CSIR NET Maths previous year question papers

Let A be a subring of the field of rationals $\mathbb{Q}$ such that for any nonzero rational $r \in \mathbb{Q}, r \in A$ or 1/r ∈ A. Which of the following statements is FALSE?

For a finite group G, let S(G) denote the number of subgroups of G. Which of the following statements is necessarily true?

Let h be a holomorphic function on $\mathbb{C}${0} such that $lim_{|z|\to\infty}$ h(z) = 0. For every n ≥ 1, consider $f_{n}(z) = \sum_{k=1}^n$ h(zᵏ). Let 𝔻 denote the open unit disc {$z \in \mathbb{C}$ : |z| < 1}. Which of the following statements is necessarily true?

For z = x + iy $\in \mathbb{C}$, let f(z) = u(x, y) + iv(x, y) define an entire function. Consider the function g(z) = u(x, −y) − iv(x, −y), for z = x + iy $\in \mathbb{C}$. Suppose that v(x, 0) = 0 for all $x \in \mathbb{R}$. Define E = {$z \in \mathbb{C}$ | f(z) = g(z)}. Which of the following statements is true?

Let p be a prime number. An element a of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})$ˣ is said to be a primitive root in $(\mathbb{Z}/p\mathbb{Z})$ˣ if the order of a in $(\mathbb{Z}/p\mathbb{Z})$ˣ is p − 1. Let S_p be the number of primitive roots in $(\mathbb{Z}/p\mathbb{Z})$ˣ and $\varphi$ denote the Euler $\varphi-$function. Which of the following statements is true?

Let G be a group of order n > 3 and H be a subgroup with 1 < |H| < n. Consider the set X = ⋃(g∈G) gHg⁻$^{1}$. 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)?

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?