CSIR NET Maths previous year question papers

Let u = u(x, t) be a solution of the wave equation $\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0$ satisfying the condition u(0, t) = 0, ∀t ≥ 0. Then which of the following statements is true?

Let $f : \mathbb{R} \to \mathbb{R}$ be such that sup(x≠y) |f(x) − f(y)|/|x − y| = L, where $1 < L < \infty$. Let $h : \mathbb{R} \to \mathbb{R}$ be a differentiable function satisfying |h′(x)| ≤ 3/4 for all $x \in \mathbb{R}$. For $\alpha > 0$, define $g(x) = \alpha{}f(x) + h(x)$ for $x \in \mathbb{R}$. Consider the sequence {$x_{k}$}(k≥0) defined by $x_(k+1) = g(x_{k}), k = 0, 1, \dots$, where $x_{0} \in \mathbb{R}$. The sequence {$x_{k}$}(k≥0) converges to the solution of the equation x = g(x) if

Let S denote the set of all solutions of the Euler-Lagrange equation of the variational problem: minimize $J[y] = \int_{0}^{1}(y^{2} + (y')^{2})$dx, subject to $y(0) = 0, y(1) = 0, \int_{0}^{1}y^{2}$dx = 1. Then the set {$\varphi(1/2) : \varphi \in S$} is equal to

Let $\lambda \in \mathbb{R}$, and $K : [0, 1] \times [0, 1] \to \mathbb{R}$ be a function such that every solution of the boundary value problem $(d^{2}u/$dx$^{2})(x) + \lambda{}u(x) = 0; ($du/dx)(0) = u(0), (du/dx)(1) = 0 satisfies the integral equation $u(x) + \lambda\int_{0}^{1}K(x, t)u(t)$dt = 0. Then

Two blocks of equal mass m are connected by a flexible inelastic cord of mass M. One block is placed on a smooth horizontal table, the other block hangs over the edge. The total potential energy of the entire cord is given by (−Mg$/2l)x^{2}$, where x is the distance of the hanging block from the edge of the table, l is the length of the cord, and g is the gravitational acceleration. Then

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

Let $\varphi$ denote the solution to the boundary value problem (BVP) (xy′)′ − 2y′ + y/x = 1 for $1 < x < e^{4}$, with y(1) = 0 and $y(e^{4}) = 4e^{4}$. Then the value of $\varphi(e)$ is

Let u = u(x, t) be the solution of the following initial value problem: $u_{t} + 2024u_{x} = 0$ for $x \in \mathbb{R}, t > 0$, and $u(x, 0) = u_{0}(x)$ for $x \in \mathbb{R}$, where $u_{0} : \mathbb{R} \to \mathbb{R}$ is an arbitrary $C^{1}$ function. Consider the following statements: $S_{1}$: If $A_{t} :=$ {$x \in \mathbb{R} : u(x, t) < 1$} and |$A_{t}$| denotes the Lebesgue measure of $A_{t}$ for every t ≥ 0, then |$A_{t}$| = |$A_{0}$|, $\forall{}t > 0. S_{2}$: If $u_{0}$ is Lebesgue integrable, then for every t > 0, the function x ↦ u(x, t) is Lebesgue integrable. Then

If u = u(x, t) is the solution of the initial value problem $u_{t} = u_{xx}$ for $x \in \mathbb{R}, t > 0$, with u(x, 0) = sin(4x) + x + 1 for $x \in \mathbb{R}$, satisfying |u(x, t)| $< 3e^(x^{2})$ for all $x \in \mathbb{R}$ and t > 0, then

If the value of the approximate solution of the initial value problem $y'(x) = x(y(x) + 1), x \in \mathbb{R}, y(0) = \beta$ at x = 0.2 using the forward Euler method with step size 0.1 is 1.02, then the value of $\beta$ is

Let B(0,1) = {$(x, y) \in \mathbb{R}^{2}$ | $x^{2} + y^{2} < 1$} be the open unit disc in $\mathbb{R}^{2}, \partial{}B(0,1)$ denote the boundary of B(0,1), and $\nu$ denote the unit outward normal to $\partial{}B(0,1)$. Let $f : \mathbb{R}^{2} \to \mathbb{R}$ be a given continuous function. The Euler–Lagrange equation of the minimization problem min { ½∬_B(0,1) |$\nabla{}u$|$^{2}$ dxdy + ½∬$_B(0,1) e^(u^{2})$ dxdy $+ \int_\partial{}B(0,1) f u$ ds } subject to $u \in C^{1}($closure of B(0,1)) is

Let u be the solution of the Volterra integral equation $\int_{0}$ᵗ [½ $+ \sin(t - \tau)] u(\tau) d\tau = \sin t$. Then the value of u(1) is

Consider a solid circular cylinder of radius 2 meters and height 3 meters of uniform density. If the density of the cylinder is $\rho$ kg/meter$^{3}$, then the moment of inertia (in kg meter$^{2})$ of the cylinder about a diameter of its base is

Consider the ordinary differential equation (ODE$) d^{2}y/$dx$^{2} + 2(e^(2x) - 1)($dy/dx$) + \alpha{}e^(4x)y = 0, x \in \mathbb{R}$, where $\alpha \in \mathbb{R}$. Note that the ODE transforms into an equation with constant coefficients under the change of independent variable given by t = e^(2x)/2. 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 + dy/dt = 2x − 3y + eᵗ, dx/dt + 2(dy/dt) = 3x − 4y + 2eᵗ, x(0) = −1, y(0) = −1/2, then which of the following statements are true?

