Crey | If cos2x – asinx = 2a – 7, then range of a is –2 ≤ a ≤ 0, 2 ≤ a ≤ 6, a ≥ 6, 6 ≤ a ≤ 8

If cos2x – asinx = 2a – 7, then range of a is

Options

–2 ≤ a ≤ 0

2 ≤ a ≤ 6

a ≥ 6

6 ≤ a ≤ 8

Correct Answer :

2 ≤ a ≤ 6

Solution :

The correct option is 2 ≤ a ≤ 6.

To find the range of the parameter $a$ that satisfies the equation shown in the image, we start with the given trigonometric equation:

$cos 2 x - a sin x = 2 a - 7$

Step 1: Express the equation in terms of $sin x$
Recall the trigonometric double-angle identity for cosine:

$cos 2 x = 1 - 2 {sin}^{2} x$

Substituting this identity into the original equation, we obtain the expression visible at the top of the image:

$1 - 2 {sin}^{2} x - a sin x = 2 a - 7$

Step 2: Rearrange into a quadratic equation form
Move all terms to one side of the equation to form a standard quadratic equation in terms of $sin x$ :

$2 {sin}^{2} x + a sin x + 2 a - 8 = 0$

Step 3: Solve for $sin x$ using the quadratic formula
Applying the quadratic formula $y = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ where $y = sin x$ , we have:

$sin x = \frac{- a \pm \sqrt{a^{2} - 4 (2) (2 a - 8)}}{4}$

Simplify the terms under the square root:

$a^{2} - 8 (2 a - 8) = a^{2} - 16 a + 64 = {(a - 8)}^{2}$

Thus, the expression simplifies as shown in the image:

$sin x = \frac{- a \pm | a - 8 |}{4}$

Step 4: Determine the valid case for $sin x$
This gives us two possible values:
1. Case 1 (taking the positive root):
$sin x = \frac{- a + (a - 8)}{4} = \frac{- 8}{4} = - 2$
Since the range of the sine function is restricted to $- 1 \leq sin x \leq 1$ , a value of $- 2$ is impossible.

2. Case 2 (taking the negative root):
$sin x = \frac{- a - (a - 8)}{4} = \frac{8 - 2 a}{4}$