Question

The sum of roots of the equation $s i n^{4}x-c o s^{2}xsinx+2s i n^{2}x+sinx=0$ for $0\leq x\leq 3\pi $ is equal to

• A. $\frac{3 \pi }{4}$
• B. $2\pi $
• C. $4\pi $
• D. $6\pi $

Answer 1

Answer: (D)

$\sin x\left(\sin ^{3} x-\cos ^{2} x+2 \sin x+1\right)=0$
$\sin x\left(\sin ^{3} x+2 \sin x+1-\cos ^{2} x\right)=0$
$\sin x\left(\sin ^{3} x+\sin ^{2} x+2 \sin x\right)=0$
$\sin ^{2} x\left(\sin ^{2} x+\sin x+2\right)=0$
$\Rightarrow \sin x=0$ as $\sin ^{2} x+\sin x+2 \neq 0$
$\Rightarrow x=n \pi, n \in I$
$x \in[0,3 \pi], x=0, \pi, 2 \pi, 3 \pi$
$\Rightarrow $ sum of roots $=6 \pi$