CSIR NET Maths previous year question papers

Let G be a finite group. For any prime number p, let E(p) = {g ∈ G | g^p = 1}. Which of the following statements are true?

Which of the following quotient rings are fields?

Consider $\mathbb{R}^{2}$ with the Euclidean metric d. Let $O = (0, 0), P_{0} = (0, 1) \in \mathbb{R}^{2}$, and for any integer $n \ge 1, P_{n} = (1/n, 1) \in \mathbb{R}^{2}$. Let X = ⋃$(n\ge0) L_{n}$, where $L_{n}$ is the line segment joining $P_{n}$ and O. Define $d_X : X \times X \to \mathbb{R}$ as follows: d_X(a, b) = d(a, b) when $a, b \in L_{n}$ for some n, and d_X(a, b) = d(a, O) + d(b, O) otherwise. Let $\tau$ be the smallest topology such that the sets $B_(a,\varepsilon) =$ {$b \in X : d_X(a, b) < \varepsilon$} are open in $\tau$ for all a ∈ X and $\varepsilon > 0$. 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?

If x = x(t), y = y(t) is the solution of the initial value problem dx/dt + dy/dt = 2x − 3y + eᵗ, dx/dt + 2(dy/dt) = 3x − 4y + 2eᵗ, x(0) = −1, y(0) = −1/2, then which of the following statements are true?

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

Let $X_{1}, X_{2} \dots$ be a sequence of independent random variables with $X_{i}$ following N(0, 1 + 1/i) distribution for all $i \in \mathbb{N}$. Let X be N(0, 1)-random variable independent of {$X_{n} : n \ge 1$}. Then, which of the following statements are true?

Let $U_{1}, U_{2}, \dots$ be a sequence of independent and identically distributed (i.i.d.) U(0, 1) random variables. Let $G_{n} = (\prod(i=1..n) U_{i})^(1/n)$ be the geometric mean of $U_{1}, U_{2}, \dots, U_{n}$ for $n \in \mathbb{N}$. Let $X_{1}$ and $X_{2}$ be degenerate random variables such that $P(X_{1} = 0) = 1$ and $P(X_{2} = 1/e) = 1$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed (i.i.d.) random variables such that for $x = 3, 4, \dots, P(X_{1} = \pm{}x) = 1/(2$cx$^{2}\ln(x))$, where $c = \sum(x=3..\infty) 1/(x^{2}\ln(x))$. Then, which of the following statements are true?

Consider the Markov chain with state space {0, 1, 2, 3, 4} and the transition probability matrix P = [[0.4, 0.3, 0.3, 0, 0], [0, 0.5, 0, 0.5, 0], [0.5, 0, 0.5, 0, 0], [0, 0.5, 0, 0.5, 0], [0, 0.3, 0, 0.3, 0.4]]. Then, which of the following statements are true?

Let X and Y be independent random variables with the moment generating functions M_X(t) = (1/9)(2 + eᵗ$)^{2}, t \in \mathbb{R}$ and M_Y(t) = exp[eᵗ $- 1], t \in \mathbb{R}$, respectively. 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 manufacturing unit producing three-component series systems, where component's lifetimes are independent and identically distributed random variables with hazard function $h(x) = \lambda$ if x > 0, and 0 otherwise. Here, $\lambda > 0$ is an unknown parameter. Let $X_{1}, X_{2}, X_{3}$ be a random sample of size 3 on lifetimes of systems produced by the manufacturing unit. Let $\theta = P_\lambda(X_{1} > 1/2)$. If $\theta$̂ is the maximum likelihood estimate of $\theta$ based on the realization $x_{1} = 1/2, x_{2} = 1/6, x_{3} = 1/3$ of the random sample, then which of the following statements are true?

The life (in hours) of an electrical component is exponentially distributed with mean $\theta > 0$. For testing $H_{0} : \theta = 5/(\ln(10) - \ln(9))$ against $H_{1} : \theta = 5/\ln(2)$, three such components are chosen at random. Suppose that we reject the null hypothesis $H_{0}$ if and only if two or more of these three components survive for less than five hours. 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?

Consider a linear regression model $Y = X\beta + \varepsilon$, with r regressors and an intercept. Random error $\varepsilon$ ~ $N_{n}(0, \sigma^{2}I_{n})$ and X has full column rank. Here $I_{n}$ denotes the identity matrix of order n. Regression coefficients are estimated by the least squares estimation method. Let $\sigma$̂$^{2}$ and $\sigma$̂$^{2}($MLE), respectively, be the mean squares residuals and the maximum likelihood estimator of $\sigma^{2}$. Then, which of the following statements are true?

If $(X_{1}, X_{2})$ᵗ ~ $N_{2}((0, 0)$ᵗ, [[3, 1], [1, 3]]), then which of the following statements are true?

Let $f : [0, 1] \to \mathbb{R}$ be a Riemann integrable function. Define $F(x) = \int_{0}$ˣ f(t)dt, ∀x ∈ [0, 1]. Which of the following statements are necessarily true?

Let $\mu$ denote the Lebesgue measure on $\mathbb{R}$ and $f : \mathbb{R} \to [0, \infty)$ be a Lebesgue measurable function. For n ≥ 1, let $E_{n} =$ {$x \in \mathbb{R} : n - 1 \le f(x) < n$}. Suppose that $\sum(n\ge1) n^{2}\mu(E_{n}) < \infty$. Which of the following statements are true?

Consider the real vector space X = {$f : [0, 1] \to \mathbb{R}$ | f is continuous}, along with the norms ‖·‖$_{1}$ and ‖·‖$_{2}$ defined by ‖f‖$_{1} = \int_{0}^{1}$|f(x)|dx and ‖f‖$_{2} = (\int_{0}^{1}$|f(x)|$^{2}$dx)^(1/2). For n ≥ 1 and x ∈ [0, 1], let $f_{n}(x) =$ nx$^{n}$. Which of the following statements are true?