MHTCET 19 Apr 2026 Shift 2 Paper

Concept:The powers of $i$ repeat every 4 terms, and the sum of one full cycle is $0$.Explanation:We know $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$.So one complete cycle has sum:$$1 + i - 1 - i = 0.$$The series runs from $n=0$ to $n=2026$, giving $2027$ terms in total.The first $2024$ terms form $506$ complete cycles, so their sum is $0$.The remaining three terms are $n=2024, 2025, 2026$.Now $i^{2024} = 1$, $i^{2025} = i$, and $i^{2026} = -1$.Thus,$$z = 1 + i - 1 = i.$$Since $i = e^{i \pi/2}$, we have$$z = e^{i \pi/2}.$$Hence,$$\sqrt{z} = \left(e^{i \pi/2}\right)^{1/2} = e^{i \pi/4}.$$The other square root is $e^{i 5\pi/4}$, but one valid value is $e^{i \pi/4}$.Answer:Option A: $e^{i \pi/4}$.