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?

The problem $\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} = 0$ in {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} > 1$}, u(x, y) = 1 on {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} = 1$} has

The problem $\partial{}u/\partial{}x + \partial{}u/\partial{}y = u^{2}, u(x, 0) = x^{2}, \forall{}x \in \mathbb{R}$ has a solution on an open set containing the line {$(x, y) \in \mathbb{R}^{2}$ : ax + by = 0} if

What is the cardinality of the set of real solutions of eˣ + x = 1?

Consider the initial value problem (IVP$) y'(x) = \surd$|$y(x) + \varepsilon$|, $x \in \mathbb{R}, y(0) = y_{0}$. Consider the following statements: S1: There is an $\varepsilon > 0$ such that for all $y_{0} \in \mathbb{R}$, the IVP has more than one solution. S2: There is $a y_{0} \in \mathbb{R}$ such that for all $\varepsilon > 0$, the IVP has more than one solution. Then