MCQ
$\int_0^{\pi /2} {} \log \sin x\,dx = $
- ✓$ - \left( {\frac{\pi }{2}} \right)\log 2$
- B$\pi \log \frac{1}{2}$
- C$ - \pi \log \frac{1}{2}$
- D$\frac{\pi }{2}\log 2$
==> $2I = \int_0^{\pi /2} {\log \sin x\cos x\,dx} = \int_0^{\pi /2} {\log \sin 2x\,dx} - \int_0^{\pi /2} {\,\,\log 2dx} $
$ = \frac{1}{2}\int_0^\pi {\log \sin tdt - \frac{\pi }{2}\log 2} $, (Putting $2x = t$)
$ = \frac{1}{2}.2\int_0^{\pi /2} {\log \sin t\,dt - \frac{\pi }{2}\log 2} $
$ \Rightarrow 2I = I - \frac{\pi }{2}\log 2 $
$\Rightarrow I = \frac{{ - \pi }}{2}\log 2$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$(A)$ determinant of $\left( M ^2+ MN ^2\right)$ is $0$
$(B)$ there is a $3 \times 3$ non-zero matrix $U$ such that $\left( M ^2+ MN ^2\right) U$ is the zero matrix
$(C)$ determinant of $\left( M ^2+ MN ^2\right) \geq 1$
$(D)$ for a $3 \times 3$ matrix $U$, if $\left( M ^2+ MN ^2\right) U$ equals the zero matrix then $U$ is the zero matrix