Question

The number of solution of pair of equations $2si{n}^2\theta -cos2\theta =0$ and $2co{s}^2\theta -3sin\theta =0$ in the interval $[\theta ,2\pi ]$

• A. 1
• B. 2
• C. 4

Answer 1

The correct option is B 2

$2si{n}^2\theta -cos2\theta =0\Rightarrow 4si{n}^2\theta -1=0\Rightarrow (2sin\theta +1)(2sin\theta -1)=0....\left(1\right)$
and $2co{s}^2\theta -3sin\theta =0\Rightarrow 2-2si{n}^2\theta -3sin\theta =0\Rightarrow 2si{n}^2\theta +3sin\theta -2=0\Rightarrow (sin\theta +2)(2sin\theta -1)=0....\left(2\right)$
From (1) and (2)
$2sin\theta -1=0\Rightarrow sin\theta =\frac{1}{2}\theta \in$ I and II quadrants
$\Rightarrow$ number of solutions = 2