CSIR NET Maths previous year question papers

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

The infimum of the set {$\int_{a}$ᵇ $\sqrt{1 + (y'(t))^{2}}$dt : $y \in C^{1}[a, b], y(a) = a^{2}, y(b) = b - 5$} is

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

Consider a solid torus of constant density $\rho$, formed by revolving the disc $(y - b)^{2} + z^{2} \le a^{2}, x = 0$ about the z-axis, where 0 < a < b. Then the moment of inertia of the solid torus about the z-axis is

Consider the initial value problem (IVP$) y'(x) = \sin(y(x))/(1 + y^{4}(x))$ for $x \in \mathbb{R}$, with $y(0) = y_{0}$. Then which of the following statements are true?

Consider the boundary value problem (BVP) (e^(−5x)y′)′ + 6e^(−5x)y = −f(x), 0 < x < ln 2, with y(0) = 0, y(ln 2) = 0. If $G(x, \xi) = (e^(3x) +$ Be^(2x))(Ce$^(2\xi) +$ De$^(3\xi))$ for $0 \le \xi \le x$, and $G(x, \xi) = (e^(3\xi) +$ Be$^(2\xi))($Ce^(2x) + De^(3x)) for $x \le \xi \le \ln 2 ($Green's function) is such that $\int_{0}^(\ln 2) G(x, \xi)f(\xi)d\xi$ is the solution of the BVP, then the values of B, C and D are

Let S denote the set of all 2 × 2 matrices A such that the iterative sequence generated by the Gauss-Seidel method applied to the system of linear equations $A(x_{1}, x_{2})^{T} = (2, 3)^{T}$ converges for every initial guess. Then which of the following statements are true?

Let g(x) be the polynomial of degree at most 4 that interpolates the data (x, y) = (−1, −30), (0, 1), (2, c), (3, 10), (6, 19). If g(4) = 5, then which of the following statements are true?

The extremizer of the problem $\min[(1/2)\int$₋$_{1}^{1} ((y'(x))^{2} + (y(x))^{2})$dx] subject to $y \in C^{1}[-1, 1], \int$₋$_{1}^{1}$ xy(x)dx = 0 and y(−1) = y(1) = 1 is

For $\lambda \in \mathbb{R}$ such that |$\lambda$| < 5/32, let $R(x, t, \lambda)$ and u denote the resolvent kernel and the solution, respectively, of the Fredholm integral equation $u(x) = x + (\lambda/2)\int$₋$_{2}^{2} ($xt $+ x^{2}t^{2})u(t)$dt. Then which of the following statements are true?