JEE Mathematics 2025 Question with Solution

Given: Two natural numbers $k_1$ and $k_2$ are chosen randomly. We need the probability that $i^{k_1} + i^{k_2} \neq 0$, where $i = \sqrt{-1}$.

Find: The required probability.

The powers of $i$ repeat with period $4$:

So, according to the value of $k \pmod{4}$,

For the sum to be zero,

which means

Thus, the two values must be opposite among $\{i,-1,-i,1\}$.

The opposite pairs are:

• $i$ and $-i$
• $1$ and $-1$

Now fix $k_1 \pmod{4}$. For each of the $4$ possible residue classes of $k_1$, there is exactly one residue class of $k_2$ that makes the sum zero:

• if $k_1 \equiv 0 \pmod{4}$, then $k_2 \equiv 2 \pmod{4}$
• if $k_1 \equiv 1 \pmod{4}$, then $k_2 \equiv 3 \pmod{4}$
• if $k_1 \equiv 2 \pmod{4}$, then $k_2 \equiv 0 \pmod{4}$
• if $k_1 \equiv 3 \pmod{4}$, then $k_2 \equiv 1 \pmod{4}$

Since powers of $i$ depend only on the exponent modulo $4$, we only need to look at residue classes modulo $4$. There are $4 \times 4 = 16$ equally likely pairs of values for $\left(i^{k_1}, i^{k_2}\right)$.

The sum is zero only in these $4$ cases:

• $i + (-i) = 0$
• $(-i) + i = 0$
• $1 + (-1) = 0$
• $(-1) + 1 = 0$

Hence,

Therefore,

So, the correct option is A.