CSIR NET Maths previous year question papers

Let {$X_{n} : n \ge 0$} be any homogeneous Markov Chain on the state space S = {1,2,3,4} having the transition probability matrix $P = (p_{ij})$ given by row 1 = (1/4, 0, 2/3, 1/12), row 2 = (0, 1, 0, 0), row 3 = (1/12, 0, 1/4, 2/3), row 4 = (2/3, 0, 1/12, 1/4). Which of the following statements about stationary distributions of any such Markov Chain is true?

Let $Z_{1}$ and $Z_{2}$ be two independent discrete random variables such that $Z_{1}$ follows binomial distribution with parameters n = 2 and p = 1/2, and $Z_{2}$ follows Poisson distribution with mean 1. Consider the following system of equations with three variables $x_{1}, x_{2}$ and $x_{3}: x_{1} - 2x_{2} + x_{3} = 1; 2x_{1} - 5x_{2} + 2x_{3} = 2; x_{1} + 2x_{2} + Z_{1}x_{3} = Z_{2}$. Then, the probability that the given system of equations has infinite number of solutions equals