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 ?

Rearranging: x² − x cos θ + p = 0. For real solutions, discriminant ≥ 0: cos²θ − 4p ≥ 0, i.e., p ≤ cos²θ/4. For this to hold for EVERY θ in [0, π/4], we need p ≤ min(cos²θ/4). On [0, π/4], cos θ is minimum at θ = π/4 where cos θ = 1/√2, so cos²θ = 1/2. Hence min(cos²θ/4) = 1/8. So p ≤ 1/8.