MCQ
$\sin \left(\tan ^{-1} x\right),|x|<1=$ __________ .
- ✓$\frac{x}{\sqrt{1+x^2}}$
- B$\frac{1}{\sqrt{1-x^2}}$
- C$\frac{1}{\sqrt{1+x^2}}$
- D$\frac{x}{\sqrt{1-x^2}}$
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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 $2.$
$4\alpha=3\beta$
$3\alpha=4\beta$
$\alpha-\beta=\frac{7\pi}{12}$
$\text{none of these}$