jee-main 2023 Q61

jee-main · India · session1_24jan_shift1 Complex Numbers Arithmetic Trigonometric/Polar Form and De Moivre's Theorem

Let $p , \quad q \in \mathbb { R }$ and $( 1 - \sqrt { 3 } i ) ^ { 200 } = 2 ^ { 199 } ( p + i q ) , i = \sqrt { - 1 }$. Then, $p + q + q ^ { 2 }$ and $p - q + q ^ { 2 }$ are roots of the equation.
(1) $x ^ { 2 } + 4 x - 1 = 0$
(2) $x ^ { 2 } - 4 x + 1 = 0$
(3) $x ^ { 2 } + 4 x + 1 = 0$
(4) $x ^ { 2 } - 4 x - 1 = 0$

Let $p , \quad q \in \mathbb { R }$ and $( 1 - \sqrt { 3 } i ) ^ { 200 } = 2 ^ { 199 } ( p + i q ) , i = \sqrt { - 1 }$. Then, $p + q + q ^ { 2 }$ and $p - q + q ^ { 2 }$ are roots of the equation.\\
(1) $x ^ { 2 } + 4 x - 1 = 0$\\
(2) $x ^ { 2 } - 4 x + 1 = 0$\\
(3) $x ^ { 2 } + 4 x + 1 = 0$\\
(4) $x ^ { 2 } - 4 x - 1 = 0$

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