Q63 / 160 · AP EAMCET 2024 (20 May Shift 2) · Mathematics

The general solution of the equation $\sin ^2 \theta+3 \cos ^2 \theta=5 \sin \theta$ is

  • A
    $n \pi \pm \frac{\pi}{3}, n \in Z$
  • Correct answer
    B
    $n \pi+(-1)^{\mathrm{n}} \frac{\pi}{6}, n \in Z$
  • C
    $n \pi \pm \frac{\pi}{6}, n \in Z$
  • D
    $n \pi+(-1)^n \frac{\pi}{3}, n \in Z$

Solution

$\sin ^2 \theta+3\left(1-\sin ^2 \theta\right)=5 \sin \theta$ $\begin{aligned} & \Rightarrow 2 \sin ^2 \theta+5 \sin \theta-3=0 \\ & \Rightarrow(2 \sin \theta-1)(\sin \theta+3)=0 \\ & \sin \theta=\frac{1}{2}=\sin \frac{\pi}{6} \\ & \theta=n \pi+(-1)^n \frac{\pi}{6}, n \in \mathbf{Z}\end{aligned}$

Hence, the correct answer is option: 'B'.

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