Verified Solution • 2025–26 Session moderate

QJEEM25Ma03b8: JEE Main 2025 Mathematics - Class 12 JEEM25Ma03b8 Solution: Two number and are randomly chosen from the set of natural…

Two number k1\mathrm{k}_1 and k2\mathrm{k}_2 are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1+ik2,(i=−1)\mathrm{i}^{\mathrm{k}_1}+\mathrm{i}^{\mathrm{k}_2},(\mathrm{i}=\sqrt{-1}) is non-zero, equals

Multiple Choice Options

  • A 34\frac{3}{4}
  • B 12\frac{1}{2}
  • C 14\frac{1}{4}
  • D 23\frac{2}{3}

Step-by-Step Verified Solution

ik1+ik2≠0ik1→4i^{k_1}+i^{k_2} \neq 0 \quad i^{k_1} \rightarrow 4 option for i,−1,−i,1\mathrm{i},-1,-\mathrm{i}, 1

Total cases ⇒4×4=16\Rightarrow 4 \times 4=16

Unfovourble cases ⇒ik1+ik2=0\Rightarrow \mathrm{i}^{\mathrm{k}_1}+\mathrm{i}^{\mathrm{k}_2}=0

{1,−1−1,1i,−i−i,i}

4 Cases ⇒\Rightarrow Probability =16−416=34=\frac{16-4}{16}=\frac{3}{4}

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