CSIR NET Maths previous year question papers

For two indeterminates x, y, let R = 𝔽$_{3}[x]$ and S = R[y]. Which of the following statements are true?

For which of the following values of q, does a finite field of order q have exactly 6 subfields?

Let X denote the topological space $\mathbb{R}$ with the cofinite topology (i.e., the finite complement topology) and let Y denote the topological space $\mathbb{R}$ with the Euclidean topology. Which of the following statements are true?

Let $\tau$ be the smallest topology on the set $\mathbb{R}$ containing $\beta =$ {[a, b) | $a < b; a, b \in \mathbb{R}$}. Which of the following statements are true?

Consider the initial value problem (IVP$) y'(x) = \sin(y(x))/(1 + y^{4}(x))$ for $x \in \mathbb{R}$, with $y(0) = y_{0}$. Then which of the following statements are true?

Consider the boundary value problem (BVP) (e^(−5x)y′)′ + 6e^(−5x)y = −f(x), 0 < x < ln 2, with y(0) = 0, y(ln 2) = 0. If $G(x, \xi) = (e^(3x) +$ Be^(2x))(Ce$^(2\xi) +$ De$^(3\xi))$ for $0 \le \xi \le x$, and $G(x, \xi) = (e^(3\xi) +$ Be$^(2\xi))($Ce^(2x) + De^(3x)) for $x \le \xi \le \ln 2 ($Green's function) is such that $\int_{0}^(\ln 2) G(x, \xi)f(\xi)d\xi$ is the solution of the BVP, then the values of B, C and D are

Let S denote the set of all 2 × 2 matrices A such that the iterative sequence generated by the Gauss-Seidel method applied to the system of linear equations $A(x_{1}, x_{2})^{T} = (2, 3)^{T}$ converges for every initial guess. Then which of the following statements are true?

Let g(x) be the polynomial of degree at most 4 that interpolates the data (x, y) = (−1, −30), (0, 1), (2, c), (3, 10), (6, 19). If g(4) = 5, then which of the following statements are true?

The extremizer of the problem $\min[(1/2)\int$₋$_{1}^{1} ((y'(x))^{2} + (y(x))^{2})$dx] subject to $y \in C^{1}[-1, 1], \int$₋$_{1}^{1}$ xy(x)dx = 0 and y(−1) = y(1) = 1 is

For $\lambda \in \mathbb{R}$ such that |$\lambda$| < 5/32, let $R(x, t, \lambda)$ and u denote the resolvent kernel and the solution, respectively, of the Fredholm integral equation $u(x) = x + (\lambda/2)\int$₋$_{2}^{2} ($xt $+ x^{2}t^{2})u(t)$dt. Then which of the following statements are true?

Let {$X_{n}$}$_{n}\ge_{1}$ be a sequence of independent and identically distributed random variables with $E(X_{1}) = 0$ and $Var(X_{1}) = 1$. Which of the following statements are true?

Let $X_{1}, \dots, X_{n}$ be independent and identically distributed $U(0, \theta), \theta > 0$ random variables. Define $X_{(n)} = \max${$X_{1}, \dots, X_{n}$} and $X_{(1)} = \min${$X_{1}, \dots, X_{n}$}. Which of the following statements are true?

Let $X_{1}, \dots, X_{n} (n \ge 3)$ be a random sample from a distribution having probability density function f(x | $\theta) = \theta{}e^(-\theta{}x)$ if x > 0, and 0 otherwise, where $\theta > 0$ is an unknown parameter. Let $T_{n} = (1/n)\sum_{i}$₌$_{1}^{n}X_{i}$. Which of the following statements are true?

Consider a six faced die whose i-th face is marked with i dots, i = 1, 2, …, 6. In a single random throw of the die, let $p_{i}$ denote the probability that the obtained upper face has i dots, i = 1, 2, …, 6. The die is rolled 240 times independently and the following result is obtained — face observed 1, 2, 3, 4, 5, 6 with frequencies 40, 55, 40, 25, 35, 45 respectively. Suppose we want to test $H_{0} : p_{i} = 1/6$ for i = 1, 2, …, 6; against $H_{1} : p_{i} \ne 1/6$ for at least one i. It is given that $\chi^{2}_{5};_{0}._{05} = 11.07, \chi^{2}_{6};_{0}._{05} = 12.59, \chi^{2}_{5};_{0}._{01} = 15.09, \chi^{2}_{6};_{0}._{01} = 16.81$. Based on the asymptotic goodness of fit $\chi^{2}$ test for testing $H_{0}$ against $H_{1}$, which of the following statements are true?

Observations on the shear strength of concrete from 5 randomly selected structures are given below — structure 1, 2, 3, 4, 5 with shear strength 1718.4, 1787.4, 2562.3, 2356.9, 2153.2 respectively. The null hypothesis $H_{0}$ that the median shear strength is 2000 units is tested against the alternative hypothesis $H_{1}$ that the median shear strength is greater than 2000 units at 5% level of significance. Which of the following statements are true?

Consider the following ANOVA table for a randomized block design — Treatments: sum of squares 48, degrees of freedom 4, mean squares 12, F calculated $\beta$; Blocks: sum of squares 72, degrees of freedom 3, mean squares 24, F calculated 12; Error: sum of squares $\alpha$, degrees of freedom m, mean squares $\gamma$; Total: sum of squares 144, degrees of freedom 19. Which of the following statements are true?

Consider the series $\sum_{n=1}^\infty$ $a_{n}$, where $a_{n} =$ $(-1)^{n+1}(\sqrt{n+1} - \sqrt{n})$. Which of the following statements is true?

Which of the following assertions is correct?

Which one of the following functions is uniformly continuous on the interval (0, 1)?

Let A be a 3×3 real matrix whose characteristic polynomial p(T) is divisible by $T^{2}$. Which of the following statements is true?

Let T be a linear operator on $\mathbb{R}^{3}$. Let $f(X) \in \mathbb{R}[X]$ denote its characteristic polynomial. Consider the following statements. (a) Suppose T is non-zero and 0 is an eigenvalue of T. If we write f(X) = X·g(X) in $\mathbb{R}[X]$, then the linear operator g(T) is zero. (b) Suppose 0 is an eigenvalue of T with at least two linearly independent eigenvectors. If we write f(X) = X·g(X) in $\mathbb{R}[X]$, then the linear operator g(T) is zero. Which of the following is true?

Let $x = (x_{1}, \dots, x_{n})$ and $y = (y_{1}, \dots, y_{n})$ denote vectors in $\mathbb{R}^{n}$ for a fixed n ≥ 2. Which of the following defines an inner product on $\mathbb{R}^{n}$?

Let p be a prime number. Let G be a group such that for each g ∈ G there exists an $n \in \mathbb{N}$ such that $g^{p^{n}}$ = 1. Which of the following statements is FALSE?

The number of solutions of the equation $x^{2} = 1$ in the ring $\mathbb{Z}/105\mathbb{Z}$ is

How many real roots does the polynomial $x^{3} + 3x - 2023$ have?