JEE Advanced 2019 · Paper 2 · Mathematics Q15

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Passage

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

Let $f(x) = \sin(\pi\cos x)$ and $g(x) = \cos(2\pi\sin x)$ be two functions defined for $x > 0$ . Define the following sets whose elements are written in the increasing order:

$X = \{x : f(x) = 0\},\qquad Y = \{x : f'(x) = 0\},$

$Z = \{x : g(x) = 0\},\qquad W = \{x : g'(x) = 0\}.$

List I contains the sets $X$ , $Y$ , $Z$ and $W$ . List II contains some information regarding these sets.

Which of the following is the only CORRECT combination?

List-I

I
$X$
II
$Y$
III
$Z$
IV
$W$

List-II

P
$\supseteq\left\{\frac\pi2, \frac{3\pi}2, 4\pi, 7\pi\right\}$
Q
an arithmetic progression
R
NOT an arithmetic progression
S
$\supseteq\left\{\frac\pi6, \frac{7\pi}6, \frac{13\pi}6\right\}$
T
$\supseteq\left\{\frac\pi3, \frac{2\pi}3, \pi\right\}$
U
$\supseteq\left\{\frac\pi6, \frac{3\pi}4\right\}$

(I), (P), (R)

(II), (Q), (T)

(I), (Q), (U)

(II), (R), (S)

Answer and worked solution

Official answer: B

X. $\sin(\pi\cos x)=0$ needs $\cos x\in\{-1,0,1\}$ , so $X=\{\frac{k\pi}2\}$ . That is an arithmetic progression and contains $\frac\pi2,\frac{3\pi}2,4\pi,7\pi$ .

Y.
$f'(x)=-\pi\sin x\cos(\pi\cos x)=0\iff\sin x=0\ \text{ or }\ \cos x=\pm\tfrac12.$
So $Y=\{\frac{k\pi}3\}$ . That is an arithmetic progression and contains $\frac\pi3,\frac{2\pi}3,\pi$ .

The options.

(A) fails because X is an AP, so not (R).
(B) (II), (Q), (T) is correct.
(C) fails: X does not contain $\frac\pi6$ .
(D) fails: Y is an AP.

(B)

Topics:Trigonometric equations Arithmetic progression

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JEE Advanced 2019 · Paper 2, Mathematics question 15