2024 · Q11
If the equation x cos θ = x² + p has a real solution for every θ where 0 ≤ θ ≤ (π)/(4), then which one of the following is correct?
Elementary MathematicsAlgebra
- Ap = 1/8
- Bp ≤ 1/8
- Cp ≥ 1/8
- Dp ≤ 1/4
Explanation
The quadratic x² − xcosθ + p = 0 needs discriminant cos²θ − 4p ≥ 0 for every θ in [0, π/4]. Since cos²θ is smallest (1/2) at θ = π/4, we need 4p ≤ 1/2. Hence p ≤ 1/8, option (b).