Consider a system of two equations: p*x² + q*x = 0 and r*x² + s*x = 0, where p, q, r, s are each independently chosen from the set {2, 3, 4, 6}. Statement I: The probability that the system has exactly two solutions is 9/64. Statement II: Given that the system has exactly one solution, the

Question

Consider a system of two equations: p*x² + q*x = 0 and r*x² + s*x = 0, where p, q, r, s are each independently chosen from the set {2, 3, 4, 6}. Statement I: The probability that the system has exactly two solutions is 9/64. Statement II: Given that the system has exactly one solution, the probability that p = r is 12/55. Which of the following is/are correct?

Accepted Answer

Statement I is true. Each equation factors as x(px+q)=0, giving x=0 and x=-q/p. The system always shares x=0. It has two solutions total when q/p = s/r (so the nonzero roots coincide). Statement I asks for probability of exactly two solutions: need q/p = s/r. Count pairs (p,q,r,s) with qr = ps out of 4⁴=256. Statement II conditions on exactly one solution (qr != ps) and asks P(p=r).

Solution

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