Let $z$ be a complex variable, and write $x = \Re(z)$ and $y = \Im(z)$ for the real and the imaginary parts, respectively. Let $f(z)$ be a complex polynomial. Let $R > 0$ be a real number and $\gamma$ the circle in $\mathbb{C}$ of radius $R$ and centre at 0, oriented in the counterclockwise direction. What is the value of $$\frac{1}{2\pi \imath R} \int_{\gamma} \left( \Re(f(z))\,\mathrm{d}x + \Im(f(z))\,\mathrm{d}y \right)$$

Fix a non-negative integer $d$. Let $$\mathcal{A}_d := \{A \subseteq \mathbb{C} : A \text{ is the zero-set of a polynomial of degree } \leq d \text{ in } \mathbb{C}[X]\}.$$ Let $\mathcal{T}$ be the coarsest topology on $\mathbb{C}$ in which $A$ is closed for every $A \in \mathcal{A}_d$.
(A) Determine whether $\mathcal{T}$ is Hausdorff.
(B) Show that for every polynomial $f(X) \in \mathbb{C}[X]$, the function $\mathbb{C} \longrightarrow \mathbb{C}$ defined by $z \mapsto f(z)$ is continuous, where $\mathbb{C}$ (on both the sides) is given the topology $\mathcal{T}$.

Let $a_n,\ n \geq 0$ be complex numbers such that $\lim_n a_n = 0$.
(A) Show that $F(z) := \sum_{n \geq 0} a_n z^n$ is a holomorphic function on $\{z \in \mathbb{C} : |z| < 1\}$.
(B) Let $G(z)$ be a meromorphic function on $\{z \in \mathbb{C} : |z| < 2\}$, with a pole at 1. Show that $G \neq F$ on $\{z \in \mathbb{C} : |z| < 1\}$. (Hint: consider the function $(1-z)F(z)$ as $z \longrightarrow 1$.)

Let $|\cdot| : \mathbb{R} \longrightarrow \mathbb{R}_{\geq 0}$ be a function such that for every $x, y \in \mathbb{R}$, (i) $|x| = 0$ if and only if $x = 0$; (ii) $|x + y| \leq |x| + |y|$; (iii) $|xy| = |x||y|$. Show that the following are equivalent:
(A) The set $\{|n| : n \in \mathbb{Z}\}$ is bounded;
(B) $|x + y| \leq \max\{|x|, |y|\}$ for every $x, y \in \mathbb{R}$.

Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Show that the sequence $$\left[\int_0^1 |f(x)|^n\,\mathrm{d}x\right]^{\frac{1}{n}}$$ is convergent.

Let $V$ be a subspace of the complex vector space $M_n(\mathbb{C})$. Suppose that every non-zero element of $V$ is an invertible matrix. Show that $\dim_{\mathbb{C}} V \leq 1$.

Let $n$ be a positive integer such that every group of order $n$ is cyclic. Show the following.
(A) For all prime numbers $p$, $p^2$ does not divide $n$.
(B) If $p$ and $q$ are prime divisors of $n$, then $p$ does not divide $q - 1$. (Hint: Consider $2 \times 2$ matrices $$\left[\begin{array}{ll} x & y \\ 0 & 1 \end{array}\right]$$ with $x, y \in \mathbb{Z}/q\mathbb{Z}$ and $x^p = 1$.)
(C) Show that $(n, \phi(n)) = 1$, where $\phi(n)$ is the number of integers $m$ such that $1 \leq m \leq n$ with $\gcd(n, m) = 1$.

Let $F$ be a field and $G = \mathrm{GL}_n(F)$. For $g \in G$, write $C_g = \{hgh^{-1} \mid h \in G\}$. Let $X = \{C_g \mid g \in G,\ \text{the order of } g \text{ is } 2\}$. Determine $|X|$.

A compactification of a topological space $X$ is a compact topological space $Y$ which contains a dense subspace homeomorphic to $X$. Let $X = (0,1]$, in the subspace topology of $\mathbb{R}$ and $f : X \longrightarrow \mathbb{R},\ x \mapsto \sin\frac{1}{x}$. Show the following:
(A) $Y := [0,1]$ is a compactification of $X$, but $f$ does not extend to a continuous function $Y \longrightarrow \mathbb{R}$, i.e., there does not exist a continuous function $g : Y \longrightarrow \mathbb{R}$ such that $\left.g\right|_X = f$.
(B) $X$ is homeomorphic to the set $X_1 := \left\{\left.\left(t, \sin\frac{1}{t}\right)\right\rvert\, t \in X\right\} \subseteq \mathbb{R}^2$.
(C) The closure $Y_1$ of $X_1$ in $\mathbb{R}^2$ is a compactification of $X$.
(D) $f$ extends to a continuous function $Y_1 \longrightarrow \mathbb{R}$.

