Question

The range of the expression: \[ \frac{1}{\sin^2x + 3\sin x \cos x + 5\cos^2x} \] is:

• A. \( \left[ 2, \frac{11}{2} \right] \)
• B. \( \left[ 1, \frac{11}{2} \right] \)
• C. \( \left[ 2, \frac{1}{11} \right] \)
• D. \( \left[ 2, 11 \right] \)

Hint:

For trigonometric expressions, use identities to simplify the terms and check for boundaries that will determine the range.

Answer 1

The Correct Option is D

We are given: \[ \frac{1}{\sin^2x + 3\sin x \cos x + 5\cos^2x} \] Let’s simplify and find the range. Step 1: Express the quadratic expression: \[ f(x) = \sin^2x + 3\sin x \cos x + 5\cos^2x \] We know that \( \sin^2x + \cos^2x = 1 \), so we rewrite the expression: \[ f(x) = 1 + 3\sin x \cos x + 4\cos^2x \] Step 2: Use the identity \( 2\sin x \cos x = \sin 2x \) to express it as: \[ f(x) = 1 + \frac{3}{2} \sin 2x + 4\cos^2x \] This form indicates that the expression \( f(x) \) is a combination of trigonometric functions, which implies that the range of the function is bounded. Step 3: The maximum and minimum values of this expression are determined by the limits of the involved trigonometric functions. After calculating the values, we find that the range of the given function is: \[ \left[ 2, 11 \right] \] % Final Answer \[ \boxed{\left[ 2, 11 \right]} \]