JEE Mathematics 2023 Question with Solution

Given: $f: \{1,2,3,4\} \to \{a\in \mathbb{Z}: |a| \le 8\}$ and

Find: The number of such functions.

From the solution table provided, the valid functions are obtained by solving the relation successively for $n=3,2,1$:

The extracted working concludes that only two tuples satisfy all conditions within $|f(k)|\le 8$:

1. $\big(f(1),f(2),f(3),f(4)\big)=(1,0,2,-3)$
2. $\big(f(1),f(2),f(3),f(4)\big)=(0,1,0,3)$

Hence, the number of functions is $2$. The solution explicitly states: The correct answer is (A) : 2. Since option A on the page is labeled with value $3$, there is a source discrepancy between the listed options and the worked answer. Following the solution, the correct option label is A.

Using the table shown in the solution:

• For $f(4)=-3$, we get $f(3)=2$, $f(2)=0$, $f(1)=1$.
• For $f(4)=3$, we get $f(3)=0$, $f(2)=1$, $f(1)=0$.

Both satisfy

for $n=1,2,3$ and all values lie in $[-8,8]$. Therefore exactly $2$ functions exist.