JEE Advanced 2021 · Paper 2, Mathematics question 3

(A) Let $h(x)=f(x)-3\cos3x$. Then
$$\int_0^{\pi/3}h=0-[\sin3x]_0^{\pi/3}=0.$$
A continuous function with zero integral over an interval vanishes somewhere inside it. True.

(B) Let $h(x)=f(x)-3\sin3x+\frac6\pi$. Then
$$\int_0^{\pi/3}h=0-[-\cos3x]_0^{\pi/3}+\frac6\pi\cdot\frac\pi3=0-2+2=0,$$
so again $h$ vanishes somewhere in $(0,\frac\pi3)$. True.

(C) Near 0, $\int_0^xf\approx xf(0)=x$, so the numerator is about $x^2$ while $1-e^{x^2}\approx-x^2$. The limit is $-1$. True.

(D) Here $\sin x\int_0^xf\approx x^2$, so the limit is $+1$, not $-1$. False.

(A), (B), (C)