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Range of a real parameter for which an equation in z has a solution

This type of question, asked 2 times: 2021 Paper 1, 6 Apr 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 1

    89
    If the least and the largest real values of α\alpha, for which the equation z+α∣z−1∣+2i=0z+\alpha|z- 1| + 2i= 0 (z∈C(z\in C and i=−1)i=\sqrt{- 1}) has a solution, are pp and qq respectively; then 4(p2+q2)4\left( p^{2}+q^{2} \right) is equal to

  2. Question 2

    6 Apr 2026 S1

    54
    Let the set of all values of k∈Rk \in R such that the equation z(z‾+2+i)+k(2+3i)=0,z∈Cz(\overline{z}+ 2 + i) + k(2 + 3i) = 0,z \in C, has at least one solution, be the interval [α,β]\lbrack\alpha,\beta\rbrack. Then 9(α+β)9(\alpha+\beta) is equal to:

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