JEE Advanced 2020 · Paper 1 · Mathematics Q12

MathematicsMultiple correct+4 / -2Question 48 of 54

Which of the following inequalities is/are TRUE?

\displaystyle\int_0^1 x\cos x\,dx \ge \frac38

\displaystyle\int_0^1 x\sin x\,dx \ge \frac3{10}

\displaystyle\int_0^1 x^2\cos x\,dx \ge \frac12

\displaystyle\int_0^1 x^2\sin x\,dx \ge \frac29

Answer and worked solution

Official answer: A, B, D

Integrate by parts, using $\sin1\approx0.8415$ and $\cos1\approx0.5403$ .

(A) $\int_0^1x\cos x\,dx=\sin1+\cos1-1\approx0.382\ge0.375$ . True.
(B) $\int_0^1x\sin x\,dx=\sin1-\cos1\approx0.301\ge0.300$ . True.
(C) $\int_0^1x^2\cos x\,dx=2\cos1-\sin1\approx0.239<0.5$ . False.
(D) $\int_0^1x^2\sin x\,dx=\cos1+2\sin1-2\approx0.2232\ge\frac29\approx0.2222$ . True.

(A), (B), (D)

Topics:Properties of definite integrals Integration by parts

JEE Advanced 2020 · Paper 1, Mathematics question 12