Let $f(X) \in \mathbb{Z}[X]$ be a monic polynomial. Suppose that $\alpha \in \mathbb{C}$ and $3\alpha$ are roots of $f$.
(A) Show that $f(0) \neq 1$. (Hint: if $\zeta$ and $\zeta'$ are complex numbers satisfying monic polynomials in $\mathbb{Z}[X]$, then $\zeta\zeta'$ satisfies a monic polynomial in $\mathbb{Z}[X]$.)
(B) Assume that $f$ is irreducible. Let $K$ be the smallest subfield of $\mathbb{C}$ containing all the roots of $f$. Let $\sigma$ be a field automorphism of $K$ such that $\sigma(\alpha) = 3\alpha$. Show that $\sigma$ has finite order and that $\alpha = 0$.

[12 points] For sets $S$ and $T$, a relation from $S$ to $T$ is just a subset $R$ of $S \times T$. If $(x, y)$ is in $R$, we say that $x$ is related to $y$. Answer the following. Part (i) is independent of (ii) and (iii).
(i) A relation $R$ from $S$ to $S$ is called antisymmetric if it satisfies the following condition: if $(a, b)$ is in $R$, then $(b, a)$ must NOT be in $R$. For $S = \{1, 2, \ldots, k\}$, how many antisymmetric relations are there from $S$ to $S$?
(ii) Write a recurrence equation for $f(k, n) =$ the number of non-crossing relations from $\{1, 2, \ldots, k\}$ to $\{1, 2, \ldots, n\}$ that have no isolated elements in either set. Your recurrence should have only a fixed number of terms on the RHS.
(iii) Using your recurrence in (ii) or otherwise, find a formula for $f(3, n)$.
Definition 1: We say that a relation from $S$ to $T$ has no isolated elements if each $s$ in $S$ is related to some $t$ in $T$ and if for each $t$ in $T$, some $s$ in $S$ is related to $t$.
Definition 2: We say that a relation $R$ from $\{1, 2, \ldots, k\}$ to $\{1, 2, \ldots, n\}$ is non-crossing if the following never happens: $(i, x)$ and $(j, y)$ are both in $R$ with $i < j$ but $x > y$.
Visual meaning: one can visualise a relation $R$ very similarly to a function. List 1 to $k$ as dots arranged vertically in increasing order on the left and similarly list 1 to $n$ on the right. For each $(s, t)$ in $R$, draw a straight line segment from $s$ on the left to $t$ on the right. In the situation one wants to avoid for non-crossing relations, the segments connecting $i$ with $x$ and $j$ with $y$ would cross. Having no isolated elements also has an obvious visual meaning.

Let $G$ be a group and $N$ be a proper normal subgroup. Pick the true statement(s) from below.
(A) If $N$ and the quotient $G / N$ is finite, then $G$ is finite.
(B) If the complement $G \backslash N$ of $N$ in $G$ is finite, then $G$ is finite.
(C) If both $N$ and the quotient $G / N$ are cyclic, then $G$ is cyclic.
(D) $G$ is isomorphic to $N \times G / N$.

Let $R$ denote the ring of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$, where addition and multiplication are given, respectively, by $(f + g)(x) = f(x) + g(x)$ and $(fg)(x) = f(x)g(x)$ for every $f, g \in R$ and $x \in \mathbb{R}$. A zero-divisor in $R$ is a non-zero $f \in R$ such that $fg = 0$ for some non-zero $g \in R$. Pick the true statement(s) from below:
(A) $R$ has zero-divisors.
(B) If $f$ is a zero-divisor, then $f^{2} = 0$.
(C) If $f$ is a non-constant function and $f^{-1}(0)$ contains a non-empty open set, then $f$ is a zero-divisor.
(D) $R$ is an integral domain.

Let $U = \left\{(x, y) \in \mathbb{R}^{2} \mid x < y^{2} < 4\right\}$ and $V = \left\{(x, y) \in \mathbb{R}^{2} \mid 0 < xy < 4\right\}$, both taken with the subspace topology from $\mathbb{R}^{2}$. Which of the following statement(s) is/are true?
(A) There exists a non-constant continuous map $V \longrightarrow \mathbb{R}$ whose image is not an interval.
(B) Image of $U$ under any continuous map $U \longrightarrow \mathbb{R}$ is bounded.
(C) There exists an $\epsilon > 0$ such that given any $p \in V$ the open ball $B_{\epsilon}(p)$ with centre $p$ and radius $\epsilon$ is contained in $V$.
(D) If $C$ is a closed subset of $\mathbb{R}^{2}$ which is contained in $U$, then $C$ is compact.

