If the range of f(θ)=(sin ⁴ θ+3 cos ² θ)/(sin ⁴ θ+cos ² θ), θ ∈ ℝ is [α, β], then the sum of the infinite G.P., whose first term is 64 and the common ratio is α/β, is equal to __________.

Solution

Maths sir · 4 steps

Write everything in cos² θ and split off the 1; the leftover fraction is easy to bound.

1. Split off 1. The numerator is the denominator plus 2 cos² θ, so f(θ) = 1 + (2 cos² θ)/(sin⁴ θ + cos² θ).
2. One variable. Put t = cos² θ, so t ∈ [0, 1] and sin⁴ θ = (1-t)². The denominator becomes (1-t)² + t = t² - t + 1.

   f = 1 + 2t/(t² - t + 1) = 1 + 2/(t + 1/t - 1) (t ≠ 0)

3. Bound the fraction. At t = 0 the fraction is 0, so f = 1. For 0 < t ≤ 1 we have t + 1/t ≥ 2, with equality at t = 1. So the fraction is at most 2/(2-1) = 2, giving f = 3. Since f is continuous in t, it takes every value from 1 to 3.
4. So. The range is [1, 3], so α = 1, β = 3 and the common ratio is 1/3. The sum is 64/(1 - 1/3) = (64 × 3)/2 = 96.

The idea to keep. When numerator and denominator share most of their terms, split off the common part first. Then bound what is left using t + 1/t ≥ 2 for t > 0.

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