Sets, Relations and Functions – Functions – JEE Main 23 Jan 2026 Shift 2

Solution:

First, we find the elements of Set $A$:
$$ ||x-3|-3| \le 1 \Rightarrow -1 \le |x-3|-3 \le 1 $$

$$ 2 \le |x-3| \le 4 $$

This gives two cases:
Case 1: $2 \le x-3 \le 4 \Rightarrow 5 \le x \le 7$. Integer values: $\{5, 6, 7\}$.
Case 2: $-4 \le x-3 \le -2 \Rightarrow -1 \le x \le 1$. Integer values: $\{-1, 0, 1\}$.
$$ A = \{-1, 0, 1, 5, 6, 7\} \Rightarrow n(A) = 6 $$

Next, we find the elements of Set $B$:
$$ \frac{(x-2)(x-4)}{x-1}\ln(|x-2|) = 0 $$

The defined domain is $\mathbb{R} – \{1, 2\}$.
Condition for zero: Numerator is zero or log term is zero.
1. $(x-2)(x-4) = 0 \Rightarrow x=2, x=4$. Since $x \neq 2$, only $x=4$ is valid.
2. $\ln(|x-2|) = 0 \Rightarrow |x-2| = 1 \Rightarrow x-2 = \pm 1$.
$x=3$ or $x=1$. Since $x \neq 1$, only $x=3$ is valid.
$$ B = \{3, 4\} \Rightarrow n(B) = 2 $$

We need the number of onto functions from $A$ to $B$.
Total functions = $n(B)^{n(A)} = 2^6 = 64$.
Into functions are those where the range is a proper subset of $B$ (i.e., range is $\{3\}$ or $\{4\}$). There are 2 such constant functions.

Number of Onto functions = Total functions – Into functions
$$ = 2^6 – 2 = 64 – 2 = 62 $$

Ans. (1)