CSIR NET Maths previous year question papers

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?

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?

Let $f : \mathbb{R} \to [0, \infty)$ be a bijective function. Which of the following statements is true?

Suppose that the probability density function of the random variable X is $f(x) = 2(\theta - x)/\theta^{2}$ if $0 \le x \le \theta$, and 0 otherwise, where $\theta > 0$ is an unknown parameter. Based on a single observation X, the confidence coefficient of the confidence interval [(2/5)X, (5/2)X] for $\theta$ is

Let u = u(x, t) be a solution of the wave equation $\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0$ satisfying the condition u(0, t) = 0, ∀t ≥ 0. Then which of the following statements is true?

Let S denote the set of all solutions of the Euler-Lagrange equation of the variational problem: minimize $J[y] = \int_{0}^{1}(y^{2} + (y')^{2})$dx, subject to $y(0) = 0, y(1) = 0, \int_{0}^{1}y^{2}$dx = 1. Then the set {$\varphi(1/2) : \varphi \in S$} is equal to

Suppose that X ~ binomial$(9, \theta), 0.7 < \theta < 1$, and Y ~ Poisson$(\lambda), \lambda > 0$. If 3E(Y) = E(X), then which of the following is true?

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 2)$ be a random sample from Uniform$(\theta, 1 - \theta)$ distribution, where $-\infty < \theta < 1/2$. The maximum likelihood estimator of $\theta$ is

Consider the set A = {$x \in \mathbb{Q} : 0 < (\sqrt{2} - 1)x < \sqrt{2} + 1$} as a subset of $\mathbb{R}$. Which of the following statements is true?

Let B(0,1) = {$(x, y) \in \mathbb{R}^{2}$ | $x^{2} + y^{2} < 1$} be the open unit disc in $\mathbb{R}^{2}, \partial{}B(0,1)$ denote the boundary of B(0,1), and $\nu$ denote the unit outward normal to $\partial{}B(0,1)$. Let $f : \mathbb{R}^{2} \to \mathbb{R}$ be a given continuous function. The Euler–Lagrange equation of the minimization problem min { ½∬_B(0,1) |$\nabla{}u$|$^{2}$ dxdy + ½∬$_B(0,1) e^(u^{2})$ dxdy $+ \int_\partial{}B(0,1) f u$ ds } subject to $u \in C^{1}($closure of B(0,1)) is

Let X be a random variable with cumulative distribution function given by F(x) = 0 if x < 0; F(x) = (x + 1)/3 if 0 ≤ x < 1; F(x) = 1 if x ≥ 1. Then the value of P(1/3 < X < 3/4) + P(X = 0) is equal to

Let $X_{1}, X_{2}$ be a random sample from a population having probability density function f ∈ {$f_{0}, f_{1}$} where $f_{0}(x) = 1/2$ if 0 ≤ x ≤ 2 and 0 otherwise, and $f_{1}(x) = 1/4$ if 0 ≤ x ≤ 4 and 0 otherwise. For testing the null hypothesis $H_{0} : f = f_{0}$ against the alternate hypothesis $H_{1} : f = f_{1}$, the power of a most powerful test of size $\alpha = 0.05$ is equal to

Consider the following functions on the interval $[0, 1] : g(x) = \sin(\pi{}x), h(x) = \cos(\pi(x - 1))$. Which of the following statements are true?

Consider the ordinary differential equation (ODE$) d^{2}y/$dx$^{2} + 2(e^(2x) - 1)($dy/dx$) + \alpha{}e^(4x)y = 0, x \in \mathbb{R}$, where $\alpha \in \mathbb{R}$. Note that the ODE transforms into an equation with constant coefficients under the change of independent variable given by t = e^(2x)/2. Then which of the following statements are true?

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?

Let $X_{1}, X_{2}, \dots, X_{10}$ be a random sample of size 10 from a Bernoulli distribution with success probability $1/(1 + e^\theta)$, where $\theta \in \mathbb{R}$ is an unknown parameter. If $M = \sum(i=1..10)X_{i}$, then which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample of size n from $U(\theta, \theta + 1)$ distribution, where $\theta \in \mathbb{R}$ is an unknown parameter. If $T_{n} = \max${$X_{1}, X_{2}, \dots, X_{n}$}, n = 1, 2, …, then which of the following statements are true?

Consider a linear model $Y_{i} = \beta_{1} + \beta_{2} + \cdots + \beta_{i} + \varepsilon_{i}, 1 \le i \le n$, where the errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Then, which of the following statements are true?

Let ‖·‖ denote the Euclidean norm on $\mathbb{R}^{2}$ and $\alpha \ge 0$. Let $f : \mathbb{R}^{2} \to \mathbb{R}$ be a function such that |f(x)| ≤ ‖x‖$^\alpha$. Which of the following statements are necessarily true?

Let $f : \mathbb{C}$ \ {0} $\to \mathbb{C}$ be a non-zero holomorphic function such that |f(z)| ≤ |z|^(5/2) + 1/|z|$^(1/2), z \in \mathbb{C}$ \ {0}. Which of the following statements are true?

Consider maximizing the objective function $P = x_{1} + x_{2}$ subject to $x_{1} + x_{2} + 2x_{3} \le 5, 2x_{1} - 3x_{3} \ge 1, x_{2} + x_{3} \le 0, x_{1} \ge 0, x_{2} \ge 0, x_{3} \ge 0$. Then, which of the following statements are true?

For a variable x, consider the $\mathbb{Q}-$vector space V = {ax$^{3} +$ bx$^{2} +$ cx + d | $a, b, c, d \in \mathbb{Q}$}. Further, let A = {$f : V \to \mathbb{Q}$ | f is $a \mathbb{Q}-$linear transformation} and B = {f ∈ A | f(1) = 0}. Which of the following statements are true?

For a 4 × 4 positive definite real symmetric matrix A and real numbers a, b, c, d, consider the 5 × 5 matrix B whose first row is (0, a, b, c, d), whose first column is (0, a, b, c, d)ᵗ, and whose lower-right 4 × 4 block is A. Which of the following statements are necessarily 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?