CSIR NET Maths previous year question papers

Consider $\mathbb{R}$ and $\mathbb{Q}[x]$ as vector spaces over $\mathbb{Q}$. Which of the following statements are true?

Let $T : \mathbb{R}^{4} \to \mathbb{R}^{4}$ be a linear map with four distinct eigenvalues and satisfying $T^{4} - 15T^{2} + 10T + 24I = 0$. Which of the following statements are necessarily true?

Let A be a 4 × 4 real matrix whose minimal polynomial is $x^{2} + x + 1$ and let $B = A + I_{4}$. Which of the following statements are necessarily true?

Let $M_{5}(\mathbb{C})$ be the complex vector space of 5 × 5 matrices with entries in $\mathbb{C}$. Let V be a non-zero subspace of $M_{5}(\mathbb{C})$ such that every non-zero A ∈ V is invertible. Which among the following are possible values for the dimension of V?

Let V (≠ {0}) be a finite dimensional vector space over $\mathbb{R}$ and T : V → V be a linear operator. Suppose that the kernel of T equals the image of T. Which of the following statements are necessarily true?

Let $q_{1}(x_{1}, x_{2})$ and $q_{2}(y_{1}, y_{2})$ be real quadratic forms such that there exist $(u_{1}, u_{2}), (v_{1}, v_{2}) \in \mathbb{R}^{2}$ such that $q_{1}(u_{1}, u_{2}) = 1 = q_{2}(v_{1}, v_{2})$. Define $q(x_{1}, x_{2}, y_{1}, y_{2}) = q_{1}(x_{1}, x_{2}) - q_{2}(y_{1}, y_{2})$. Which of the following statements are necessarily true?

Suppose that f is an entire function such that |f(z)| ≥ 2024 for all $z \in \mathbb{C}$. Which of the following statements are necessarily true?

Let f be an entire function such that for every integer k ≥ 1 there is an infinite set X_k such that f(z) = 1/k for all z ∈ X_k. Which of the following statements are necessarily true?

Which of the following numbers are order of some element of the symmetric group $S_{5}$?

Let R and S be non-zero commutative rings with multiplicative identities 1_R, 1_S, respectively. Let f : R → S be a ring homomorphism with f(1_R) = 1_S. Which of the following statements are true?

Let I be an ideal of the ring 𝔽$_{2}[t]/(t^{2}(1 - t)^{2})$. Which of the following are the possible values for the cardinality of I?

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

The infimum of the set {$\int_{a}$ᵇ $\sqrt{1 + (y'(t))^{2}}$dt : $y \in C^{1}[a, b], y(a) = a^{2}, y(b) = b - 5$} is

For $c \in \mathbb{R}$, consider the following Fredholm integral equation y(x) = 1 + x + cx$^{2} + 2\int_{0}^{1} (1 - 3$xt)y(t)dt. Then the values of c for which the integral equation admits a solution are

Consider a solid torus of constant density $\rho$, formed by revolving the disc $(y - b)^{2} + z^{2} \le a^{2}, x = 0$ about the z-axis, where 0 < a < b. Then the moment of inertia of the solid torus about the z-axis is

Let X and Y be independent random variables with X ~ N(2, 4) and Y ~ N(−4, 9) where $N(\mu, \sigma^{2})$ denotes a normal distribution with mean $\mu$ and variance $\sigma^{2}$. Given $\Phi(1) = 0.8413, \Phi(2) = 0.9772$ and $\Phi(3) = 0.9987$ where $\Phi(\cdot)$ is the cumulative distribution function of a standard normal random variable. Which of the following statements are true?

Transition probability matrix of a homogeneous Markov chain with states 0, 1, 2, 3 is P = [[1/4, 3/4, 0, 0], [1, 0, 0, 0], [2/3, 0, 1/3, 0], [0, 0, 2/5, 3/5]], where rows and columns are indexed 0, 1, 2, 3. Which of the following statements are true?

Let X and Y be jointly distributed continuous random variables with joint probability density function f(x, y) = x/y if 0 < x < y < 2, and 0 otherwise. Which of the following statements are true?

Let $X_{1}, X_{2}$ denote lifetimes (in years) of 2 components of an electronic system. Let $Y_{1} = X_{1} + X_{2}, Y_{2} = \max${$X_{1}, X_{2}$} and $Y_{3} = \min${$X_{1}, X_{2}$}. Assume that $X_{1}$ and $X_{2}$ are independent, each following exponential distribution with probability density function f(x) = (1/2)e^(−x/2) if x > 0, and 0 otherwise. Which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}$ be a random sample from a continuous distribution having cumulative distribution function F(t), probability density function f(t), and failure rate function r(t) = f(t)/(1 − F(t)), t > 0, where F(0) = 0. If r(t) = 1 for all t > 0, then which of the following statements are true?

Let $X_{1}, \dots, X_{n} (n \ge 2)$ be a random sample from $a U(-\theta, 2\theta)$ distribution, where $\theta > 0$ is an unknown parameter. Let ${\bar{X}} = (1/n)\sum_{i}$₌$_{1}^{n}X_{i}, X_{(1)} = \min${$X_{1}, \dots, X_{n}$} and $X_{(n)} = \max${$X_{1}, \dots, X_{n}$}. Which of the following statements are true?

Let $Y_{1}, \dots, Y_{n} (n \ge 2)$ be independent observations; $Y_{i}$ ~ $N(\beta{}x_{i}, \sigma^{2}), i = 1, \dots, n$; where $x_{1}, \dots, x_{n}$ and $\sigma^{2}(> 0)$ are known constants and $\beta \in \mathbb{R}$ is an unknown parameter. Consider $N(\beta_{0}, \tau^{2})$ prior for the parameter $\beta$, where $\beta_{0}$ and $\tau^{2}(> 0)$ are known constants, and $N(\mu, \lambda^{2})$ denotes a normal distribution with mean $\mu$ and variance $\lambda^{2}$. Suppose ȳ $= (1/n)\sum_{i}$₌$_{1}^{n}y_{i}$ and ${\bar{x}} = (1/n)\sum_{i}$₌$_{1}^{n}x_{i}$ are observed sample means. Under squared error loss function, which of the following statements are true?

Let $X_{1}, \dots, X_{12}$ be a random sample from the N(2, 4) distribution and $Y_{1}, \dots, Y_{15}$ be a random sample from the N(−2, 5) distribution, where $N(\mu, \sigma^{2})$ denotes a normal distribution with mean $\mu$ and variance $\sigma^{2}$. Assume that the two random samples are mutually independent. Let ${\bar{X}} = (1/12)\sum_{i}$₌$_{1}^{12}X_{i}, S_{1}^{2} = (1/11)\sum_{i}$₌$_{1}^{12}(X_{i} - {\bar{X}})^{2}$, Ȳ $= (1/15)\sum_{j}$₌$_{1}^{15}Y_{j}, S_{2}^{2} = (1/14)\sum_{j}$₌$_{1}^{15}(Y_{j} -$ Ȳ$)^{2}$. Which of the following statements are true?