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Quadratic with random coefficients: probability it is positive for all x

This type of question, asked 3 times: 2021 Paper 11, 25 Jul 2022, 2 Apr 2026 S1. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 11

    60
    The probability of selecting integers a∈[−5,30]a \in \lbrack - 5,30\rbrack such that x2+2(a+4)x−5a+64>0x^{2}+ 2(a+ 4)x- 5a+ 64 > 0, for all x∈Rx \in R, is:

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  2. Question 2

    25 Jul 2022

    77
    If the numbers appeared on the two throws of a fair six faced die are α\alpha and β\beta, then the probability that x2+αx+β>0x^{2}+ \alpha x + \beta > 0, for all x∈Rx \in R, is:

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  3. Question 3

    2 Apr 2026 S1

    73
    Let a,b,c∈{1,2,3,4}a,b,c \in \{ 1,2,3,4\}. If the probability, that ax2+22bx+c>0{ax}^{2} + 2\sqrt{2}bx + c > 0 for all x∈Rx \in R, is mn\frac{m}{n}, gcd(m,n)=1gcd(m,n) = 1, then m+nm + n is equal to ____\_\_\_\_ .