JEE Main 2016 — Sequences And Series Question with Solution
From: JEE Main 2016 (Online) 10th April Morning Slot
Question
Let z = 1 + ai be a complex number, a > 0, such that z3 is a real number.
Then the sum 1 + z + z2 + . . . . .+ z11 is equal to :
Then the sum 1 + z + z2 + . . . . .+ z11 is equal to :
Choose an option
Show full solutionCorrect option: D
Correct answer
D
Step-by-step explanation
z = 1 + ai
z2 = 1 a2 + 2ai
z2 . z = {(1 a2) + 2ai} {1 + ai}
= (1 a2) + 2ai + (1 a2) ai 2a2
z3 is real 2a + (1 a2) a = 0
a (3 a2) = 0 a = (a > 0)
1 + z + z2 . . . . . . . z11 =
=
(1 + )12 = 212
= 212 (cos + isin)12 = 212 (cos4 + isin4) = 212
z2 = 1 a2 + 2ai
z2 . z = {(1 a2) + 2ai} {1 + ai}
= (1 a2) + 2ai + (1 a2) ai 2a2
z3 is real 2a + (1 a2) a = 0
a (3 a2) = 0 a = (a > 0)
1 + z + z2 . . . . . . . z11 =
=
(1 + )12 = 212
= 212 (cos + isin)12 = 212 (cos4 + isin4) = 212
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This is a previous-year question from JEE Main 2016, covering the Sequences And Series chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.