JEE Advanced 2019 · Paper 2 · Mathematics Q16

← JEE Advanced 2019 · Paper 2

MathematicsList match+3 / -1Question 52 of 54

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
$\supseteq\left\{\frac\pi6, \frac{3\pi}4\right\}$

(III), (R), (U)

(IV), (P), (R), (S)

(III), (P), (Q), (U)

(IV), (Q), (T)

Answer and worked solution

Official answer: B

Z. $\cos(2\pi\sin x)=0$ means $\sin x=\pm\frac14,\pm\frac34$ . That is not an arithmetic progression, and Z contains neither $\frac\pi6$ nor $\frac{3\pi}4$ .

W.
$g'(x)=-2\pi\cos x\sin(2\pi\sin x)=0\iff\cos x=0\ \text{ or }\ \sin x\in\{0,\pm\tfrac12,\pm1\}.$
So W contains $\frac\pi2,\frac{3\pi}2,4\pi,7\pi$ (P) and $\frac\pi6,\frac{7\pi}6,\frac{13\pi}6$ (S). Its gaps alternate between $\frac\pi3$ and $\frac\pi6$ , so it is not an arithmetic progression (R).

(B)

Topics:Trigonometric equations Arithmetic progression

← Previous question Next question →

JEE Advanced 2019 · Paper 2, Mathematics question 16