CSIR NET Maths previous year question papers

Consider the Cauchy problem (CP$) x \partial{}u/\partial{}x + \partial{}u/\partial{}y = 1, u(0, y) = e^y, y \in \mathbb{R}$. 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?

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

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

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?