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.15 which of the following is the only CORRECT combination?

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

Answer 1

The correct option is B $\left(II\right),\left(Q\right),\left(T\right)$

$f\left(x\right)=0\Rightarrow \sin \left(\pi \cos x\right)=0$
$\pi \cos x=n_1\pi ,n_1\in I$
$\cos x=-1,\cos x=0,\cos x=1$
$x=\frac{n_2\pi }{2},n_2\in I$
$\Rightarrow X=\{\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi ,.....\}$
$f^{\prime }\left(x\right)=0\Rightarrow -\cos \left(\pi \cos x\right)\pi \sin x=0$
$\sin x=0or\cos \left(\pi \cos x\right)=0$
$x=n_3\pi$ or $\pi \cos x=(2n_4+1)\frac{\pi }{2}$
$\cos x=\frac{-1}{2},\frac{1}{2}$
$x=n_5\pi \pm \frac{\pi }{3}$
$Y=\{\frac{\pi }{3},\frac{2\pi }{3},\frac{3\pi }{3},......\}$
Hence, set $Y$ elements are in arithmetic progression.