Let g_ i :[(π)/(8), (3 π)/(8)] arrow ℝ, i=1,2 , and f:[(π)/(8), (3 π)/(8)] arrow ℝ be functions such that g_ 1 (x)=1, g_ 2 (x)=|4 x-π| and f(x)= ^ 2 x , for all x ∈[(π)/(8), (3 π)/(8)] Define S_ i =∫_ (π)/(8) ^ (3

Correct answer: 0 S _ 2 = ∫ _ (π)/(8) ^ (3π)/(8) ^ 2 x|4x - π|dx (i) S _ 2 = ∫ _ (π)/(8) ^ (3π)/(8) ^ 2 ((3π)/(8) + (π)/(8) - x)|4((3π)/(8) + (π)/(8) - x) - π|dx S _ 2 = ∫ _ (π)/(8) ^ (3π)/(8) ^ 2 x|π - 4x|dx (ii) From (i) and (ii), we get 2 S _ 2 = ∫ _ (π)/(8) ^ (3π)/(8) |4x - π|dx 2 S _ 2 = π ^ 2 16 S _ 2 = π ^ 2 32 ⇒ 48 S _ 2 π ^ 2 = (48)/(32) ⇒ 48 S _ 2 π ^ 2 = (3)/(2) = 1 . 5

JEE Advanced 2021Paper 2

Exam: JEE Advanced
Year: 2021
Paper: Paper 2
Subject: Mathematics
Chapter: Definite Integration

JEE Advanced 2021 · Paper 2 · Mathematics · Definite Integration

Let g_ i :[(π)/(8), (3 π)/(8)] arrow ℝ, i=1,2 , and f:[(π)/(8), (3...

Let

g_{i}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, i=1,2

, and

f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}

be functions such that

g_{1}(x)=1, g_{2}(x)=|4 x-\pi| and

f(x)=\sin ^{2} x

, for all

x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]

Define

S_{i}=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, \quad i=1,2 The value of

\dfrac{48{S}_{2}}{{\pi}^{2}}

is ____.

Let g_ i :[(π)/(8), (3 π)/(8)] arrow ℝ, i=1,2 , and f:[(π)/(8), (3 π)/(8)] arrow ℝ be functions such that g_ 1 (x)=1,

Correct answer

Step-by-step solution

${S}_{2} = {∫}_{\dfrac{\pi}{8}}^{\dfrac{3\pi}{8}}\sin^{2}x\left|4x - \pi\right|dx \ldots \ldots \ldots \left(i\right)$

${S}_{2} = {∫}_{\dfrac{\pi}{8}}^{\dfrac{3\pi}{8}}\sin^{2}\left(\dfrac{3\pi}{8} + \dfrac{\pi}{8} - x\right)\left|4\left(\dfrac{3\pi}{8} + \dfrac{\pi}{8} - x\right) - \pi\right|dx$

${S}_{2} = {∫}_{\dfrac{\pi}{8}}^{\dfrac{3\pi}{8}}\cos^{2}x\left|\pi - 4x\right|dx \ldots \ldots \ldots \left(ii\right)$

From (i) and (ii), we get

$2{S}_{2} = {∫}_{\dfrac{\pi}{8}}^{\dfrac{3\pi}{8}}\left|4x - \pi\right|dx$

Let g_ i :[(π)/(8), (3 π)/(8)] arrow ℝ, i=1,2 , and f:[(π)/(8), (3 π)/(8)] arrow ℝ be functions such that g_ 1 (x)=1,

$2{S}_{2} = \dfrac{{\pi}^{2}}{16}$

${S}_{2} = \dfrac{{\pi}^{2}}{32}$

$⇒ \dfrac{48{S}_{2}}{{\pi}^{2}} = \dfrac{48}{32}$

$⇒ \dfrac{48{S}_{2}}{{\pi}^{2}} = \dfrac{3}{2} = 1 . 5$

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