Question

Let \( \alpha \) and \( \beta \) respectively be the maximum and the minimum values of the function \( f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right) \). Then \( \alpha + 2\beta \) is equal to :

• A. 4
• B. 3
• C. 5
• D. 6

Hint:

The expression \( a(\sin^4 x + \cos^4 x) + b(\sin^6 x + \cos^6 x) \) is a very common JEE pattern. Always simplify it down to a function of \( \sin^2(2x) \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Simplify the trigonometric expression using allied angle rules to find a manageable function for which max/min values can be determined.
Step 2: Key Formula or Approach:
1. \( \sin^4 \theta + \cos^4 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \).
2. \( \sin^6 \theta + \cos^6 \theta = 1 - 3\sin^2 \theta \cos^2 \theta \).
Step 3: Detailed Explanation:
Simplifying the function components: \[ f(\theta) = 4(\cos^4 \theta + \sin^4 \theta) - 2(\cos^6 \theta + \sin^6 \theta) \] Substitute the identities: \[ f(\theta) = 4(1 - 2\sin^2 \theta \cos^2 \theta) - 2(1 - 3\sin^2 \theta \cos^2 \theta) \] \[ f(\theta) = 4 - 8\sin^2 \theta \cos^2 \theta - 2 + 6\sin^2 \theta \cos^2 \theta \] \[ f(\theta) = 2 - 2\sin^2 \theta \cos^2 \theta = 2 - \frac{1}{2}\sin^2(2\theta) \] Max value \( \alpha \) (when \( \sin^2 2\theta = 0 \)): \( \alpha = 2 \). Min value \( \beta \) (when \( \sin^2 2\theta = 1 \)): \( \beta = 2 - 1/2 = 3/2 \). \[ \alpha + 2\beta = 2 + 2(3/2) = 5 \] Step 4: Final Answer:
(3) 5.