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Quadratic with real roots, no real roots, or always of one sign: range of the parameter from the discriminant

This type of question, asked 5 times: 2021 Paper 3, 2021 Paper 18, 2021 Paper 21, 31 Jan 2024, 29 Jan 2025. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 3

    68
    The integer ' kk ', for which the inequality x2−2(3k−1)x+8k2−7>0x^{2}- 2(3k- 1)x+ 8k^{2}- 7 > 0 is valid for every xx in RR is:

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  2. Question 2

    2021 Paper 18

    82
    The sum of all integral values of k(k≠0)k(k\neq 0) for which the equation 2x−1−1x−2=2k\frac{2}{x- 1}-\frac{1}{x- 2}=\frac{2}{k} in xx has no real roots, is

  3. Question 3

    2021 Paper 21

    71
    The set of all values of k>−1k> - 1, for which the equation (3x2+4x+3)2−(k+1)(3x2+4x+3)\left( 3x^{2}+ 4x + 3 \right)^{2}- (k + 1)\left( 3x^{2}+ 4x + 3 \right) (3x2+4x+2)+k(3x2+4x+2)2=0\left( 3x^{2}+ 4x+ 2 \right)+k\left( 3x^{2}+ 4x+ 2 \right)^{2}= 0 has real roots, is:

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  4. Question 4

    31 Jan 2024

    80
    Let SS be the set of positive integral values of a for which ax2+2(a+1)x+9a+4x2−8x+32<0,∀x∈R\frac{ax^{2}+ 2(a + 1)x + 9a + 4}{x^{2}- 8x + 32}< 0,\forall x \in\mathbb{R}. Then, the number of elements in SS is:

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  5. Question 5

    29 Jan 2025

    126
    If the set of all a∈Ra \in R, for which the equation 2x2+(a−5)x+15=3a2x^{2}+ (a - 5)x + 15 = 3a has no real root, is the interval (α,β)(\alpha,\beta), and X={x∈Z:α<x<β}X = \{ x \in Z:\alpha < x < \beta\}, then ∑x∈Xx2\sum_{x \in X} x^{2} is equal to

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