Let u = u(x, t) be the solution to the initial value problem $\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0, u(x, 0) = f(x), x \in \mathbb{R}, (\partial{}u/\partial{}t)(x, 0) = 1/\sqrt{1 + x^{2}}, x \in \mathbb{R}$, where f is a twice continuously differentiable function defined on $\mathbb{R}$ satisfying f(x) → 0 as |x| $\to \infty$. Then which of the following statements are true?

Define S := {$y \in C^{1}[0, 1] : y(0) = y(1) = 0$}, ‖y‖$(\infty) := \max(x\in[0,1])$|y(x)| for every $y \in S, B_{0}(0, \varepsilon) :=$ {y ∈ S : ‖y‖$(\infty) < \varepsilon$}, $B_{1}(0, \varepsilon) :=$ {y ∈ S : ‖y‖$(\infty) +$ ‖y′‖$(\infty) < \varepsilon$}. Consider the functional $J : S \to \mathbb{R}$ given by $J[y] = \int_{0}^{1}[(y')^{2} - 2x(y')^{4}]$dx, then there exists an $\varepsilon > 0$ such that

Consider the variational problem minimize $J[y] = \int_{0}^{1}[(y')^{2} + 2025$ yy′]dx, where y(0) and y(1) are free. Then which of the following statements are true?

Consider the boundary value problem (BVP$) d^{2}y/$dx$^{2} + 4($dy/dx$) + (\lambda + 2)y = 0, y(0) = y(\pi) = 0$. Then which of the following statements are true?

Consider a particle of mass m which is moving on a surface due to gravity. Suppose the Lagrangian of the particle in the cylindrical coordinates is given by $L = (m/2)[(1 + 4\alpha^{2}r^{2})$ṙ$^{2} + r^{2}\theta$̇$^{2}] - m\alpha$gr$^{2}$, where $\alpha$ is a positive constant, and g is the acceleration due to gravity. If the particle is in circular motion, then

Consider the partial differential equation $4(\partial^{2}u/\partial{}x^{2}) + 6(\partial^{2}u/\partial{}x\partial{}y) + 3(\partial^{2}u/\partial{}y^{2}) = 0$. Which of the following partial differential equations can be obtained by a change of independent variables given by $(\xi, \eta)$ᵗ = A(x, y)ᵗ for some 2 × 2 invertible matrix A with real entries?

Consider the following statements: $S_{1}$: There exists an $\varepsilon > 0$ such that $\forall{}x_{0} \in (2 - \varepsilon, 2 + \varepsilon)$, the iterative sequence defined by $x(n+1) = 6/(5 - x_{n}), n = 0, 1, 2, 3, \dots$, converges to $2. S_{2}$: There exists an $\varepsilon > 0$ such that $\forall{}x_{0} \in (3 - \varepsilon, 3 + \varepsilon)$, the iterative sequence defined by $x(n+1) = (x_{n}^{2} + 6)/5, n = 0, 1, 2, 3, \dots$, converges to $3. S_{3}$: There exists an $\varepsilon > 0$ such that $\forall{}x_{0} \in (3 - \varepsilon, 3 + \varepsilon)$, the iterative sequence defined by $x(n+1) = (5x_{n} - 6)/x_{n}, n = 0, 1, 2, 3, \dots$, converges to 3. Then which of the following statements are true?

Let $f : [0, 1] \times \mathbb{R} \to \mathbb{R}$ be a function such that f and its partial derivatives of orders less than or equal to 3 are continuous and bounded. Let $y : [0, 1] \to \mathbb{R}$ be the solution of dy/dt $= f(t, y(t)), 0 \le t \le 1, y(0) = y_{0}$, where $y_{0} \in \mathbb{R}$. For h > 0, denote t_j = jh. Let Y_j be an approximation of y(t_j), defined by Y_(j+1) = Y_j + ahf(t_j, Y_j) + bhf(t_(j+1), Y_j + chf$(t_j, Y_j)), 0 < (j+1)h < 1, Y_{0} = y_{0}$, where $a, b, c \in \mathbb{R}$. If there exists an M > 0 such that |y(t_(j+1)) − y(t_j) − ahf(t_j, y(t_j)) − bhf(t_(j+1), y(t_j) + chf(t_j, y(t_j)))| ≤ Mh$^{3}$ for every h > 0, and every j with 0 ≤ (j+1)h < 1, 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 R(x, t) and u(x) denote the resolvent kernel and the solution, respectively, of the Volterra integral equation u(x) = eˣ $+ 2\int_(\ln 6)^x e^(2(t-x))u(t)$dt. Then which of the following statements are true?