Question

The maximum value of 12 $\sin \, \theta - 9 \, \sin^2 \, \theta$ is

• A. 3
• B. 4
• C. 5
• D. none of these

Answer 1

Answer: (B)

$12 \sin \theta - 9 \sin^{2} \theta = - 9 \left(\sin^{2} \theta - \frac{12}{9}\sin\theta\right) $
$=- 9 \left(\sin^{2} \theta - \frac{4}{3} \sin\theta\right) $
$= - 9 \left[\sin^{2} \theta - \frac{4}{3} \sin\theta + \left(\frac{2}{3}\right)^{2} - \left(\frac{2}{3}\right)^{2}\right]$
$ = - 9 \left(\sin\theta - \frac{2}{3}\right)^{2} + 9 \times\frac{4}{9} \le 4 $