2021 · Q58
If p = sin²θ + cos⁴θ, for 0 ≤ θ ≤ (π/2), then consider the following statements:
- 1p can be less than (3/4);
- 2p can be more than 1.
Which of the above statements is/are correct?
Elementary MathematicsTrigonometry
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Put u = sin²θ ∈ [0, 1]: p = u + (1−u)² = u² − u + 1. This quadratic has minimum 3/4 at u = 1/2 and value 1 at both endpoints, so 3/4 ≤ p ≤ 1. Thus p can be neither less than 3/4 nor more than 1: neither statement is correct. Hence option D.