CSIR NET Maths previous year question papers

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?

Consider the half planes R₊ = {z = x + iy $\in \mathbb{C} : x > 0$} and R₋ = {z = x + iy $\in \mathbb{C} : x < 0$}, and the fractional linear transformations $T_{1}(z) = (z - 1)/(z + 1)$ and $T_{2}(z) = (z + 1)/(z - 1)$. Let disc 𝔻 = {$z \in \mathbb{C}$ : |z| < 1}. Which of the following statements is true?

Let $\gamma$ be the circle {$z \in \mathbb{C}$ : |z| = 3} oriented counterclockwise. Let f be an entire function. What is the value of A for which $\int_\gamma (A/(z - 1) - f(z)/(z - 2)^{2})$dz = 0 holds?

Consider the entire function $f(z) = z^{2}(e$ᶻ − e⁻ᶻ) and the meromorphic function $g(z) = z^{2}/(e$ᶻ − e⁻ᶻ) on $\mathbb{C}$. 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?

Consider the ring homomorphism $\psi : \mathbb{C}[x, y] \to \mathbb{C}[t]$ defined by $\psi(f(x, y)) = f(t^{2}, t^{3})$. Which of the following statements is true?

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

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?

Which of the following statements is true?

Let f be an entire function. Which of the following statements is FALSE?

Consider the contour $\gamma$ given by $\gamma(\theta) = e^(2i\theta)$ for $\theta \in [0, \pi/2]; \gamma(\theta) = 1 + 2e^(2i\theta)$ for $\theta \in [\pi/2, 3\pi/2]; \gamma(\theta) = e^(2i\theta)$ for $\theta \in [3\pi/2, 2\pi]$. Then what is the value of $\int_\gamma$ dz / (z(z − 2))?

Let a, b be two real numbers such that a < 0 < b. For a positive real number r, define $\gamma_r(t) =$ re^(it) (where $t \in [0, 2\pi])$ and $I_r = (1/2\pi{}i) \int_(\gamma_r) (z^{2} + 1) / ((z - a)(z - b))$ dz. Which of the following statements is necessarily true?

For a complex number a such that 0 < |a| < 1, which of the following statements is true?

How many arrangements of the digits of the number 1234567 are there, such that exactly three of them occur in their original position? (E.g., in the arrangement 5214763, exactly the digits 2, 4 and 6 are in their original positions. In the arrangement 1243576, exactly the digits 1, 2 and 5 are in their original positions.)

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?

For z = x + iy $\in \mathbb{C}$, let f(z) = u(x, y) + iv(x, y) be an entire function such that u(x, y) + v(x, y) = 1. Which of the following statements are FALSE?

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?