CSIR NET Maths previous year question papers

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?

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?

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

For a real number x, let [x] denote the largest integer ≤ x. Which of the following sets are uncountable?

Consider the disc 𝔻 = {$z \in \mathbb{C}$ : |z| < 1} and a non-constant holomorphic function f : 𝔻 → 𝔻. Suppose that f(0) = 0. For each n ≥ 1 and z ∈ 𝔻, define $f_{n}(z) = f(z^{n})$. Which of the following statements are true?

Which of the following quotient rings are fields?

Consider $\mathbb{R}^{2}$ with the Euclidean metric d. Let $O = (0, 0), P_{0} = (0, 1) \in \mathbb{R}^{2}$, and for any integer $n \ge 1, P_{n} = (1/n, 1) \in \mathbb{R}^{2}$. Let X = ⋃$(n\ge0) L_{n}$, where $L_{n}$ is the line segment joining $P_{n}$ and O. Define $d_X : X \times X \to \mathbb{R}$ as follows: d_X(a, b) = d(a, b) when $a, b \in L_{n}$ for some n, and d_X(a, b) = d(a, O) + d(b, O) otherwise. Let $\tau$ be the smallest topology such that the sets $B_(a,\varepsilon) =$ {$b \in X : d_X(a, b) < \varepsilon$} are open in $\tau$ for all a ∈ X and $\varepsilon > 0$. Which of the following statements are true?

Let X and Y be independent random variables with the moment generating functions M_X(t) = (1/9)(2 + eᵗ$)^{2}, t \in \mathbb{R}$ and M_Y(t) = exp[eᵗ $- 1], t \in \mathbb{R}$, respectively. Then, which of the following statements are true?

Which of the following rings are integral domains?

For a finite group G, let S(G) denote the set of all its Sylow subgroups. A finite group G is said to have property (J) if G is isomorphic to the direct product $\prod(H\in{}S(G)) H$. Which of the following statements are true?

Let R(x, t) and u(x) denote the resolvent kernel and the solution, respectively, of the Volterra integral equation u(x) = eˣ $+ 2\int_(\ln 6)^x e^(2(t-x))u(t)$dt. Then which of the following statements are true?

Let $g : \mathbb{R} \to \mathbb{R}$ be a continuous function. Define $f(x) = \int_{0}$ˣ(x − t)g(t)dt, $x \in \mathbb{R}$. Which of the following statements are true?