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Both roots of a quadratic positive or both negative: range of the parameter

This type of question, asked 3 times: 2 Apr 2025, 7 Apr 2025, 4 Apr 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    2 Apr 2025

    150
    If the set of all a∈R−{1}a \in R - \left\{ 1 \right\}, for which the roots of the equation (1−a)x2+2(a−3)x+9=0(1 - a)x^{2} + 2(a - 3)x + 9 = 0 are positive is (−∞,−α]∪[β,γ)( - \infty, - \alpha\rbrack \cup \lbrack\beta,\gamma), then 2α+β+γ2\alpha + \beta + \gamma is equal to

  2. Question 2

    7 Apr 2025

    64
    Let the set of all values of p∈Rp\in\mathbb{R}, for which both the roots of the equation x2−(p+2)x+(2p+9)=x^{2}- (p+ 2)x+ (2p+ 9) = 0 are negative real numbers, be the interval (α,β](\alpha,\beta\rbrack. Then β−2α\beta- 2\alpha is equal to.

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  3. Question 3

    4 Apr 2026 S2

    55
    If the quadratic equation (λ+2)x2−3λx+4λ=0(\lambda+ 2)x^{2}- 3\lambda x+ 4\lambda= 0, λ≠−2\lambda\neq - 2, has two positive roots, then the number of possible integral values of λ\lambda is :

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