QJEEAdv21Ma156e: JEE Advanced 2021 Mathematics - Class 12 JEEAdv21Ma156e Solution: Let , and be functions such that and , for all . Define , i =…

Question

Verified Solution • 2025–26 Session moderate

Let

{g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2

, and

f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R

be functions such that

{g_1}(x) = 1,{g_2}(x) = |4x - \pi |

and

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

, for all

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

. Define

{S_i} = \int\limits_{{\pi \over 8}}^{{{3\pi } \over 8}} {f(x).{g_i}(x)dx}

, i = 1, 2

The value of

{{48{S_2}} \over {{\pi ^2}}}

is ___________.

Accepted Answer

{S_2} = \int\limits_{\pi /8}^{3\pi /8} {{{\sin }^2}x\left| {4x - \pi } \right|dx}

.... (i)

{S_2} = \int_{\pi /8}^{3\pi /8} {{{\sin }^2}\left( {{{3\pi } \over 8} + {\pi \over 8} - x} \right)\left| {4\left( {{{3\pi } \over 8} + {\pi \over 8} - x} \right) - \pi } \right|dx}

[\int_a^b {f(x)dx = \int_a^b {f(a + b - x)dx]} }

\Rightarrow {S_2} = \int_{\pi /8}^{3\pi /8} {{{\cos }^2}x\left| {\pi - 4x} \right|dx}

.... (ii)

Adding Eqs. (i) and (ii), we get

2{S_2} = \int_{\pi /8}^{3\pi /8} {\left| {4x - \pi } \right|dx}

From figure,

{A_1} = {1 \over 2} \times {\pi \over 8} \times {\pi \over 2} = {{{\pi ^2}} \over {32}} = {A_2}

\therefore

2{S_2} = 2{A_1} = {{{\pi ^2}} \over {16}} \Rightarrow {S_2} = {{{\pi ^2}} \over {32}}

Hence,

{{48{S_2}} \over {{\pi ^2}}} = {{48} \over {32}} = {3 \over 2} = 1.5

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