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Quadratic with random coefficients: probability of equal or real roots

This type of question, asked 3 times: 2021 Paper 3, 5 Apr 2024, 9 Apr 2024. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 3

    61
    The coefficients a,ba,b and cc of the quadratic equation, ax2+bx+c=0ax^{2}+bx+c= 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is:

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  2. Question 2

    5 Apr 2024

    70
    The coefficients a,b,ca,b,c in the quadratic equation ax2+bx+c=0ax^{2}+ bx + c = 0 are chosen from the set {1,2,3,4,5,6,7,8}\mathbf{\{ 1,2,3,4,5,6,7,8\}}. The probability of this equation having repeated roots is:

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  3. Question 3

    5 Apr 2024

    165
    The coefficients a,b,ca,b,c in the quadratic equation ax2+bx+c=0ax^{2}+ bx + c = 0 are from the set {1,2,3,4,5,6}\mathbf{\{ 1,2,3,4,5,6\}}. If the probability of this equation having one real root bigger than the other is pp, then 216 p equals:

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  4. Question 4

    9 Apr 2024

    81
    Let a,ba,b and cc denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1,2,3,41,2,3,4. If the probability that ax2+bx+c=0ax^{2}+ bx + c = 0 has all real roots is mn\frac{m}{n}, gcd(m,n)=1gcd(m,n) = 1, then m+nm + n is equal to