IIT-JEE 2007 · Paper 1 · Mathematics Q53

MathematicsSingle correct+3 / -1Question 53 of 66

The number of solutions of the pair of equations
$2\sin^2\theta-\cos2\theta=0,\qquad2\cos^2\theta-3\sin\theta=0$
in the interval $[0,2\pi]$ is

zero

one

two

four

Answer and worked solution

Official answer: C

First equation. $2\sin^2\theta-(1-2\sin^2\theta)=0\implies\sin^2\theta=\frac14$ .

Second equation. $2(1-\sin^2\theta)-3\sin\theta=0\implies2\sin^2\theta+3\sin\theta-2=0\implies\sin\theta=\frac12$ (the root $-2$ is impossible).

Common solutions. $\sin\theta=\frac12$ in $[0,2\pi]$ : $\theta=\frac\pi6,\frac{5\pi}6$ , two solutions.

(C)

Topics:Trigonometric equations

IIT-JEE 2007 · Paper 1, Mathematics question 53