CSIR NET Maths previous year question papers

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?

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?

Let $u : \mathbb{R}^{2} \to \mathbb{R}$ solve $\partial_{x}u + 2\partial_yu = 0$ on $\mathbb{R}^{2}$ with u(x, y) = sin x on the line y = 3x + 1, and let $v : \mathbb{R}^{2} \to \mathbb{R}$ solve $\partial_{x}v + 2\partial_yv = 0$ with v(x, 0) = sin x. Let S = [0,1] × [0,1]. Which of the following statements are true?

Let y(x) and z(x) be the stationary functions (extremals) of $J(y, z) = \int_{0}^{1} [(y')^{2} + (z')^{2} + y'z']$ dx subject to y(0) = 1, y(1) = 0, z(0) = −1, z(1) = 2. Which of the following statements are correct?

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 initial value problem $x^{2}y'' - 2x^{2}y' + (4x - 2)y = 0, y(0) = 0$. Suppose $y = \varphi(x)$ is a polynomial solution satisfying $\varphi(1) = 1$. 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?

The coefficient of $x^{3}$ in the interpolating polynomial for the data x = 0, 1, 2, 3, 4 with y = 1, 2, 1, 3, 5 is

For the IVP dy/dx $= f(x, y), y(x_{0}) = y_{0}$, with step size h, let the first iterate of a second-order scheme be $y_{1} = y_{0} +$ Pk$_{1} +$ Qk$_{2}$, where $k_{1} = h f(x_{0}, y_{0}), k_{2} = h f(x_{0} + \alpha_{0}h, y_{0} + \beta_{0}k_{1})$ and $P, Q, \alpha_{0}, \beta_{0} \in \mathbb{R}$. Which of the following statements are correct?

Among the curves connecting the points (1, 2) and (2, 8), let $\gamma$ be the curve on which an extremal of $J[y] = \int_{1}^{2} (1 + x^{3}y') y'$ dx can be attained. Which of the following points lie on the curve $\gamma$?

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

Let y be the solution to the Volterra integral equation y(x) = eˣ $+ \int_{0}$ˣ $((1 + x^{2})/(1 + t^{2})) y(t)$ dt. Which of the following statements are true?

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?

Let $q_{1}, q_{2}$ be generalized coordinates and $p_{1}, p_{2}$ their conjugate momenta. Let a and b be such that $Q_{1} = q_{1}, P_{1} = a p_{1} + 16 p_{2}, Q_{2} = p_{2}, P_{2} = 2q_{1} + b q_{2}$ is a canonical transformation. Which of the following statements are true?