CSIR NET Maths previous year question papers

Let u(x, y) be the solution of the Cauchy problem $u\cdot{}u_{x} + u_y = 0$ for $x \in \mathbb{R}, y > 0$, with u(x, 0) = x for $x \in \mathbb{R}$. Which of the following is the value of u(2, 3)?

Consider the variational problem $(P): J(y) = \int_{0}^{1} [(y')^{2} - y$|y|·y′ + xy] dx, y(0) = 0, y(1) = 0. Which of the following statements is correct?

For the unknown $y : [0,1] \to \mathbb{R}$, consider the boundary value problem y″(x) + 2y(x) = 0 for x ∈ (0,1), y(0) = y(1) = 0. It is given that it corresponds to the integral equation $y(x) = 2\int_{0}^{1} K(x, t) y(t)$ dt. Which of the following is the kernel K(x, t)?

The smallest real number $\lambda$ for which the problem $-y'' + 3y = \lambda{}y, y(0) = 0, y(\pi) = 0$ has a non-trivial solution is

The partial differential equation $x^{2} u_{xx} - 2$xy $u_{x}$ᵧ $- 3y^{2} u_yy + u_{x} - u_y = 0$ is

Consider the Cauchy problem for the wave equation $u_tt - 4u_{xx} = 0$ on $\mathbb{R} \times (0, \infty)$, with u(x, 0) = $e^{-1/x^{2}}$ for x ≠ 0 (and 0 at x = 0), and u_t(x, 0) = x $e^{-x^{2}}$. Which one of the following is true?

Using Euler's method with step size 0.05, the approximate value of the solution of dy/dx $= \sqrt{3x + 2y + 1}, y(1) = 1$, at x = 1.1 (rounded to two decimal places) is

The cardinality of the set of extremals of $J[y] = \int_{0}^{1} (y')^{2}$ dx subject to $y(0) = 1, y(1) = 6, \int_{0}^{1} y$ dx = 3 is

The value of $\lambda$ for which the integral equation $y(x) = \lambda\int_{0}^{1} x^{2}$ $e^{x+t}$ y(t) dt has a non-zero solution is

Let g denote the acceleration due to gravity and a > 0. A particle of mass m glides without friction on the cycloid $x = a(\theta - \sin \theta), y = a(1 + \cos \theta), 0 \le \theta \le 2\pi$. The equation of motion of the particle is

Let $f \in C^{1}(\mathbb{R})$ be bounded. Consider the initial-value problem (P): x′(t) = f(x(t)), t > 0, x(0) = 0. Which of the following statements are true?

Let $u : \mathbb{R}^{2} \to \mathbb{R}$ solve $\partial_{x}u + 2\partial_yu = 0$ on $\mathbb{R}^{2}$ with u(x, y) = sin x on the line y = 3x + 1, and let $v : \mathbb{R}^{2} \to \mathbb{R}$ solve $\partial_{x}v + 2\partial_yv = 0$ with v(x, 0) = sin x. Let S = [0,1] × [0,1]. Which of the following statements are true?

Let y(x) and z(x) be the stationary functions (extremals) of $J(y, z) = \int_{0}^{1} [(y')^{2} + (z')^{2} + y'z']$ dx subject to y(0) = 1, y(1) = 0, z(0) = −1, z(1) = 2. Which of the following statements are correct?

Consider the initial value problem y′ = y + ½|$\sin(y^{2})$|, x > 0, y(0) = −1. Which of the following statements are true?

Consider the problem $y' = (1 - y^{2})^{10} \cos y, y(0) = 0$. Let J be the maximal interval of existence and K be the range of the solution. Which of the following statements are true?

Consider the initial value problem $x^{2}y'' - 2x^{2}y' + (4x - 2)y = 0, y(0) = 0$. Suppose $y = \varphi(x)$ is a polynomial solution satisfying $\varphi(1) = 1$. Which of the following statements are true?

Consider the Cauchy problem $u\cdot{}u_{x} + u_y = 1$ for $(x, y) \in \mathbb{R} \times (0, \infty)$, with u(x, 0) = kx, $x \in \mathbb{R}$, for a real parameter k. For which of the following values of k does the problem have a solution defined on $\mathbb{R} \times (0, \infty)$?

Let B be the open unit disc in $\mathbb{R}^{2}, \partial{}B$ its boundary and ${\bar{B}} = B \cup \partial{}B$. For $\lambda \in (0, \infty)$, let $S_\lambda$ be the set of twice continuously differentiable functions in B, continuous on B̄, satisfying $(u_{x})^{2} + \lambda(u_y)^{2} = 1$ in B and u = 0 on $\partial{}B$. Which of the following statements are true?

The coefficient of $x^{3}$ in the interpolating polynomial for the data x = 0, 1, 2, 3, 4 with y = 1, 2, 1, 3, 5 is

For the IVP dy/dx $= f(x, y), y(x_{0}) = y_{0}$, with step size h, let the first iterate of a second-order scheme be $y_{1} = y_{0} +$ Pk$_{1} +$ Qk$_{2}$, where $k_{1} = h f(x_{0}, y_{0}), k_{2} = h f(x_{0} + \alpha_{0}h, y_{0} + \beta_{0}k_{1})$ and $P, Q, \alpha_{0}, \beta_{0} \in \mathbb{R}$. Which of the following statements are correct?

Among the curves connecting the points (1, 2) and (2, 8), let $\gamma$ be the curve on which an extremal of $J[y] = \int_{1}^{2} (1 + x^{3}y') y'$ dx can be attained. Which of the following points lie on the curve $\gamma$?

Define S = {$y \in C^{1}[0, \pi] : y(0) = y(\pi) = 0$}, ‖f‖$_\infty = \max$ |f|, $B_{0}(0, \varepsilon) =$ {f ∈ S : ‖f‖$_\infty < \varepsilon$} and $B_{1}(0, \varepsilon) =$ {f ∈ S : ‖f‖$_\infty +$ ‖f′‖$_\infty < \varepsilon$}. For the functional $J[y] = \int_{0}^\pi (1 - (y')^{2}) y^{2}$ dx, there exists $\varepsilon > 0$ such that

Let y be the solution to the Volterra integral equation y(x) = eˣ $+ \int_{0}$ˣ $((1 + x^{2})/(1 + t^{2})) y(t)$ dt. Which of the following statements are true?

Consider the Fredholm integral equation $y(x) - 3\int_{0}^{1} t x y(t)$ dt = f(x), with f continuous on [0, 1]. Which of the following choices for f(x) have the property that the equation admits at least one solution?

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?