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 D = {$(x, y) \in \mathbb{R}^{2} : -1 \le x \le 1, -1 \le y \le 1$}, and $f : D \to \mathbb{R}$ be the function defined by $f(x, y) = 1 + \surd(y$₊), where y₊ = max{y, 0}. Consider the initial value problem (IVP) dy/dx = f(x, y), y(0) = 0. Then which of the following statements are true?

Let $\lambda\in\mathbb{R}$ be such that the integral equation $y(x) = \lambda\int$₋$_{1}^{1}(5$xt$^{3}+4x^{2}t+3$xt)y(t)dt admits a non-trivial solution y(x) such that y(1)=5/2. Then which of the following statements are 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

The integral equation $u(x) = f(x) + (2/\pi)\int_{0}^\pi \cos(x + t)u(t)$dt has infinitely many solutions if

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

Let B(0,2) = {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} < 4$}, and $\partial{}B$ denote the boundary of B(0,2). Assume $(\alpha, \beta) \ne (0,0), k \in \mathbb{R}$, and u is any solution to $-\Delta{}u = 0$ in $B(0,2), \alpha{}u(x, y) + \beta(\partial{}u/\partial\nu)(x, y) = 1 + (x^{2} + y^{2})k$ on $\partial{}B$, where $\nu(x, y)$ is the unit outward normal to B(0,2) at $(x, y) \in \partial{}B$. Consider the following statements: $S_{1}$: If $\beta = 0$, then there exists $a (x_{0}, y_{0}) \in B(0,2)$ such that |$u(x_{0}, y_{0})$| = |1 + 4k|/|$\alpha$|. $S_{2}$: If $\alpha = 0$, then k = −1/4. Then

For $c \in \mathbb{R}$, consider the following Fredholm integral equation y(x) = 1 + x + cx$^{2} + 2\int_{0}^{1} (1 - 3$xt)y(t)dt. Then the values of c for which the integral equation admits a solution are

Let $f \in C^{1}(\mathbb{R})$ be bounded. Consider the initial-value problem (P): x′(t) = f(x(t)), t > 0, x(0) = 0. Which of the following statements are true?

Consider the initial value problem y′ = y + ½|$\sin(y^{2})$|, x > 0, y(0) = −1. Which of the following statements are true?

Consider the problem $y' = (1 - y^{2})^{10} \cos y, y(0) = 0$. Let J be the maximal interval of existence and K be the range of the solution. Which of the following statements are true?

Consider the Cauchy problem $u\cdot{}u_{x} + u_y = 1$ for $(x, y) \in \mathbb{R} \times (0, \infty)$, with u(x, 0) = kx, $x \in \mathbb{R}$, for a real parameter k. For which of the following values of k does the problem have a solution defined on $\mathbb{R} \times (0, \infty)$?

Let B be the open unit disc in $\mathbb{R}^{2}, \partial{}B$ its boundary and ${\bar{B}} = B \cup \partial{}B$. For $\lambda \in (0, \infty)$, let $S_\lambda$ be the set of twice continuously differentiable functions in B, continuous on B̄, satisfying $(u_{x})^{2} + \lambda(u_y)^{2} = 1$ in B and u = 0 on $\partial{}B$. Which of the following statements are true?

Define S = {$y \in C^{1}[0, \pi] : y(0) = y(\pi) = 0$}, ‖f‖$_\infty = \max$ |f|, $B_{0}(0, \varepsilon) =$ {f ∈ S : ‖f‖$_\infty < \varepsilon$} and $B_{1}(0, \varepsilon) =$ {f ∈ S : ‖f‖$_\infty +$ ‖f′‖$_\infty < \varepsilon$}. For the functional $J[y] = \int_{0}^\pi (1 - (y')^{2}) y^{2}$ dx, there exists $\varepsilon > 0$ such that

Consider the Fredholm integral equation $y(x) - 3\int_{0}^{1} t x y(t)$ dt = f(x), with f continuous on [0, 1]. Which of the following choices for f(x) have the property that the equation admits at least one solution?