CSIR NET Maths previous year question papers

Consider the non-homogeneous ordinary differential equation (ODE$) d^{2}y/$dx$^{2} + 5$dy/dx + 6y = sin(e^(−5x)), x > 0. Then which of the following statements are true?

If x = x(t), y = y(t) is the solution of the initial value problem dx/dt = x − 4e^(−2t)y, dy/dt = e^(2t)x − y, x(0) = 1, y(0) = 1, then which of the following statements are true?

Let u = u(x, y) be the solution of the boundary value problem $\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} = 0$ on (0, 1) × (0, 1), with $u(x, 0) = e^(\pi{}x), u(x, 1) = -e^(\pi{}x)$ for $x \in [0, 1], u(0, y) = \cos(\pi{}y) + \sin(\pi{}y)$ and $u(1, y) = e^\pi(\cos(\pi{}y) + \sin(\pi{}y))$ for y ∈ [0, 1]. Then there exists a point $(x_{0}, y_{0}) \in (0, 1) \times (0, 1)$ such that

Let u = u(x, t) be the solution of the initial-boundary value problem $\partial{}u/\partial{}t = \partial^{2}u/\partial{}x^{2}$ on $(0, 1) \times (0, \infty)$, with u(x, 0) = 4x(1 − x) for x ∈ [0, 1] and u(0, t) = u(1, t) = 0 for t ≥ 0. Then which of the following statements are true?

Consider the system of two particles with total kinetic energy T = (5/2)ẋ$^{2} + l^{2}\theta$̇$^{2} + 2l$ẋ$\theta$̇$\cos \theta$, and Lagrangian L = (5/2)ẋ$^{2} + l^{2}\theta$̇$^{2} + 2l$ẋ$\theta$̇$\cos \theta + 2$gl $\cos \theta$, where $x, \theta$ are generalized coordinates, and g, l are positive constants. Then the non-zero frequency of the normal mode of the system with small oscillations (|$\theta$| ≪ 1) is

For $b \in \mathbb{R}$, let y_b = y_b(x) be the unique solution of the initial value problem dy/dx $= y^{5} + y^{4} + y^{3} + y^{2} + y + 1, y(0) = b$ defined on its maximal interval of existence I_b. Then which of the following statements are true?

If $\lambda \in \mathbb{R}$ and $p \in \mathbb{R}$ are such that the quadrature formula $\int_(x_{0})^(x_{0}+h) f(x)$dx $\approx \lambda{}h(f(x_{0}) + f(x_{0} + h)) +$ ph$^{3}(f''(x_{0}) + f''(x_{0} + h))$ is exact for all polynomials of degree as high as possible, then

Let f(x) be the polynomial of degree at most 2 that interpolates the data (−1, 2), (0, 1), and (1, 2). If g(x) is a polynomial of degree at most 3 such that f(x) + g(x) interpolates the data (−1, 2), (0, 1), (1, 2), and (2, 17), then

For any $b \in \mathbb{R}$, let S(b) denote the set of all broken extremals with one corner of the variational problem: minimize $J[y] = \int_{0}^{1}((y')^{4} - 3(y')^{2})$dx, subject to y(0) = 0, y(1) = b. Then which of the following statements are true?

Define S := {$y \in C^{1}[-1, 1] : y(-1) = -1, y(1) = 3$}. Let $\varphi$ be the extremal of the functional $J : S \to \mathbb{R}$ given by $J[y] = \int$₋$_{1}^{1}[(y')^{3} + (y')^{2}]$dx. Define ‖y‖$(\infty) := \max(x\in[-1,1])$|y(x)| for every y ∈ S and let $B_{0}(\varphi, \varepsilon) :=$ {y ∈ S : ‖$y - \varphi$‖$(\infty) < \varepsilon$}, $B_{1}(\varphi, \varepsilon) :=$ {y ∈ S : ‖$y - \varphi$‖$(\infty) +$ ‖$y' - \varphi'$‖$(\infty) < \varepsilon$}. Then which of the following statements are true?

The integral equation $u(x) = f(x) + (2/\pi)\int_{0}^\pi \sin(x - t)u(t)$dt has a unique solution if

If u is the solution of the Volterra integral equation $u(x) = 3 + \sin x + \int_{0}$ˣ [(3 + sin x)/(3 + sin t)]u(t)dt, then

If $x_{1} = x_{1}(t), x_{2} = x_{2}(t)$ is the solution of the initial value problem e^(−t) dx$_{1}/$dt $= -x_{1} + x_{2}, e^(-t)$ dx$_{2}/$dt $= -x_{1} - x_{2}, x_{1}(0) = 1, x_{2}(0) = 0$, and $r(t) = \sqrt{x_{1}^{2}(t) + x_{2}^{2}(t)}$, then which of the following statements are true?

Consider the initial boundary value problem (IBVP$) u_{t} + u_{x} = 2u$ for x > 0, t > 0; u(0, t) = 1 + sin t for t > 0; u(x, 0) = eˣ cos x for x > 0. If u is the solution of the IBVP, then the value of $u(2\pi, \pi)/u(\pi, 2\pi)$ is

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?

Suppose x(t) is the solution of the initial value problem in $\mathbb{R}^{2}$: ẋ = Ax, $x(0) = x_{0}$, where A = [[5, 4], [1, 2]]. Which of the following statements is true?