Question

The set of all values of $a$ for which the equation $cos2x+asinx=2a-7$ has a solution is

• A. $\left(- \infty , 2\right)$
• B. $\left[2 , 6\right]$
• C. $\left(6 , \infty \right)$
• D. $\left(- \infty , \infty \right)$

Answer 1

Answer: (B)

$cos2x+asinx=2a-7$
i.e. $2sin^{2}x-asinx+2a-8=0$
$sinx=\frac{a \pm \sqrt{a^{2} - 8 \left(2 a - 8\right)}}{4}=\frac{a \pm \left(a - 8\right)}{4}$
$sinx=\frac{a - 4}{2}$ or $2$
Hence, $-1\leq \frac{\left(a - 4\right)}{2}\leq 1$
The range of $a$ is $\left[2 , 6\right]$ .