If p=\sin^{2}\theta+\cos^{4}\theta for 0 \leq \theta \leq \frac{\pi}{2}, consider the following statements:<br>1. p can be less than \frac{3}{4}.<br>2. p can be more than 1.<br>Which of the above statements is/are correct?
- A. 1 <strong>ONLY</strong>
- B. 2 <strong>ONLY</strong>
- C. <strong>BOTH</strong> 1 and 2
- D. <strong>NEITHER</strong> 1 <strong>NOR</strong> 2 ✓
Correct Answer: D. <strong>NEITHER</strong> 1 <strong>NOR</strong> 2
Explanation
Rewrite p = 1 - \cos^2 \theta + \cos^4 \theta = (\cos^2 \theta - \frac{1}{2})^2 + \frac{3}{4}. The minimum value is \frac{3}{4} and the maximum is 1. So p cannot be less than \frac{3}{4} nor more than 1. Neither statement is correct.
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