JEE Main 2016MathematicsComplex NumberMediumMCQ

JEE Main 2016 — Complex Number Question with Solution

JEE Main 2016 (10 Apr Online)

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 :

Choose an option

Show full solutionCorrect option: B
Correct answer
B-1365 3i

Step-by-step explanation

z=1+ai
Cubing both sides,
z3=1-a2+2ai+1-a2ai-2a2
z3=1-3a2+(3a-a3)i
∵ z3 is a real number.
a3-a2=0 ⇒  a= 3 a>0

1+z+z2…..z11=z12-1z-1=1+ 3i12-11+ 3i-1=1+ 3i12-13i

1+ 3i12=21212+32i12=212cos⁡π3+isin⁡π312=212 cos⁡4π+isin⁡4π=212

 =212-13i=-13653i

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About this question

This is a previous-year question from JEE Main 2016, covering the Complex Number 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.