CSIR NET Maths previous year question papers

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?

For a complex number a such that 0 < |a| < 1, which of the following statements is true?

How many arrangements of the digits of the number 1234567 are there, such that exactly three of them occur in their original position? (E.g., in the arrangement 5214763, exactly the digits 2, 4 and 6 are in their original positions. In the arrangement 1243576, exactly the digits 1, 2 and 5 are in their original positions.)

The number of group homomorphisms from $\mathbb{Z}/150\mathbb{Z}$ to $\mathbb{Z}/90\mathbb{Z}$ is

Consider the ring R = { $\sum_(n\in\mathbb{Z}) a_{n}X^{n}$ | $a_{n} \in \mathbb{Z}$, and $a_{n} \ne 0$ only for finitely many $n \in \mathbb{Z}$ }, with the usual addition and multiplication of such sums. Which of the following statements is true?