JEE Advanced 2026 · Paper 2 · Mathematics Q14

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MathematicsNumerical+4Question 14 of 54

For a real number $\alpha$ , let $[\alpha]$ denote the greatest integer less than or equal to $\alpha$ . For a finite set $S$ , let $|S|$ denote the number of elements in the set $S$ .

Consider the functions $f:(-3,3)\to(-\infty,\infty)$ and $g:(-3,3)\to(-\infty,\infty)$ defined by
$f(x)=\left[x^3\right]\log_e\left(1+\sin^2\left(\pi(x-[x])\right)\right)$
and
$g(x)=x^3\sin^2\left(\pi\log_e(1+x-[x])\right).$
Let
$A=\{x\in(-3,3):f\text{ is discontinuous at }x\}$
and
$B=\{x\in(-3,3):g\text{ is discontinuous at }x\}.$
Then the value of $|A|+2|B|-|A\cap B|$ is ________.

Your answer

Answer and worked solution

Official answer: 56

$f$ . $\sin^2(\pi(x-[x]))=\sin^2\pi x$ , so $h(x)=\ln(1+\sin^2\pi x)$ is continuous and vanishes exactly at integers. $f=[x^3]\,h$ can only jump where $[x^3]$ jumps, i.e. where $x^3=k$ is an integer: $k=-26,\dots,26$ , which is 53 points in $(-3,3)$ . At the five integer points $x=0,\pm1,\pm2$ we have $h=0$ , so $f\to0=f(x)$ from both sides and $f$ is continuous there. At the other 48 points $h\ne0$ and $f$ jumps. $|A|=48$ .

$g$ . $\ln(1+\{x\})$ is continuous except at integers $n$ , where it jumps from $\ln2$ to $0$ . Since $\sin^2(\pi\ln2)\ne0$ , $g$ jumps at $n$ unless $n^3=0$ . So $B=\{-2,-1,1,2\}$ , $|B|=4$ .

Overlap. $A$ contains no integers, so $A\cap B=\varnothing$ .
$|A|+2|B|-|A\cap B|=48+8-0=56.$

Topics:Continuity Graphs of functions

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JEE Advanced 2026 · Paper 2, Mathematics question 14