Q71 / 155 · WBJEE 2020 · Mathematics
$\cos (2 x+7)=a(2-\sin x)$ can have a real solution for
- Aall real values of a
- Correct answerB$a \in[2,6]$
- C$\mathrm{a} \in[-\infty, 2] \backslash\{0\}$
- D$\mathrm{a} \in(0, \infty)$
Solution
Step 1. find the solution of given equation:
\(\begin{aligned}
&\cos 2 x+a \sin x=2 a-7 \\
&\Rightarrow 1-2 \sin ^2 x+a \sin x=2 a-7 ;\left[\because \cos 2 x=1-2 \sin ^2 x\right] \\
&\Rightarrow 2 \sin ^2 x-a \sin x+2 a-8=0
\end{aligned}\)
Step 2. Find the roots of above equation:
\(\begin{aligned}
&\Rightarrow \sin x=\frac{a \pm \sqrt{a^2-16(a-4)}}{4} ;\left[\because \boldsymbol{x}=\frac{-\mathbf{b} \pm \sqrt{\mathbf{b}^2-4 \mathbf{a c}}}{2 \mathbf{a}}\right] \\
&=\frac{a \pm(a-8)}{4} \\
&\Rightarrow \sin x=\frac{a-4}{2} \text { or } \sin x=2(\text { not possible }) \\
&\Rightarrow \sin x=\frac{a-4}{2} \\
&\because-1 \leq \sin x \leq 1 \\
&\therefore-1 \leq \frac{a-4}{2} \leq 1 \\
&\Rightarrow 2 \leq a \leq 6 \\
&\Longrightarrow a \in[2,6]
\end{aligned}\)
Hence, the correct answer is option: 'B'.