JEE Main 2021 · 17 March · morning · Single correct · +4 / −1
Which of the following statements is correct for the function g(α) for α ∈ R such that
g(α ) = ∫_(π/6)^(π/3) (sin ^α x)/(cos ^α x + sin ^α x)dx
- (a) g(α ) is a strictly increasing function
- (b) g(α ) is an even function
- (c) g(α ) has an inflection point at α = -1/2
- (d) g(α ) is a strictly decreasing function
Answer and solution
Answer (b) g(α ) is an even function
Solution
Maths sir · 4 steps
Use the King property on [π/6, π/3] and watch α disappear completely.
- Apply King property. Use ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx with a + b = π/2. Replacing x by π/2 - x swaps sin x and cos x, so g(α) = ∫_(π/6)^(π/3) (cos^α x)/(cos^α x + sin^α x) dx.
- Add both forms. Adding the original and the new form, the numerators add up to the denominator, so the integrand becomes 1. On this interval sin x and cos x are both positive, so this works for every real α.
2g(α) = ∫_(π/6)^(π/3) 1 dx = π/3 - π/6 = π/6
- Read the function. So g(α) = π/12 for every α. A constant function is neither strictly increasing nor strictly decreasing. Its concavity never changes, so it has no inflection point.
- So. Since g(-α) = π/12 = g(α), the function g is even, which is option (b).
The idea to keep. When the limits add to π/2 and the integrand has the form (sinⁿ x)/(sinⁿ x + cosⁿ x), the King property makes the answer half the interval length, whatever n is.
Answer from the NTA final answer key. Solution by PrepWiser.