JEE Main 2022 · 24 Jun · Shift 2 · Mathematics Q13

MathematicsSingle correct+4 / -1Question 13 of 90

The number of solutions of the equation $\cos\left(x+\dfrac\pi3\right)\cos\left(\dfrac\pi3-x\right)=\dfrac14\cos^22x$ , $x\in[-3\pi,3\pi]$ is:

Answer and worked solution

Official answer: D

Product formula. $\cos(A+B)\cos(A-B)=\cos^2A-\sin^2B$ with $A=\frac\pi3$ , $B=x$ : the left side is $\frac14-\sin^2x=\frac{\cos2x}2-\frac14$ .

Equation. $\frac{\cos2x}2-\frac14=\frac14\cos^22x$ gives $(\cos2x-1)^2=0$ , so $\cos2x=1$ , $x=k\pi$ .

Count in $[-3\pi,3\pi]$ : $k=-3,\dots,3$ , i.e. 7 solutions.

(D)

JEE Main 2022 · 24 Jun · Shift 2, Mathematics question 13