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A given relation between the roots (their difference or reciprocals) fixes the parameter

This type of question, asked 3 times: 1 Feb 2024, 22 Jan 2026 S2, 28 Jan 2026 S1. All 3 questions below, oldest first.

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  1. Question 1

    1 Feb 2024

    152
    Let α\alpha and β\beta be the roots of the equation px2+qx−px^{2}+ qx - r=0r = 0, where p≠0p \neq 0. If p,qp,q and rr be the consecutive terms of a non-constant G.P and 1α+1β=34\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}, then the value of (α−β)2(\alpha - \beta)^{2} is :

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  2. Question 2

    22 Jan 2026 S2

    63
    Let α,β\alpha,\beta be the roots of the quadratic equation 12x2−20x+3λ=0,λ∈Z12x^{2}- 20x + 3\lambda = 0,\lambda \in Z. If 12≤∣β−α∣≤32\frac{1}{2}\leq |\beta - \alpha| \leq\frac{3}{2}, then the sum of all possible values of λ\lambda is :

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  3. Question 3

    28 Jan 2026 S1

    60
    If α,β\alpha,\beta, where α<β\alpha<\beta, are the roots of the equation λx2−(λ+3)x+3=0\lambda x^{2}- (\lambda+ 3)x + 3 = 0 such that 1α−1β=13\frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}, then the sum of all possible values of λ\lambda is :

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