JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Probability

Two number

k_{1}

and

k_{2}

are randomly chosen from the set of natural numbers. Then, the probability that the value of

i^{k_{1}} + i^{k_{2}}, (i = - 1)

is non-zero, equals

Answer: (C)

Solution

The powers of

i

repeat with period 4:

i^{1} = i, i^{2} = - 1, i^{3} = - i, i^{4} = 1

, and this pattern continues.

For a randomly chosen natural number

k

i^{k}

is equally likely to be any of

{i, - 1, - i, 1}

. So for two independently chosen numbers

k_{1}, k_{2}

, there are

4 \times 4 = 16

equally likely combinations of

(i^{k_{1}}, i^{k_{2}})

i^{k_{1}} + i^{k_{2}} = 0

exactly when the two values are negatives of each other:

(1, - 1), (- 1, 1), (i, - i), (- i, i)

— 4 combinations.

So the number of combinations giving a non-zero sum is

16 - 4 = 12

, and

P (i^{k_{1}} + i^{k_{2}} \neq = 0) = \frac{12}{16} = \frac{3}{4}

JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Single correct option