MCQ
${d \over {dx}}\left[ {{{\sin }^2}{{\cot }^{ - 1}}\left\{ {\sqrt {{{1 - x} \over {1 + x}}} } \right\}} \right]$ equals
- A$ - 1$
- ✓${1 \over 2}$
- C$ - {1 \over 2}$
- D$1$
Put $x = \cos \theta \Rightarrow \theta = {\cos ^{ - 1}}x$
==> $y = {\sin ^2}{\cot ^{ - 1}}\left\{ {\sqrt {\frac{{1 - \cos \theta }}{{1 + \cos \theta }}} } \right\} = {\sin ^2}{\cot ^{ - 1}}\left( {\tan \frac{\theta }{2}} \right)$
==> $y = {\sin ^2}\left( {\frac{\pi }{2} - \frac{\theta }{2}} \right)$
==> $\frac{{dy}}{{d\theta }} = 2\sin \left( {\frac{\pi }{2} - \frac{\theta }{2}} \right)\,.\,\cos \left( {\frac{\pi }{2} - \frac{\theta }{2}} \right)\,\left( { - \frac{1}{2}} \right)$
==> $\frac{{dy}}{{d\theta }} = - \frac{{\sin (\pi - \theta )}}{2} = - \frac{{\sin \theta }}{2} = \frac{{ - 1}}{2}\,\sqrt {1 - {x^2}} $
==> $\frac{{dy}}{{dx}} = \frac{{dy}}{{d\theta }}\,.\,\frac{{d\theta }}{{dx}} = \frac{{ - 1}}{2}\sqrt {1 - {x^2}} \,.\,\frac{d}{{dx}}({\cos ^{ - 1}}x) = \frac{1}{2}$.
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$12x + by + cz = 0$ ; $ax + 24y + cz = 0$ ; $ax + by + 36z = 0$ .
(where $a$ , $b$ , $c$ are real numbers, $a \ne 12$ , $b \ne 24$ , $c \ne 36$ ).
If system of equation has solution and $z \ne 0$, then value of $\frac{1}{{a - 12}} + \frac{2}{{b - 24}} + \frac{3}{{c - 36}}$ is
$f(x)=\left\{\begin{array}{cc}e^{\min \left[x^2, x-[x]\right\}}, & x \in[0,1) \\e^{\left[x-\log _e x\right]}, & x \in[1,2]\end{array}\right.$
where [t] denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int \limits_0^2 x f(x) d x$ is