JEE Main 2016 · 10 Apr · Shift 1 · Mathematics Q63

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MathematicsSingle correct+4 / -1Question 63 of 90

Let z = 1 + ai be a complex number, a > 0, such that $z^3$ is a real number. Then the sum $1+z+z^2+\dots+z^{11}$ is equal to :

-1250\sqrt3\,i

1250\sqrt3\,i

1365\sqrt3\,i

-1365\sqrt3\,i

Answer and worked solution

Official answer: D

$\operatorname{Im}(1+ai)^3=3a-a^3=0$ gives $a=\sqrt3$ , so $z=2e^{i\pi/3}$ and $z^{12}=4096$ . Then
$\sum_{k=0}^{11}z^k=\frac{z^{12}-1}{z-1}=\frac{4095}{\sqrt3\,i}=-1365\sqrt3\,i.$

Topics:Algebra of complex numbers Geometric progression

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JEE Main 2016 · 10 Apr · Shift 1, Mathematics question 63