Question

The total number of solutions of $ln\left|\sin x\right|=-x^2+2x$ in $[-\frac{\pi }{2},\frac{\pi }{2}]$ is equal to

• A. $1$
• B. $3$
• C. $4$
• D. $6$

Answer 1

Answer: (A)

$1$

$ln\left|\sin x\right|=-x(x-2)$
To find the number of solutions, let us consider the graphs of the L.H.S and R.H.S
At $x=-\frac{\pi }{2}$ , $L.H.S=0$
|$\sin x$| $\le 1$. Hence, $L.H.S\le 0$
The L.H.S is discontinuous at $x=0$.
For $x>0$, the $L.H.S$ is negative. At $x=\frac{\pi }{2}$, $L.H.S=0$
The $R.H.S$ is a parabolic expression and $\frac{\pi }{2}<2$. Hence, the graph is as shown below.

$ln\left|\sin x\right|$ and $y=-x(x-2)$ meet exactly once in $[-\frac{\pi }{2},\frac{\pi }{2}]$ as shown in the graph.