JEE Mathematics 2026 Question with Solution

Given: $A = \{x \in Z : ||x-3|-3| \le 1\}$ and $B$ is the set of roots of the equation

$$\frac{(x-2)(x-4)}{x-1}\ln|x-2| = 0$$

with domain $x \ne 1, 2$.

Find: Number of onto functions $A \to B$.

For $A$:

$$-1 \le |x-3|-3 \le 1$$

which gives

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

So,

$$2 \le x-3 \le 4 \implies x \in \{5,6,7\}$$

and

$$-4 \le x-3 \le -2 \implies x \in \{-1,0,1\}$$

Hence, set $A$ has $6$ elements.

For $B$:

$$\frac{(x-2)(x-4)}{x-1}\ln|x-2| = 0$$

Since $x \ne 1,2$, the roots come from

$$x-4 = 0 \implies x=4$$

and

$$\ln|x-2| = 0 \implies |x-2|=1 \implies x=3$$

where $x=1$ is rejected by the domain restriction.

Thus, $B = \{3,4\}$ and set $B$ has $2$ elements.

Total number of functions from a $6$-element set to a $2$-element set is

$$2^6 = 64$$

The non-onto functions are the into functions with range size less than $2$. These are only:

1. all elements mapped to $3$
2. all elements mapped to $4$

So the number of non-onto functions is $2$. Therefore, the number of onto functions is

$$64 - 2 = 62$$

Hence, the correct option is B.

Given: $|A| = 6$ and $|B| = 2$.

Find: Number of onto functions $A \to B$.

Using the surjection formula from an $m$-element set to an $n$-element set:

$$\sum_{k=0}^{n} (-1)^k \binom{n}{k}(n-k)^m$$

Here $m=6$ and $n=2$. So,

$$\sum_{k=0}^{2} (-1)^k \binom{2}{k}(2-k)^6$$ $$= \binom{2}{0}2^6 - \binom{2}{1}1^6 + \binom{2}{2}0^6$$ $$= 64 - 2 + 0 = 62$$

Therefore, the number of onto functions is $62$, so the correct option is B.