First slide

Trigonometric equations

Question

The set of values of ‘a’ for which the equation $\cos 2 x + a \sin x = 2 a - 7$ possesses a solution is $[c, d]$ , then $c + d$ =

Moderate

A

2
B

4
C

6
D

8

Solution

$W e h a v e \cos 2 x + a \sin x = 2 a - 7$
$\Rightarrow 1 - 2 \sin^{2} x + a \sin x = 2 a - 7$
$\Rightarrow 2 \sin^{2} x - a \sin x + 2 a - 8 = 0$
$\Rightarrow \sin x = \frac{a \pm \sqrt{a^{2} - 8 (2 a - 8)}}{4} = \frac{a \pm \sqrt{{(a - 8)}^{2}}}{4}$
$\Rightarrow \sin x = \frac{a \pm (a - 8)}{4} = \frac{2 a - 8}{4} o r 2$
Now $- 1 \leq \sin x \leq 1 \Rightarrow - 1 \leq \frac{2 a - 8}{4} \leq 1$
$\Rightarrow - 4 \leq 2 a - 8 \leq 4 \Rightarrow 4 \leq 2 a \leq 12 \Rightarrow 2 \leq a \leq 6$
$∴ c = 2$ , $d = 6$ $\Rightarrow c + d = 8$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App