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Quadratic with complex coefficients: its roots and an expression in them

This type of question, asked 3 times: 28 Jan 2025, 3 Apr 2025, 5 Apr 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    28 Jan 2025

    129
    If α+iβ\alpha+i\beta and γ+iδ\gamma+i\delta are the roots of x2−(3−2i)x−(2i−2)=0,i=−1x^{2}- (3 - 2i)x- (2i- 2) = 0,i=\sqrt{- 1}, then αγ+βδ\alpha\gamma+\beta\delta is equal to :

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  2. Question 2

    3 Apr 2025

    64
    Let z∈Cz\in C be such that z2+3iz−2+i=2+3i\frac{z^{2}+ 3i}{z- 2 +i}= 2 + 3i. Then the sum of all possible values of z2z^{2} is.

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  3. Question 3

    5 Apr 2026 S2

    52
    Let z1,z2∈Cz_{1},z_{2}\in C be the distinct solutions of the equation z2+4z−(1+12i)=0z^{2}+ 4z- (1 + 12i) = 0. Then ∣z1∣2+∣z2∣2\left| z_{1} \right|^{2}+\left| z_{2} \right|^{2} is equal to :

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