QJEEAdv20Maa626: JEE Advanced 2020 Mathematics - Class 12 JEEAdv20Maa626 Solution: Which of the following inequalities is/are TRUE?

Question

Verified Solution • 2025–26 Session moderate

Which of the following inequalities is/are TRUE?

Accepted Answer

\because

\cos x = 1 - {{{x^2}} \over {2!}} + {{{x^4}} \over {4!}} - {{{x^6}} \over {6!}} + ...

and

\sin x = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^7}} \over {7!}} + ....

\therefore

\int_0^1 {x\cos xdx} \ge \int_0^1 {\left( {x - {{{x^3}} \over {2!}}} \right)} \,dx

= \left[ {{{{x^2}} \over 2} - {{{x^4}} \over 8}} \right]_0^1 = {1 \over 2} - {1 \over 8} = {3 \over 8}

\Rightarrow \int_0^1 {x\cos xdx\, \ge \,{3 \over 8}}

and,

\int_0^1 {x\sin dx \ge \int_0^1 {\left( {{x^2} - {{{x^4}} \over 6}} \right)} \,dx}

\left[ {{{{x^3}} \over 3} - {{{x^5}} \over {30}}} \right]_0^1 = {1 \over 3} - {1 \over {30}} = {9 \over {30}} = {3 \over {10}}

\Rightarrow \int_0^1 {x\sin xdx\, \ge \,{3 \over {10}}}

and,

\int_0^1 {{x^2}\cos xdx\, \ge \,\int_0^1 {\left( {{x^3} - {{{x^5}} \over 2}} \right)\,dx} }

= \left[ {{{{x^4}} \over 4} - {{{x^6}} \over {12}}} \right]_0^1 = {1 \over 4} - {1 \over {12}} = {2 \over {12}} = {1 \over 6}

\therefore

\int_0^1 {{x^2}\cos xdx\, \ge \,{1 \over 6}}

and,

\int_0^1 {{x^2}\sin xdx\, \ge \int_0^1 {\left( {{x^3} - {{{x^5}} \over 6}} \right)\,dx} }

= \left[ {{{{x^4}} \over 4} - {{{x^6}} \over {36}}} \right]_0^1 = {1 \over 4} - {1 \over {36}} = {8 \over {36}} = {2 \over 9}

\therefore

\int_0^1 {{x^2}\sin xdx\, \ge \,{2 \over 9}}

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