JEE Main 2026 · 6 Apr · Shift 1 · Mathematics Q20

← JEE Main 2026 · 6 Apr · Shift 1

MathematicsSingle correct+4 / -1Question 20 of 75

Let $e$ be the base of natural logarithm and let $f:\{1,2,3,4\}\to\{1,e,e^2,e^3\}$ and $g:\{1,e,e^2,e^3\}\to\left\{1,\dfrac12,\dfrac13,\dfrac14\right\}$ be two bijective functions such that $f$ is strictly decreasing and $g$ is strictly increasing. If $\phi(x)=\left[f^{-1}\left\{g^{-1}\left(\dfrac12\right)\right\}\right]^x$ , then the area of the region $R=\{(x,y):x^2\le y\le\phi(x),\ 0\le x\le1\}$ is:

\dfrac{3-\log_e(2)}{3\log_e(2)}

\dfrac1{3\log_e(2)}

3+\log_e(2)

\dfrac{3+\log_e(2)}{2+\log_e(3)}

Answer and worked solution

Official answer: A

The functions.

f decreasing: $f(2)=e^2$ .
g increasing: $g(e^2)=\tfrac12$ .

So $g^{-1}\left(\tfrac12\right)=e^2$ , $f^{-1}(e^2)=2$ , and $\phi(x)=2^x$ .

Area.
$\int_0^1(2^x-x^2)\,dx=\frac1{\ln2}-\frac13=\frac{3-\ln2}{3\ln2}.$

(A)

Topics:Area between curves Composition and inverse functions

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JEE Main 2026 · 6 Apr · Shift 1, Mathematics question 20