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(a + bi)^n = 2^m (p + iq): find the quadratic whose roots are built from p and q

The same question, asked 2 times: 2021 Paper 19*, 24 Jan 2023. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 19

    74
    If (3+i)100=299(p+iq)(\sqrt{3}+ i)^{100}=2^{99}(p + iq), then pp and qq are roots of the equation:

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  2. Question 2

    24 Jan 2023

    65
    Let p,q∈Rp,q \in R and (1−3i)200=2199(p+iq),i=−1(1 -\sqrt{3}i)^{200}=2^{199}(p + iq),i =\sqrt{- 1} Then p+q+q2p + q +q^{2} and p−q+q2p - q +q^{2} are roots of the equation

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