Question

The number of solutions of the equation ${\int }_{-2}^x\left|\cos x\right|dx=0$, $0<x<\frac{\pi }{2}$, is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

Answer: (A)

$0$

${\int }_{-2}^x\left|\cos x\right|dx=00<x<\frac{\pi }{2}$
If $0<x<\frac{\pi }{2}$ then $\cos x$ is positive
$-2<x<0,\cos x$ is positive $[\because \cos (-x)=\cos x;\cos (-2)=\cos 2]$
$\therefore {\int }_{-2}^x\cos xdx=0$
${\left|\sin x\right|}_{-2}^x=0$
$\sin x-\sin (-2)=0$
$\sin x+\sin 2=0$
$\sin x=-\sin 2$
$\sin x=\sin (-2)$
$\therefore x=-2$ But $0<x<\frac{\pi }{2}$
$\therefore$ No solution
Number of solutions=0