CSIR NET Maths previous year question papers

Suppose that the probability density function of the random variable X is $f(x) = 2(\theta - x)/\theta^{2}$ if $0 \le x \le \theta$, and 0 otherwise, where $\theta > 0$ is an unknown parameter. Based on a single observation X, the confidence coefficient of the confidence interval [(2/5)X, (5/2)X] for $\theta$ is

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from distribution Poisson$(\theta), \theta > 0$. Let $\pi(\theta) = e^(-\theta), \theta > 0$, be the prior distribution of $\theta$. Under the squared error loss function, which of the following is the Bayes estimator of $e^(-2\theta)$?

Let X be a random variable with the probability density function $f(x) = 2\theta{}x + 2(1 - \theta)(1 - x)$ if 0 < x < 1, and 0 otherwise, where $\theta \in [0, 1]$. Based on single observation x, the critical region of the most powerful test for testing null hypothesis $H_{0} : \theta = 1/2$ against alternative hypothesis $H_{1} : \theta = 1$, at level of significance $\alpha = 0.25$, is

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 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 independent and identically distributed (i.i.d.) random variables with distribution N(0, 1). Define U = 2X, V = 3X + Y, W = X + 4Z. Then the partial correlation coefficient of V and W, given U is

Consider the following design where the columns represent blocks and the letters represent treatments — block 1: A, B; block 2: C, D; block 3: A, C; block 4: B, D; block 5: A, D; block 6: B, C; block 7: A, E; block 8: B, E; block 9: C, E; block 10: D, E. Then, which of the following statements is NOT true?

Let A, B, and C be sets. Which of the following sets is equal to A \ (B \ C)?

What is the number of injective functions from {1, 2, …, 7} to {1, 2, …, 10}?

For integers n ≥ 0, let $f_{n} : [-1, 0] \to \mathbb{R}$ be defined by $f_{n}(x) = x/(1 - x)^{n}$. Which of the following statements is true about the series $\sum_(n=0)^\infty f_{n}$?

Consider the sequences $(a_{n})_{n}\ge_{1}$ and $(b_{n})_{n}\ge_{1}$ defined by $a_{n} = (e^{n} + e$⁻$^{n})/2$ and $b_{n} = a_{n}$₊$_{1}/a_{n}$. Which of the following statements is true?

Let $f : [0, 1] \to \mathbb{R}$ be defined by $f(x) = \sin(x^{2})$. Let $A = \lim(n\to\infty) (\sum_(k=1)^n f(k/n) - n\int_{0}^{1}f(x)$dx). Which of the following statements is true?

Consider the power series $\sum_(n=1)^\infty [n^(n^{2})/(n + 1)^(n^{2})]x^{n}$ with coefficients in real numbers $\mathbb{R}$. Which of the following statements is true?

Let U denote the span of {eᵗ, $e^{2}$ᵗ, $e^{3}$ᵗ} in the real vector space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Consider the $\mathbb{R}-$vector spaces V = {$f : U \to \mathbb{R}$ | f is an $\mathbb{R}-$linear transformation} and W = {f ∈ V | $f(e^{3}$ᵗ) = 0}. Which of the following statements is true?

Let $v = (a, b, c) \in \mathbb{R}^{3}$ be a nonzero vector that lies in the orthogonal complement (with respect to the standard inner product) of the row-space of the matrix A = [[2, 2, 7], [3, 1, 4]]. If a, b, c are all integers, then what is the smallest possible value of |a + b + c|?

Let A = [[0, a, 0], [0, 0, b], [c, 0, 0]], where a, b, c are real numbers with abc = 1. If $B = A + A^{2} + A^{3}$, then which of the following statements is true?

For a variable x, consider the $\mathbb{R}-$vector space V = {$a_{0} + a_{1}x + a_{2}x^{2}$ | $a_{1}, a_{2}, a_{3} \in \mathbb{R}$}. Let T : V → V be the linear transformation defined by T(f) = f + df/dx, where df/dx denotes the derivative of f with respect to x. Which of the following statements is true?

Let V be the $\mathbb{R}-$vector space of 5 × 5 real matrices. Let S = {AB − BA | A, B ∈ V} and W denote the subspace of V spanned by S. Let $T : V \to \mathbb{R}$ be the linear transformation mapping a matrix A to its trace. Which of the following statements is true?

Consider the bilinear form $B : \mathbb{R}^{4} \times \mathbb{R}^{4} \to \mathbb{R}$ defined by $B(x, y) = x_{1}y_{3} + x_{2}y_{4} - x_{3}y_{1} - x_{4}y_{2}$, where $x = (x_{1}, x_{2}, x_{3}, x_{4})$ and $y = (y_{1}, y_{2}, y_{3}, y_{4})$ in $\mathbb{R}^{4}$. Let A denote the matrix of B with respect to the standard ordered basis of $\mathbb{R}^{4}$. Which of the following statements is true?

Let $f : \mathbb{C} \to \mathbb{C}$ be the function defined by f(z) = e^((cos(1+i)) sin z). For z = x + iy $\in \mathbb{C}$, write f(z) as u(x, y) + iv(x, y), where u, v are real-valued functions. Which of the following is the value of $(\partial{}u/\partial{}x)(0, 0)$?

Let 𝔻 = {z = x + iy $\in \mathbb{C}$ : |z| < 1} be the open unit disc and f : 𝔻 $\to \mathbb{C} a$ holomorphic function such that f(0) = 0. Let $\psi(z) =$ |f(z)|$^{2}$, and $\partial^{2}\psi/\partial{}x^{2} + \partial^{2}\psi/\partial{}y^{2} \equiv 0$. Which of the following statements is FALSE?

Let ℍ = {z = x + iy $\in \mathbb{C}$ | y > 0} and f : ℍ $\to \mathbb{C}$ be a non-constant holomorphic function satisfying |f(z)| < 1 for all z ∈ ℍ. Which of the following statements is true?

For integers m, n ≥ 1, let $I_(m,n) = (1/2\pi{}i)\int_C z^{m} {\bar{z}}^{n}$ dz, where C is the circle {$z \in \mathbb{C}$ : |z| = 1} oriented counterclockwise. Which of the following statements is true?

For integers n > 1, let G(n) denote the number of groups of order n, up to isomorphism, i.e. G(n) is the number of isomorphism classes of groups of order n. Which of the following statements is true?

We say that a group G has property (A) if every non-trivial homomorphism from G to any group is injective. Which of the following groups has property (A)?

Let $\mathbb{C}[x, y]$ be the polynomial ring in two variables over $\mathbb{C}$. For which of the following ideals I, the quotient ring $\mathbb{C}[x, y]/I$ is NOT an integral domain?