pi$
$\Rightarrow 5$solutions