MCQ
${d \over {dx}}\log \tan \left( {{\pi \over 4} + {x \over 2}} \right) = $
- A$\cos {\rm{ec}}\,x$
- B$ - \cos {\rm{ec}}\,x$
- ✓$\sec x$
- D$ - \sec x$
$ = \frac{1}{2}.\frac{1}{{\sin \left( {\frac{\pi }{4} + \frac{x}{2}} \right)\cos \left( {\frac{\pi }{4} + \frac{x}{2}} \right)}} = \frac{1}{{\sin \left( {\frac{\pi }{2} + x} \right)}} = \frac{1}{{\cos x}} = \sec x$.
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$\lim _{n \rightarrow 0^{+}} \int_n^{1-n} t^{-3}(1-t)^{a-1} d t$
exists. Let this limit be $g(a)$. In addition, it is given that the function $g(a)$ is differentiable on $(0,1)$.
$1.$ The value of $g\left(\frac{1}{2}\right)$ is
$(A)$ $\pi$ $(B)$ $2 \pi$ $(C)$ $\frac{\pi}{2}$ $(D)$ $\frac{\pi}{4}$
$2.$ The value of $g ^{\prime}\left(\frac{1}{2}\right)$ is
$(A)$ $\frac{\pi}{2}$ $(B)$ $\pi$ $(C)$ $-\frac{\pi}{2}$ $(D)$ $0$
Give the answer question $1$ and $2.$