JEE MAIN - Mathematics (2025 - 28th January Morning Shift - No. 10)
Two number $\mathrm{k}_1$ and $\mathrm{k}_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $\mathrm{i}^{\mathrm{k}_1}+\mathrm{i}^{\mathrm{k}_2},(\mathrm{i}=\sqrt{-1})$ is non-zero, equals
$\frac{3}{4}$
$\frac{1}{2}$
$\frac{1}{4}$
$\frac{2}{3}$
Explanation
$i^{k_1}+i^{k_2} \neq 0 \quad i^{k_1} \rightarrow 4$ option for $\mathrm{i},-1,-\mathrm{i}, 1$
Total cases $\Rightarrow 4 \times 4=16$
Unfovourble cases $\Rightarrow \mathrm{i}^{\mathrm{k}_1}+\mathrm{i}^{\mathrm{k}_2}=0$
$$\left\{\begin{array}{c} 1,-1 \\ -1,1 \\ i,-i \\ -i, i \end{array}\right\}$$
4 Cases $\Rightarrow$ Probability $=\frac{16-4}{16}=\frac{3}{4}$
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