Test Academy | Which of the following is NOT true for all possible non-zero choices of integers m, n; m ≠ n, or all possible non-zero choices of real numbers p, q; p ≠ q, as applicable? 1 π ∫ 0 π ⁡ sin ⁡ m θ sin ⁡ n θ d θ = 0 , 1 2 π ∫ − π / 2 π / 2 ⁡ sin ⁡ p θ sin ⁡ q θ d θ = 0 , 1 2 π ∫ − π π ⁡ sin ⁡ p θ cos ⁡ q θ d θ = 0 , lim α → ∞ ⁡ 1 2 α ∫ − α α ⁡ sin ⁡ p θ sin ⁡ q θ d θ = 0

Question

Which of the following is NOT true for all possible non-zero choices of integers m, n; m ≠ n, or all possible non-zero choices of real numbers p, q; p ≠ q, as applicable?

Options :

$\frac{1}{π} \int_{0}^{π} \sin m θ \sin n θ d θ = 0$
$\frac{1}{2 π} \int_{- π / 2}^{π / 2} \sin p θ \sin q θ d θ = 0$
$\frac{1}{2 π} \int_{- π}^{π} \sin p θ \cos q θ d θ = 0$
$lim_{α \to \infty} \frac{1}{2 α} \int_{- α}^{α} \sin p θ \sin q θ d θ = 0$

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