MCQ
The domain of function $\cos ^{-1}(2 x-3)$ is :
- A$[-1,1]$
- B$(1,2)$
- C(-1,1)
- ✓$[1,2]$
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and consider the statements
$I\,:$ $I_1 < I_2$
$II\,:$ $I_2 < I_3$
$III\,:$ $I_1 = I_3$
Which of the following is $(are)$ true?
The value of objective function is maximum under linear constraints
$S_n(x)=\sum_{k=1}^n \cot ^{-1}\left(\frac{1+k(k+1) x^2}{x}\right)$
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 $x >0$
$\frac{\pi}{6}$
$\frac{\pi}{4}$
$\frac{\pi}{3}$
$\frac{\pi}{2}$