CSIR NET Maths previous year question papers

Consider an M/G/1 queuing system with arrival rate $\lambda = 1$ and independent and identically distributed successive service times having probability density function g(x) = xe⁻ˣ if x > 0, and 0 otherwise. Define, for $i = 1, 2, \dots, I_{i} = 1$ if the first transition is from i to i − 1, and $I_{i} = 0$ if the first transition from i is i + 1. Then, $Var(I_{2})$ equals

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 a gamma distribution with shape parameter $\alpha > 0$ and scale parameter $\beta = 1$. For a suitable constant C, the rejection region of the most powerful test for testing $H_{0} : \alpha = 1$ against $H_{1} : \alpha = 2$ is of the form

Suppose $\varphi$ is a most powerful test of size 0.05 for testing a simple null hypothesis $H_{0}$ against a simple alternative hypothesis $H_{1}$. If the power of the test is 0.4, 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

Let $X_{1}, X_{2}, \dots, X_{5}$ be a random sample of size 5 from an absolutely continuous distribution having median M. Let S denote the number of $X_{i}'s$ greater than 0. For testing $H_{0} : M = 0$ against $H_{1} : M > 0$, let $\varphi(X) = 1$ if $S > c, \nu$ if S = c, and 0 if S < c be a test of size $\alpha = 0.05$, where $\nu \in [0, 1]$ and c ∈ {−1, 0, …, 5} are fixed constants. Then $c + \nu$ equals

Consider a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1)$, where the errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Here, $Y_{i}$ is the i-th response. Let Ŷ$_{i}$ be the i-th predicted response by the least squares estimation method, and let $\varepsilon$̂$_{i} = Y_{i} -$ Ŷ$_{i}, 1 \le i \le n$. Then, which of the following statements is true?

Let X, Y and Z be random variables such that S = [[X, Y], [Y, Z]] ~ $W_{2}(10, \sum)$, where $W_{2}$ denotes the Wishart distribution and $\sum = [[1, 1/2], [1/2, 1]]$. Define $T = Z - Y^{2}/X$. Then, Var(T) equals

Let P be the population proportion of units possessing a certain attribute in a population of N units. Let p be the sample proportion in a simple random sample (without replacement) of n units, (2 ≤ n < N). Then an unbiased estimator of P(1 − P) 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 S = {$x \in \mathbb{R} : x > 1$ and $(1 - x^{4})/(1 - x^{3}) > 22$}. Which of the following is true about S?

Let C be the collection of all sets S such that the power set of S is countably infinite. Which of the following statements is true?

Let $(a_{n})_{n}\ge1$ be a bounded sequence in $\mathbb{R}$. Which of the following statements is FALSE?

What is the cardinality of the set of real solutions of eˣ + x = 1?

Let S be a dense subset of $\mathbb{R}$ and $f : \mathbb{R} \to \mathbb{R} a$ given function. Define $g : S \to \mathbb{R}$ by g(x) = f(x). Which of the following statements is necessarily true?

For each n ≥ 1 define $f_{n} : \mathbb{R} \to \mathbb{R}$ by $f_{n}(x) = x^{2} / \sqrt{x^{2} + 1/n}, x \in \mathbb{R}$, where $\surd$ denotes the non-negative square root. Wherever $lim_{n}\to\infty f_{n}(x)$ exists, denote it by f(x). Which of the following statements is true?

Let $A : \mathbb{R}^{m} \to \mathbb{R}^{n}$ be a non-zero linear transformation. Which of the following statements is true?

Let A be a 10 × 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true?

Let ((2, a), (b, c)) be a 2 × 2 real matrix for which 6 is an eigenvalue. Which of the following statements is necessarily true?

Let V be the real vector space of 2 × 2 matrices with entries in $\mathbb{R}$. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where A = ((2, 0), (0, 1)). What is the characteristic polynomial of T?

Let A = ((0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 1, 1), (0, 0, 1, 1)), and consider the symmetric bilinear form on $\mathbb{R}^{4}$ given by ⟨v, w⟩ = vᵗAw, for $v, w \in \mathbb{R}^{4}$. Which of the following statements is true?

For a quadratic form $f(x, y, z) \in \mathbb{R}[x, y, z]$, we say that $(a, b, c) \in \mathbb{R}^{3}$ is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)?

Let f be an entire function. Which of the following statements is FALSE?

Consider the contour $\gamma$ given by $\gamma(\theta) = e^(2i\theta)$ for $\theta \in [0, \pi/2]; \gamma(\theta) = 1 + 2e^(2i\theta)$ for $\theta \in [\pi/2, 3\pi/2]; \gamma(\theta) = e^(2i\theta)$ for $\theta \in [3\pi/2, 2\pi]$. Then what is the value of $\int_\gamma$ dz / (z(z − 2))?

Let a, b be two real numbers such that a < 0 < b. For a positive real number r, define $\gamma_r(t) =$ re^(it) (where $t \in [0, 2\pi])$ and $I_r = (1/2\pi{}i) \int_(\gamma_r) (z^{2} + 1) / ((z - a)(z - b))$ dz. Which of the following statements is necessarily true?