MCQ
Differential coefficient of ${\sin ^{ - 1}}x$ $w.r.t$ ${\cos ^{ - 1}}\sqrt {1 - {x^2}} $ is
- ✓$1$
- B${1 \over {1 + {x^2}}}$
- C$2$
- DNone of these
Differentiating w.r.t. $x$ of ${y_1}$ and ${y_2}$, we get
$\frac{{d{y_1}}}{{dx}} = \frac{1}{{\sqrt {1 - {x^2}} }}$
$\frac{{d{y_2}}}{{dx}} = - \frac{1}{{\sqrt {1 - (1 - {x^2})} }}\frac{{1( - 2x)}}{{2\sqrt {1 - x} }} $
$= \frac{1}{{\sqrt {1 - {x^2}} }} \Rightarrow \frac{{d{y_2}}}{{d{y_1}}} = 1.$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$(A)$ $y-x+3=0$ $(B)$ $y+3 x-33=0$ $(C)$ $y+x-15=0$ $(D)$ $y-2 x+12=0$
$S_n(x)=\sum_{k=1}^n \cot ^{-1}\left(\frac{1+k(k+1) x^2}{x}\right)$
where for any $x \in R , \cot ^{-1} x \in(0, \pi)$ and $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then which of the following
statements is (are) $TRUE$?
$(A)$ $S _{10}( x )=\frac{\pi}{2}-\tan ^{-1}\left(\frac{1+11 x ^2}{10 x }\right)$, for all $x >0$
$(B)$ $\lim _{n \rightarrow \infty} \cot \left(S_n(x)\right)=x$, for all $x>0$
$(C)$ The equation $S_3(x)=\frac{\pi}{4}$ has a root in $(0, \infty)$
$(D)$ $\tan \left( S _{ n }( x )\right) \leq \frac{1}{2}$, for all $n \geq 1$ and $x >0$