Let $A$ and $B$ be $5 \times 5$ real matrices with $A^{2} = B^{2}$. Which of the following statements is/are correct?
(A) Either $A = B$ or $A = -B$.
(B) $A$ and $B$ have the same eigen spaces.
(C) $A$ and $B$ have the same eigen values.
(D) $A^{13} B^{3} = A^{3} B^{13}$.

Consider the function $f : \mathbb{R}^{2} \longrightarrow \mathbb{R}$ given by
$$f(x, y) = \left(1 - \cos \frac{x^{2}}{y}\right) \sqrt{x^{2} + y^{2}}$$
for $y \neq 0$ and $f(x, 0) = 0$. (The square root is chosen to be non-negative). Pick the correct statement(s) from below:
(A) $f$ is continuous at $(0,0)$.
(B) $f$ is an open map.
(C) $f$ is differentiable at $(0,0)$.
(D) $f$ is a bounded function.

Which of the following is/are true for a series of real numbers $\sum a_{n}$?
(A) If $\sum a_{n}$ converges then $\sum a_{n}^{2}$ converges;
(B) If $\sum a_{n}^{2}$ converges then $\sum a_{n}$ converges;
(C) if $\sum a_{n}^{2}$ converges then $\sum \frac{1}{n} a_{n}$ converges;
(D) If $\sum |a_{n}|$ converges then $\sum \frac{1}{n} a_{n}$ converges;

Which of the following functions are uniformly continuous on $\mathbb{R}$?
(A) $f(x) = x$;
(B) $f(x) = x^{2}$;
(C) $f(x) = (\sin x)^{2}$;
(D) $f(x) = e^{-|x|}$.

Let $U$ and $V$ be non-empty open connected subsets of $\mathbb{C}$ and $f : U \longrightarrow V$ an analytic function. Which of the following statement(s) is/are true?
(A) $f^{\prime}(z) \neq 0$ for every $z \in U$.
(B) If $f$ is bijective, then $f^{\prime}(z) \neq 0$ for every $z \in U$.
(C) If $f^{\prime}(z) \neq 0$ for every $z \in U$, then $f$ is bijective.
(D) If $f^{\prime}(z) \neq 0$ for every $z \in U$, then $f$ is injective.

Let $U$ denote the unit open disc centred at 0. Let $f : U \backslash \{0\} \longrightarrow \mathbb{C}$ be an analytic function. Assume that $\lim_{z \longrightarrow 0} z f(z) = 0$.
(A) $\lim_{z \longrightarrow 0} |f(z)|$ exists and is in $\mathbb{R}$.
(B) $f$ has a pole of order 1 at 0.
(C) $zf(z)$ has a zero of order 1 at 0.
(D) There exists an analytic function $g : U \longrightarrow \mathbb{C}$ such that $g(z) = f(z)$ for every $z \in U \backslash \{0\}$.

Let $f(x) = x^{2} + ax + b \in \mathbb{F}_{3}[X]$. What is the number of non-isomorphic quotient rings $\mathbb{F}_{3}[X] / (f(X))$?

Let $(X, d)$ be a compact metric space. For $x \in X$ and $\epsilon > 0$, define $B_{\epsilon}(x) := \{y \in X \mid d(x, y) < \epsilon\}$. For $C \subseteq X$ and $\epsilon > 0$, define $B_{\epsilon}(C) := \cup_{x \in C} B_{\epsilon}(x)$. Let $\mathcal{K}$ be the set of non-empty compact subsets of $X$. For $C, C^{\prime} \in \mathcal{K}$, define $\delta\left(C, C^{\prime}\right) = \inf\{\epsilon \mid C \subseteq B_{\epsilon}\left(C^{\prime}\right)$ and $C^{\prime} \subseteq B_{\epsilon}(C)\}$. Show that $(\mathcal{K}, \delta)$ is a compact metric space.

Let $f$ be a non-constant entire function with $f(z) \neq 0$ for all $z \in \mathbb{C}$. Consider the set $U = \{z : |f(z)| < 1\}$. Show that all connected components of $U$ are unbounded.

Let $F \subseteq \mathbb{R}^{3}$ be a non-empty finite set, and $X = \mathbb{R}^{3} \backslash F$, taken with the subspace topology of $\mathbb{R}^{3}$. Show that $X$ is homeomorphic to a complete metric space. (Hint: Look for a suitable continuous function from $X$ to $\mathbb{R}$.)

Show that there is no differentiable function $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that $f(0) = 1$ and $f^{\prime}(x) \geq (f(x))^{2}$ for every $x \in \mathbb{R}$.