CSIR NET Maths previous year question papers

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?

Consider the non-homogeneous ordinary differential equation (ODE$) d^{2}y/$dx$^{2} + 5$dy/dx + 6y = sin(e^(−5x)), x > 0. 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 = x − 4e^(−2t)y, dy/dt = e^(2t)x − y, x(0) = 1, y(0) = 1, then which of the following statements are true?

Let u = u(x, y) be the solution of the boundary value problem $\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} = 0$ on (0, 1) × (0, 1), with $u(x, 0) = e^(\pi{}x), u(x, 1) = -e^(\pi{}x)$ for $x \in [0, 1], u(0, y) = \cos(\pi{}y) + \sin(\pi{}y)$ and $u(1, y) = e^\pi(\cos(\pi{}y) + \sin(\pi{}y))$ for y ∈ [0, 1]. Then there exists a point $(x_{0}, y_{0}) \in (0, 1) \times (0, 1)$ such that

Let u = u(x, t) be the solution of the initial-boundary value problem $\partial{}u/\partial{}t = \partial^{2}u/\partial{}x^{2}$ on $(0, 1) \times (0, \infty)$, with u(x, 0) = 4x(1 − x) for x ∈ [0, 1] and u(0, t) = u(1, t) = 0 for t ≥ 0. Then which of the following statements are true?

Consider the system of two particles with total kinetic energy T = (5/2)ẋ$^{2} + l^{2}\theta$̇$^{2} + 2l$ẋ$\theta$̇$\cos \theta$, and Lagrangian L = (5/2)ẋ$^{2} + l^{2}\theta$̇$^{2} + 2l$ẋ$\theta$̇$\cos \theta + 2$gl $\cos \theta$, where $x, \theta$ are generalized coordinates, and g, l are positive constants. Then the non-zero frequency of the normal mode of the system with small oscillations (|$\theta$| ≪ 1) is

For $b \in \mathbb{R}$, let y_b = y_b(x) be the unique solution of the initial value problem dy/dx $= y^{5} + y^{4} + y^{3} + y^{2} + y + 1, y(0) = b$ defined on its maximal interval of existence I_b. Then which of the following statements are true?

If $\lambda \in \mathbb{R}$ and $p \in \mathbb{R}$ are such that the quadrature formula $\int_(x_{0})^(x_{0}+h) f(x)$dx $\approx \lambda{}h(f(x_{0}) + f(x_{0} + h)) +$ ph$^{3}(f''(x_{0}) + f''(x_{0} + h))$ is exact for all polynomials of degree as high as possible, then

Let f(x) be the polynomial of degree at most 2 that interpolates the data (−1, 2), (0, 1), and (1, 2). If g(x) is a polynomial of degree at most 3 such that f(x) + g(x) interpolates the data (−1, 2), (0, 1), (1, 2), and (2, 17), then

For any $b \in \mathbb{R}$, let S(b) denote the set of all broken extremals with one corner of the variational problem: minimize $J[y] = \int_{0}^{1}((y')^{4} - 3(y')^{2})$dx, subject to y(0) = 0, y(1) = b. Then which of the following statements are true?

Define S := {$y \in C^{1}[-1, 1] : y(-1) = -1, y(1) = 3$}. Let $\varphi$ be the extremal of the functional $J : S \to \mathbb{R}$ given by $J[y] = \int$₋$_{1}^{1}[(y')^{3} + (y')^{2}]$dx. Define ‖y‖$(\infty) := \max(x\in[-1,1])$|y(x)| for every y ∈ S and let $B_{0}(\varphi, \varepsilon) :=$ {y ∈ S : ‖$y - \varphi$‖$(\infty) < \varepsilon$}, $B_{1}(\varphi, \varepsilon) :=$ {y ∈ S : ‖$y - \varphi$‖$(\infty) +$ ‖$y' - \varphi'$‖$(\infty) < \varepsilon$}. Then which of the following statements are true?

The integral equation $u(x) = f(x) + (2/\pi)\int_{0}^\pi \sin(x - t)u(t)$dt has a unique solution if

If u is the solution of the Volterra integral equation $u(x) = 3 + \sin x + \int_{0}$ˣ [(3 + sin x)/(3 + sin t)]u(t)dt, then

If $x_{1} = x_{1}(t), x_{2} = x_{2}(t)$ is the solution of the initial value problem e^(−t) dx$_{1}/$dt $= -x_{1} + x_{2}, e^(-t)$ dx$_{2}/$dt $= -x_{1} - x_{2}, x_{1}(0) = 1, x_{2}(0) = 0$, and $r(t) = \sqrt{x_{1}^{2}(t) + x_{2}^{2}(t)}$, then which of the following statements are true?

Consider the initial boundary value problem (IBVP$) u_{t} + u_{x} = 2u$ for x > 0, t > 0; u(0, t) = 1 + sin t for t > 0; u(x, 0) = eˣ cos x for x > 0. If u is the solution of the IBVP, then the value of $u(2\pi, \pi)/u(\pi, 2\pi)$ is

Let B(0,2) = {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} < 4$}, and $\partial{}B$ denote the boundary of B(0,2). Assume $(\alpha, \beta) \ne (0,0), k \in \mathbb{R}$, and u is any solution to $-\Delta{}u = 0$ in $B(0,2), \alpha{}u(x, y) + \beta(\partial{}u/\partial\nu)(x, y) = 1 + (x^{2} + y^{2})k$ on $\partial{}B$, where $\nu(x, y)$ is the unit outward normal to B(0,2) at $(x, y) \in \partial{}B$. Consider the following statements: $S_{1}$: If $\beta = 0$, then there exists $a (x_{0}, y_{0}) \in B(0,2)$ such that |$u(x_{0}, y_{0})$| = |1 + 4k|/|$\alpha$|. $S_{2}$: If $\alpha = 0$, then k = −1/4. Then