CSIR NET Maths previous year question papers

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 u = u(x, y) be the solution of the Cauchy problem $x \partial{}u/\partial{}x + y \partial{}u/\partial{}y = u, (x, y) \ne (0, 0)$, with $u(x, 1) = \sqrt{1 + x^{2}}, x \in \mathbb{R}$. Then 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?

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?

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)?

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?

If u = u(x, t) is the solution of the initial value problem $u_{t} = u_{xx}$ for $x \in \mathbb{R}, t > 0$, with u(x, 0) = sin(4x) + x + 1 for $x \in \mathbb{R}$, satisfying |u(x, t)| $< 3e^(x^{2})$ for all $x \in \mathbb{R}$ and t > 0, then

Let u be the solution of the Volterra integral equation $\int_{0}$ᵗ [½ $+ \sin(t - \tau)] u(\tau) d\tau = \sin t$. Then the value of u(1) 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