CSIR NET Maths previous year question papers

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 {$X_{n}$ | n ≥ 0} be a homogeneous Markov chain with state space S = {0, 1, 2, 3, 4} and transition probability matrix

Let $\alpha$ denote the probability that starting with state 4 the chain will eventually get absorbed in closed class {0, 3}. Then the value of $\alpha$ is

Consider a petrol pump which has a single petrol dispensing unit. Customers arrive there in accordance with a Poisson process having rate $\lambda = 1$ minutes. An arriving customer enters the petrol pump only if there are two or less customers in the petrol pump, otherwise he/she leaves the petrol pump without taking the petrol (at any point of time a maximum of three customers are present in the petrol pump). Successive service times of the petrol dispensing unit are independent exponential random variables having mean 1/2 minutes. Let X denote the average number of customers in the petrol pump in the long run. Then E(X) is equal to

Let a point P be chosen at random on the line segment AB of length $\alpha$. Let $Z_{1}$ and $Z_{2}$ denote the lengths of line segments AP and BP respectively. Then the value of E(|$Z_{1} - Z_{2}$|) is

Consider a distribution with probability mass function f(x | $\theta) = (1 - \theta)/2$ if x = 0; 1/2 if $x = 1; \theta/2$ if x = 2; and 0 otherwise, where $\theta \in (0, 1)$ is an unknown parameter. In a random sample of size 100 from the above distribution, the observed counts of 0, 1 and 2 are 20, 30 and 50 respectively. Then, the maximum likelihood estimate of $\theta$ based on the observed data is

Let $X_{1}, \dots, X_{10}$ be a random sample from a distribution with the probability density function f(x|$\theta) = \theta{}x^(\theta-1)$ if 0 < x < 1, and 0 otherwise, where $\theta > 0$ is an unknown parameter. The prior distribution of $\theta$ is given by $\pi(\theta) = \theta{}e^(-\theta)$ if $\theta > 0$, and 0 otherwise. The Bayes estimator of $\theta$ under squared error loss is

Let $X_{1}, X_{2}$ be a random sample from $N(0, \sigma^{2})$ distribution, where $\sigma > 0$ and $N(\mu, \sigma^{2})$ denotes a normal distribution with mean $\mu$ and variance $\sigma^{2}$. Suppose, for some constant $c, (c(X_{1}^{2} + X_{2}^{2}), \infty)$ is a confidence interval for variance $\sigma^{2}$ with confidence coefficient 0.95. Then the value of c 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

An analyst considers standardized values of observations on three variables, consumption (C), saving (S) and total income (TI) so that they have zero means and unit variances. She further considers disposable income (DI) where DI = C + S. In the simple linear regressions of DI on TI, DI on C and S on TI, the regression coefficients are 0.8, 0.5 and 0.4, respectively. There are 21 sample observations. Sample covariances and variances are calculated with divisor 20. Then, the value of sum of squared residuals in the regression of DI on S is

Let $X_{0}, X_{1}, \dots, X_p (p \ge 2)$ be independent and identically distributed random variables with mean 0 and variance 1. Suppose $Y_{i} = X_{0} + X_{i}, i = 1, \dots, p$. The first principal component based on the covariance matrix of $Y = (Y_{1}, \dots, Y_p)^{T}$ is

The expected number of distinct units in a simple random sample of 3 units drawn with replacement from a population of 100 units is

Consider maximizing the objective function $P = x_{1} + 2x_{2} + 3x_{3}$ subject to $x_{1} + x_{2} \le 5, x_{2} + x_{3} \le 2, x_{1} + x_{2} + x_{3} \le 6, x_{1} \ge 0, x_{2} \ge 0, x_{3} \ge 0$. Then, which of the following statements are true?

For a real number x, let [x] denote the largest integer ≤ x. Which of the following sets are uncountable?

Let A = {$1/(\sqrt{n + \sqrt{n}} - \sqrt{n})$ | $n \in \mathbb{Z}, n > 0$}. Which of the following statements are true?

Let $f : [0, \infty) \to [0, \infty)$ be a continuous function such that $\lim(x\to\infty) f(x)/x = \alpha < 1$. Let S = {$c \in [0, \infty)$ | f(c) = c}. Which of the following statements are true?

Let $f : \mathbb{R} \to \mathbb{R}$ be a uniformly continuous function. For each positive integer n and $x \in \mathbb{R}$, let $g_{n}$ and $h_{n}$ be defined by $g_{n}(x) = f(x + 1/n)$ and $h_{n}(x) = f($nx). Which of the following statements are necessarily true?

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?

For each n > 1, let V denote the $\mathbb{C}-$vector space of all n × n complex matrices and A ∈ V. Which of the following statements are necessarily true?

For each n > 1, let V be the $\mathbb{R}-$vector space of all n × n real matrices and A ∈ V be invertible. Consider the $\mathbb{R}-$linear transformation $\varphi : \mathbb{R}[x] \to V$ such that $\varphi(x^k) = A^k$ for all k ≥ 1 and $\varphi(1)$ is the identity matrix of order n. Which of the following statements are necessarily true?

Let $T : \mathbb{C}^{2} \to \mathbb{C}^{2}$ be $a \mathbb{C}-$linear transformation. For any ordered basis ℬ of $\mathbb{C}^{2}$, let [T]ℬ denote the matrix of T with respect to ℬ. Suppose that ℬ$_{1}$ and ℬ$_{2}$ are two ordered bases of $\mathbb{C}^{2}$ such that the matrix [T](ℬ$_{1})$ is upper-triangular and [T]_(ℬ$_{2}) = ([T]_($ℬ$_{1}))^{2}$. Which of the following statements are FALSE?

Let $T : \mathbb{R}^{4} \to \mathbb{R}^{4}$ be an $\mathbb{R}-$linear transformation. Assume that the characteristic polynomial of T has two distinct monic irreducible quadratic factors $q_{1}(x)$ and $q_{2}(x)$. Which of the following statements are true?

Let (V, ⟨ , ⟩) be an inner product space over $\mathbb{C}$ and T : V → V be $a \mathbb{C}-$linear transformation. Let $v_{1}$ and $v_{2}$ be non-zero vectors in V such that Tv$_{1} = c_{1}v_{1}$ and Tv$_{2} = c_{2}v_{2}$ for some $c_{1}, c_{2} \in \mathbb{C}$. Let $v_{3} = v_{2} - ($⟨$v_{2}, v_{1}$⟩/‖$v_{1}$‖$^{2})v_{1}$. Suppose that $v_{3}$ is non-zero and Tv$_{3} = c_{3}v_{3}$ for some $c_{3} \in \mathbb{C}$. Which of the following statements are true?

Let V be a two-dimensional real vector space with a basis {$v_{1}, v_{2}$}. Let $f : V \times V \to \mathbb{R}$ be a symmetric bilinear form. Let $r = f(v_{1}, v_{1}), s = f(v_{1}, v_{2}), t = f(v_{2}, v_{2})$. Which of the following statements are necessarily true?

For z = x + iy $\in \mathbb{C}$, let f(z) = u(x, y) + iv(x, y) be an entire function such that u(x, y) + v(x, y) = 1. Which of the following statements are FALSE?

Consider the disc 𝔻 = {$z \in \mathbb{C}$ : |z| < 1} and a non-constant holomorphic function f : 𝔻 → 𝔻. Suppose that f(0) = 0. For each n ≥ 1 and z ∈ 𝔻, define $f_{n}(z) = f(z^{n})$. Which of the following statements are true?