JEE Advanced 2021 · Paper 2 · Mathematics Q12

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Question Stem for Question Nos. 11 and 12

Let $g_i: \left[\frac\pi8, \frac{3\pi}8\right] \to \mathbb R$ , $i = 1, 2$ , and $f: \left[\frac\pi8, \frac{3\pi}8\right] \to \mathbb R$ be functions such that
$g_1(x) = 1,\quad g_2(x) = |4x - \pi| \quad\text{and}\quad f(x) = \sin^2 x,\quad\text{for all } x \in \left[\frac\pi8, \frac{3\pi}8\right]$
Define
$S_i = \int_{\pi/8}^{3\pi/8} f(x)\cdot g_i(x)\,dx,\qquad i = 1, 2$

The value of $\dfrac{48S_2}{\pi^2}$ is ____.

Your answer

Answer and worked solution

Official answer: 1.49

Symmetry. Substitute $x\to\frac\pi2-x$ . This maps the interval to itself, leaves $|4x-\pi|$ unchanged, and turns $\sin^2x$ into $\cos^2x$ . Adding the two forms of $S_2$ :
$2S_2=\int_{\pi/8}^{3\pi/8}|4x-\pi|\,dx=2\int_{\pi/4}^{3\pi/8}(4x-\pi)\,dx=2\left[2x^2-\pi x\right]_{\pi/4}^{3\pi/8}=\frac{\pi^2}{16}.$

So $S_2=\dfrac{\pi^2}{32}$ and $\dfrac{48S_2}{\pi^2}=1.5$ .

Topics:Properties of definite integrals Integration by parts

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JEE Advanced 2021 · Paper 2, Mathematics question 12