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?