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)

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