Question

Let $S$ be the set of all $\alpha \in\, R$ such that the equation, cos $2x + \alpha$sinx = 2$\alpha - 7$ has a solution. Then $S$ is equal to :

• A. $[2,6]$
• B. $[3,7]$
• C. $R$
• D. $[1,4]$

Answer 1

Answer: (A)

$cos2x + \alpha sinx = 2 \alpha - 7$
$\Rightarrow 2 sin^2 x - \alpha sin x + 2 \alpha - 8 = 0$
$sin^2 x - \frac{\alpha}{2} \, sinx + \alpha - 4 = 0$
$\Rightarrow \, \, sin x = 2 (rejected) \, or \, sin x = \frac{\alpha - 4}{2}$
$\Rightarrow \, \, \, \, \, |\frac{\alpha - 4}{2}| \le 1$
$\Rightarrow \, \, \, \, \, \, \alpha \, \in [2,6]$