Question

Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph.

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\{x:f(x)=0\},Y=\{x:f^{\prime }(x)=0\},$
$Z=\{x:g(x)=0\},W=\{x:g^{\prime }(x)=0\},$.
$List-I$ contains the sets $X,Y,Z$ and $W$. $List-II$ contains some information regarding these sets.

$\begin{array}{cccc}& ListI& & ListII\\ \left(i\right)& X& \left(P\right)& \supseteq \{\frac{\pi }{2},\frac{3\pi }{2},4\pi ,7\pi \}\\ \left(ii\right)& Y& \left(Q\right)& \text{an arithmetic progression}\\ \left(iii\right)& Z& \left(R\right)& \text{NOT an arithmetic progression}\\ \left(iv\right)& Z& \left(S\right)& \supseteq \{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\}\\ & & \left(T\right)& \supseteq \{\frac{\pi }{3},\frac{2\pi }{3},\pi \}\\ & & \left(U\right)& \supseteq \{\frac{\pi }{6},\frac{3\pi }{4}\}\end{array}$
Q.16 which of the following is the only CORRECT combination?

• A. $\left(III\right),\left(R\right),\left(U\right)$
• B. $\left(IV\right),\left(P\right),\left(R\right),\left(S\right)$
• C. $\left(III\right),\left(P\right),\left(Q\right),\left(U\right)$
• D. $\left(IV\right),\left(Q\right),\left(T\right)$

Answer 1

The correct option is B $\left(IV\right),\left(P\right),\left(R\right),\left(S\right)$

$g\left(x\right)=0\Rightarrow \cos \left(2\pi \sin x\right)=0$
$2\pi \sin x=((2n_6+1)\frac{\pi }{2})$
$\sin x=\frac{(2n_6+1)}{4}=\{\frac{-3}{4},\frac{-1}{4},\frac{1}{4},\frac{3}{4}\}$
$Z=\{x:\sin x=\pm \frac{1}{4},\pm \frac{3}{4};x>0\}$
$g^{\prime }\left(x\right)=0\Rightarrow -2\pi \cos x\sin \left(2\pi \sin x\right)=0$
$(\cos x=0\Rightarrow x=(2n+1)\frac{\pi }{2}$ or $\sin \left(2\pi \sin x\right)=0\Rightarrow 2\pi \sin x=n\pi$
$\sin x=\frac{n}{2}=-1,-\frac{1}{2},0,\frac{1}{2},1$
$W=\{\frac{n\pi }{2},n\pi \pm \frac{\pi }{6},n\in I\}$
$W=\{\frac{\pi }{6},\frac{\pi }{2},\frac{5\pi }{6},\pi ,\frac{7\pi }{6},\frac{3\pi }{2},.....\}$
Set $W$ elements are NOT an arithmetic progression series.