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Dice give the coefficients: probability that a system has a unique solution (or none)

This type of question, asked 3 times: 2021 Paper 19, 24 Jan 2023, 22 Jan 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 19

    71
    Two fair dice are thrown. The numbers on them are taken as λ\lambda and μ\mu, and a system of linear equations
    x+y+z=5x+2y+3z=μx+3y+λz=1\begin{matrix} & x+y+z= 5 \\ & x+ 2y+ 3z=\mu \\ & x+ 3y+\lambda z= 1 \end{matrix}
    is constructed. If pp is the probability that the system has a unique solution and qq is the probability that the system has no solution, then :

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  2. Question 2

    24 Jan 2023

    66
    Let NN denote the number that turns up when a fair die is rolled. If the probability that the system of equations
    x+y+z=1x + y + z = 1
    2x+Ny+2z=22x + Ny + 2z = 2
    3x+3y+Nz=33x + 3y + Nz = 3
    has unique solution is k6\frac{k}{6}, then the sum of value of kk and all possible values of NN is

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  3. Question 3

    22 Jan 2026 S2

    51
    Let n be the number obtained on rolling a fair die. If the probability that the system x−ny+z=6x - ny + z = 6, x+(n−2)y+(n+1)z=8x + (n - 2)y + (n + 1)z = 8, (n−1)y+z=1(n - 1)y + z = 1 has a unique solution is k6\frac{k}{6}, then the sum of k and all possible values of nn is :

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