Question

Let f(x) = sin($\pi$ cos x) and g(x) = cos(2 $\pi$ sin a) 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(x) = 0},
Z = [x : g(x) = 0}, W = [x : g'(x) = 0}.
List -1 contains the sets X, Y, Z and W. List - II contains some information regarding these sets.

List-I	List-II
(I) X	(p) ⊇ $\left\{\frac{\pi }{2},\frac{3\pi }{2},4\pi ,7\pi \right\}$
(II) Y	(Q) an arithmetic progression
(III) Z	(R) NOT an arithmetic progression
(IV) W	(S) ⊇ $\left\{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\right\}$
	(T) ⊇ $\left\{\frac{\pi }{2},\frac{2\pi }{2},\pi \right\}$
	(U) ⊇ $\left\{\frac{\pi }{6},\frac{3\pi }{4}\right\}$

Question : Wliich of the following is the only CORRECT combination?

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

Answer 1

Answer: (B)

$f\left(x\right)=sin\left(\pi cosx\right)$
$X:\left\{x:f\left(x\right)=0\right\}$
$f\left(x\right)=0\Rightarrow sin\left(\pi cosx\right)=0\Rightarrow cosx=n\Rightarrow cosx=1,-1,0\Rightarrow x=\frac{n\pi }{2}$
$X=\left\{\frac{n\pi }{2}:n\in N\right\}=\left\{\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi ....\right\}$
$g\left(x\right)=cos\left(2\pi sinx\right)$
$Z=\left\{x:g\left(x\right)=0\right\}$
$cos\left(2\pi sinx\right)=0\Rightarrow 2\pi sinx=\left(2n+1\right)\frac{\pi }{2}\Rightarrow sinx=\frac{\left(2n+1\right)}{4}$
$sinx=-\frac{1}{4},\frac{1}{4},\frac{-3}{4},\frac{3}{4}$
$Z=\left\{n\pi \pm si{n}^{-1}\left(\frac{1}{4}\right),n\pi \pm si{n}^{-1}\left(\frac{3}{4}\right),n\in I\right\}$
$Y=\left\{x:f\left(x\right)=0\right\}$
$f\left(x\right)=sin\left(\pi cosx\right)\Rightarrow f^{\prime }\left(x\right)=cos\left(\pi cosx\right).\left(-\pi sinx\right)=0$
$sinx=0\Rightarrow x=n\pi .$
$cos\left(\pi cosx\right)=0\pi cosx=\left(2n+1\right)\frac{\pi }{2}\Rightarrow cos=\frac{\left(2n+1\right)}{2}\Rightarrow cosx=-\frac{1}{2},\frac{1}{2}$
$Y=\left\{n\pi ,n\pi \pm \frac{\pi }{3}\right\}=\left\{\frac{\pi }{3},\frac{2\pi }{3},\pi ,\frac{4\pi }{3},\frac{5\pi }{3},2\pi ,......\right\}$
$W=\left\{x:g^{\prime }\left(x\right)=0\right\}$
$g\left(x\right)=cos\left(2\pi sinx\right)\Rightarrow g^{\prime }\left(x\right)=-sin\left(2\pi sinx\right).\left(2\pi cosx\right)=0$
$cosx=0\Rightarrow x=\left(2n+1\right)\frac{\pi }{2}$
$sin\left(2\pi sinx\right)=0\Rightarrow 2\pi sinx=n\pi \Rightarrow sinx=\frac{n}{2}=-1,-\frac{1}{2},0,\frac{1}{2},1$
$W=\left\{\frac{n\pi }{2},n\pi \pm \frac{\pi }{6},n\in 1\right\}=\left\{\frac{\pi }{6},\frac{\pi }{2},\frac{5\pi }{6},\pi ,\frac{7\pi }{6},\frac{3\pi }{2},.....\right\}$
Now check the options