Consider two sets $A=\left{x\in ℤ:\left|\left(\left|x−3\right|−3\right)\right|\leq 1\right}$ and $B=\left{x\in ℝ−\left{1,2\right}:\frac{\left(x−2\right)\left(x−4\right)}{x−1}{\text{log}}_{e}\left(\left|x−2\right|\right)=0\right}.$ Then the number of onto functions $f: A \rightarrow B$ is equal to [JEE Main 2026, 23 Jan (Shift 2)]

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YESPYQ · Accepted Answer

Correct answer: a) 62. $\begin{matrix}A:||x−3|−3|\leq 1 \ \Rightarrow −1\leq |x−3|−3\leq 1 \ \Rightarrow 2\leq |x−3|\leq 4 \ \Rightarrow 2\leq (x−3)\leq 4\text{ or }−4\leq (x−3)\leq −2\end{matrix}$ $\begin{matrix}\Rightarrow 5\leq x\leq 7\text{ or}−1\leq x\leq 1 \ A={−1,0,1,5,6,7}\end{matrix}$ For B: $\frac{(x-2)(x-4)}{x-1} \ln |x-2|=0$. Domain $x \neq 1,2$ Roots: $x-4=0 \Rightarrow x=4 $ $ \ln |x-2|=0 \Rightarrow |x-2|=1 \Rightarrow x=3 \text{ or } x=1 (rejected)$ Set $B$ has $2$ elements. $ B=\{3,4\}$ Number of onto functions from $A\text{to}B={2}^{6}–2=62$

Consider two sets \(A=\left{x\in ℤ:\left|\left(\left|x−3\right|−3\right)\right|\leq 1\right}\) and \(B=\left{x\in ℝ−\lef…

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