CSIR NET Maths previous year question papers

Let u(x) be the solution to the Volterra integral equation $u(x) = x^{2} + 4\int_{0}$ˣ $(t - x)^{2}u(t)$ dt. Then which of the following statements are true?

Let f and K be such that the solution of the initial value problem y″ − 3y′ + 2y = 4sin(x), y(0) = 1, y′(0) = −2 satisfies the Volterra integral equation $y(x) = f(x) + \int_{0}$ˣ K(x, t)y(t) dt. Then which of the following statements are true?

Suppose f, g are smooth functions of generalized coordinates $q_{1}, q_{2}, \dots, q_{n}$, the associated conjugate momenta $p_{1}, p_{2}, \dots, p_{n}$, and time t. Let [f, g] denote the Poisson bracket of f and g. Suppose H is a Hamiltonian of the system. Then which of the following statements are true?

Consider the ordinary differential equation (ODE) y″ + r(x)y = 0, where $r(x) = x^{3}\sin(1/x)$ for x≠0, and r(0)=0. Then which of the following statements are true?

Let y(x) be the solution of the integral equation (IE) y(x) = eˣ $+ e + 1 + (1/3)\int_{0}^{1}y(t)$dt, and R(x,t,1/3) be the resolvent kernel associated to IE. Then which of the following statements are true?

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{1} x^{2}(y')^{2}$ dx subject to $y(0)=0, y(1)=1, \int_{0}^{1}y^{2}$dx = 1/7. Then which of the following statements are true?

Let u(x,t) be the solution of the initial-boundary value problem $u_{tt} - u_{xx} + u_{t} = 0, 0<x<\pi, t>0, u(0,t)=0, u(\pi,t)=0, t>0, u(x,0)=\sin x, u_{t}(x,0)=0, 0<x<\pi$. For t≥0, define $E(t) = \int_{0}^\pi (u_{t}^{2}+u_{x}^{2})$ dx. Then which of the following statements are true?

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

A mechanical system is described using generalized position q and generalized momentum p. Let Q and P denote new generalized position and generalized momentum variables respectively, generated by the generating function $F(q,P)=q^{2}e^P$, and Q,P are canonical coordinates. Let G(p,Q) be a function such that G(2,e)=0, and it generates the same canonical coordinates Q,P. Then which of the following statements are true?

Suppose $y_{1}(x)$ and $y_{2}(x)$ are two linearly independent solutions of the differential equation $x^{2}y'' + [(1+\sin x)x/2]y' - (3/2)(\cos x)y = 0, x>0$, satisfying $y_{2}(0)=0$. Then which of the following statements are true?

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{1} ((y')^{2}\sin x + (2cos x)y)$ dx subject to y(0)=0, y(1)=1. 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?

Let {$x_{n}$} be a convergent iterative sequence generated by Newton-Raphson method for solving the equation sin x−1=0 such that $x_{n}\to\pi/2$ as $n\to\infty$. For $n\in\mathbb{N}$, let $e_{n}=x_{n}-\pi/2$. Let p>0 be such that $\lim(n\to\infty)$|$e_{n}$₊$_{1}$|/|$e_{n}$|^p exists and is non-zero. Then which of the following statements are true?

Consider the boundary value problem (BVP$) u_{xx}+u_yy=0$ in $\Omega=${$(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1$}, u(x,y)=e^(x+y) on $\partial\Omega=${$(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1$}. Then which of the following statements are true?

For any non-zero solution y = y(x) of the differential equation $(2x + 3)^{2}(d^{2}y/$dx$^{2}) + 6(2x + 3)($dy/dx) + 8y = 0, x > 0, denote S := {$x \in (0, \infty) : y(x) = 0$}. Then

The initial value problem dy/dx $= \surd$|x − 1| sin y, y(0) = 1 has

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

Consider the quadrature formula $\int$₋$_{1}^{1}$ |x|f(x)dx ≈ (1/2)(f(−1) + f(1)). Then the degree of precision (also known as order of exactness) of the quadrature formula is

If $\varphi \in C^{4}[0, 1]$ is the extremal of the variational problem minimize $J[y] = \int_{0}^{1}($yy$' + (y'')^{2})$dx, subject to y(0) = 0, y′(0) = 1, y(1) = 2, y′(1) = 4, then $\varphi(1/2)$ is equal to

If u is a solution of the integral equation $u(x) = \lambda\int_{0}^{1}K(x, t)u(t)$dt, where K(x, t) := x(1 − t) for 0 ≤ x ≤ t ≤ 1 and t(1 − x) for 0 ≤ t ≤ x ≤ 1, then

Consider a particle of mass $m_{1}$ moving along a horizontal line L that is perpendicular to a vertical wall. Let x denote the distance of the particle from the wall. Suppose a simple pendulum of length l having mass $m_{2}, m_{2} \ne m_{1}$, is attached to the particle, hanging below L with $\theta$ measured from the downward vertical. If the pendulum oscillates in a plane containing L, then the equations of motion in terms of the generalized coordinates x and $\theta$ are (g denotes the acceleration due to gravity)

Suppose that the differential equation $d^{2}y/$dx$^{2} + P(x)$dy/dx $+ e^(2x)y = 0, x \in \mathbb{R}$ transforms into a second order differential equation with constant coefficients under the change of independent variable given by s = s(x) satisfying (ds/dx)(0) = 1. Then which of the following statements is true?

Given that $y_{1}(x) = e^(2x)$ is a solution of the ordinary differential equation (ODE$) x d^{2}y/$dx$^{2} - (3 + 4x)$dy/dx + (4x + 6)y = 0, x > 0. Let $y_{2} = y_{2}(x)$ be the solution of the ODE satisfying the conditions $y_{2}(1) = e^{2}/4$ and (dy$_{2}/$dx$)(1) = 3e^{2}/2$. Then which of the following statements is true?

Let u = u(x, y) be the solution of the Cauchy problem $x \partial{}u/\partial{}x + y \partial{}u/\partial{}y = u, (x, y) \ne (0, 0)$, with $u(x, 1) = \sqrt{1 + x^{2}}, x \in \mathbb{R}$. Then which of the following statements is true?