jee-main 2020 Q52

jee-main · India · session1_08jan_shift2 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions

Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$. If $a = (1 + \alpha)\sum_{k=0}^{100}\alpha^{2k}$ and $b = \sum_{k=0}^{100}\alpha^{3k}$, then $a$ and $b$ are the roots of the quadratic equation.
(1) $x^{2} + 101x + 100 = 0$
(2) $x^{2} - 102x + 101 = 0$
(3) $x^{2} - 101x + 100 = 0$
(4) $x^{2} + 102x + 101 = 0$

Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$. If $a = (1 + \alpha)\sum_{k=0}^{100}\alpha^{2k}$ and $b = \sum_{k=0}^{100}\alpha^{3k}$, then $a$ and $b$ are the roots of the quadratic equation.\\
(1) $x^{2} + 101x + 100 = 0$\\
(2) $x^{2} - 102x + 101 = 0$\\
(3) $x^{2} - 101x + 100 = 0$\\
(4) $x^{2} + 102x + 101 = 0$

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