CSIR NET Maths previous year question papers

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 variable x, consider the real vector space V = {ax$^{3} +$ bx$^{2} +$ cx + d | $a, b, c, d \in \mathbb{R}$}. Let D : V → V be the linear transformation where D(f) is the derivative of f with respect to x, and M : V → V be the linear transformation M(f) = xD(f). Which of the following statements are true?

Let A, B be 2 × 2 matrices with real entries, and M = AB − BA. Let $I_{2}$ denote the 2 × 2 identity matrix. Which of the following statements are necessarily true?

Let $M_{2}(\mathbb{R})$ denote the $\mathbb{R}-$vector space of 2 × 2 matrices with real entries. Let A = [[1, 2], [0, 3]] and B = [[−1, 0], [1, 5]]. Define a linear transformation $T : M_{2}(\mathbb{R}) \to M_{2}(\mathbb{R})$ by T(X) = AXBᵗ, where Bᵗ denotes the transpose of the matrix B. 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?

Let R be a nonzero ring with unity such that $r^{2} = r$ for all r ∈ R. Which of the following statements are true?

Let $f \in \mathbb{R}[x]$ be a product of distinct monic irreducible polynomials $P_{1}, P_{2}, \dots, P_{n}$, where n ≥ 2. Let (f) denote the ideal generated by f in the ring $\mathbb{R}[x]$. 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

Suppose that a sequence of random variables {$X_{n}$}$_{n}\ge_{1}$ and the random variable X are defined on the same probability space. Then which of the following statements are true?

Let {(X_k, Y_k)}(k≥1) be a sequence of independent and identically distributed (i.i.d.) random vectors with common joint probability density function f(x, y) = e^(−y) if $0 < x < y < \infty$, and 0 otherwise. For n = 1, 2, 3, …, let $N_{n}$ be a random variable denoting the number of elements in the set {k : k = 1, 2, …, n; Y_k ≥ 2}. Then, which of the following statements are true?

Consider the Markov chain with state space {0, 1, 2} and the transition probability matrix P = [[0, 1/2, 1/2], [3/4, 0, 1/4], [3/4, 1/4, 0]]. Let P⁽$^{n}$⁾ = ((P⁽$^{n}$⁾$_{ij}))$ denote the n-step transition probability matrix. Then, which of the following statements are true?

Let $Z_{1}$ and $Z_{2}$ be two independent discrete random variables such that $Z_{1}$ follows binomial distribution with parameters n = 2 and p = 1/2, and $Z_{2}$ follows Poisson distribution with mean 1. Consider the following system of equations with three variables $x_{1}, x_{2}$ and $x_{3}: x_{1} - 2x_{2} + x_{3} = 1; 2x_{1} - 5x_{2} + 2x_{3} = 2; x_{1} + 2x_{2} + Z_{1}x_{3} = Z_{2}$. Then, the probability that the given system of equations has infinite number of solutions equals

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from an exponential distribution with probability density function f(x|$\theta) = e^(-(x-\theta))$ if $x \ge \theta$, and 0 otherwise, where $\theta \in \mathbb{R}$ is an unknown parameter. If $X_{(1)} = \min${$X_{1}, X_{2}, \dots, X_{n}$}, 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 the unknown parameter. If $T_{n} = \min${$X_{1}, X_{2}, \dots, X_{n}$}, n = 1, 2, …, then which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from $N(\theta, 1)$ distribution, where $\theta \in \mathbb{R}$ is unknown. Let $\delta_{n}$ be the Bayes estimator of $\theta$, under the squared error loss function $L(\theta, a) = (a - \theta)^{2}, a, \theta \in \mathbb{R}$ and the prior distribution N(1, 2). If $(1/\sqrt{n})[(2n + 1)\delta_{n} - 1 - 2n\theta]$ converges in distribution to a random variable Z, as $n \to \infty$, then which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}$ be a random sample from Uniform$[0, \theta]$ distribution, $\theta > 0$. Consider the likelihood ratio test of size 0.001 for testing $H_{0} : \theta = 3$ against $H_{1} : \theta \ne 3$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}$ be a random sample of size 3 from an absolutely continuous distribution that is symmetric about 0. For i = 1, 2, 3, let $R_{i}$ denote the rank of |$X_{i}$| among |$X_{1}$|, |$X_{2}$| and |$X_{3}$|. If T⁺ $= \sum(i : X_{i}>0) R_{i}$ is the Wilcoxon signed-rank statistic, 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 errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Let $\beta$̂$_{i}$ be the best linear unbiased estimator (BLUE) of $\beta_{i}, i = 1, 2, \dots, n$. 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?

Let $(X_{1}, X_{2})$ be a bivariate normal random vector with $E(X_{1}) = 1, E(X_{2}) = 0, Var(X_{1}) = 1, Var(X_{2}) = 1$, and correlation coefficient 1/2. Let U be a U(0, 1) random variable, which is independent of $(X_{1}, X_{2})$. If Z = (UX$_{1} + X_{2} - U)/\sqrt{U^{2} + U + 1}$, then which of the following statements are true?

Which of the following statements are true?