Question

Let the product of $ \omega_1 = (8 + i) \sin \theta + (7 + 4i) \cos \theta $ and $ \omega_2 = (1 + 8i) \sin \theta + (4 + 7i) \cos \theta $ be $ \alpha + i\beta $, where $ i = \sqrt{-1} $. Let $ p $ and $ q $ be the maximum and the minimum values of $ \alpha + \beta $ respectively.

• A. 140
• B. 130
• C. 160
• D. 150

Hint:

Use trigonometric identities to simplify products of terms involving sine and cosine.

Answer 1

The Correct Option is B

The given expressions for \( \omega_1 \) and \( \omega_2 \) are: \[ \omega_1 = (8 \sin \theta + 7 \cos \theta) + i(\sin \theta + 4 \cos \theta) \] \[ \omega_2 = (1 \sin \theta + 4 \cos \theta) + i(8 \sin \theta + 7 \cos \theta) \] Now, we calculate the product \( \omega_1 \omega_2 \): \[ \omega_1 \omega_2 = (8 \sin \theta + 7 \cos \theta)(\sin \theta + 4 \cos \theta) + i[(\sin \theta + 4 \cos \theta)(1 \sin \theta + 4 \cos \theta)] \] The product simplifies to: \[ \omega_1 \omega_2 = 65 + 60 \sin^2 \theta \]
Thus, the maximum and minimum values of \( \alpha + \beta \) are 125 and 5 respectively, and their sum is 130.
Thus, the correct answer is \( 130 \).