Let α_θ and β_θ be the distinct roots of 2 x²+(cos θ) x-1=0, θ ∈(0,2 π). If m and M are the minimum and the maximum values of α_θ⁴+β_θ⁴, then 16(M+m) equals :

Solution

Maths sir · 4 steps

Write α⁴+β⁴ through sum and product, and the whole thing depends only on cos² θ.

1. Sum and product. From 2x² + (cos θ)x - 1 = 0: α+β = -(cos θ)/2 and αβ = -1/2. The discriminant cos² θ + 8 is always positive, so the roots are always distinct.
2. Sum of squares. α²+β² = (α+β)² - 2αβ = (cos² θ)/4 + 1.

   α⁴+β⁴ = (α²+β²)² - 2(αβ)² = (1+(cos² θ)/4)² - 1/2

3. Find the extremes. This increases with cos² θ, which takes every value in [0, 1] for θ ∈ (0, 2π). At cos² θ = 0 (θ = π/2): m = 1 - 1/2 = 1/2. At cos² θ = 1 (θ = π): M = 25/16 - 1/2 = 17/16.
4. So. M + m = 17/16 + 8/16 = 25/16, so 16(M+m) = 25, option (c).

The idea to keep. For any symmetric expression in the roots, rebuild it from α+β and αβ first. Then it becomes a function of one parameter, and you only have to find that parameter's range.

Answer from the NTA final answer key. Solution by PrepWiser.