Question

Assertion :The number of solutions of the pair of equations $2{\sin }^2\theta -\cos 2\theta =$ $2{\cos }^2\theta -3\sin \theta =0$ in the interval [0, 2$\pi$] is two. Reason: If $2{\cos }^2\theta -3\sin \theta =0$, then $\theta$ can not lie in III or IV quadrant.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given system of equations
$2{\sin }^2\theta -\cos 2\theta =0$
$2{\cos }^2\theta -3\sin \theta =0$
Now, $2{\sin }^2\theta -\cos 2\theta =0$
$\Rightarrow 4{\sin }^2\theta -1=0$
$\Rightarrow \sin \theta =\pm \frac{1}{2}$
$\Rightarrow \theta =\frac{\pi }{6}$, $\frac{5\pi }{6}$ , $\frac{7\pi }{6}$ , $\frac{11\pi }{6}$
Now, $2{\cos }^2\theta -3\sin \theta =0$
$\Rightarrow 2{\sin }^2\theta +3\sin \theta -2=0$
$\Rightarrow \sin \theta =\frac{1}{2},-2$(not possible)
So $\sin \theta =\frac{1}{2}$
$\Rightarrow \theta =\frac{\pi }{6}$ ,$\frac{5\pi }{6}$
Here, $\theta$ lies in first or second quadrant
Hence, statement 2 is correct
So, the number of common solutions is 2.
Hence, statement 1 is correct. Hence, reason is the correct explanation for assertion.