Let $A \in M_{m \times n}(\mathbb{R})$ and let $b_0 \in \mathbb{R}^m$. Suppose the system of equations $Ax = b_0$ has a unique solution. Which of the following statement(s) is/are true?
(A) $Ax = b$ has a solution for every $b \in \mathbb{R}^m$.
(B) If $Ax = b$ has a solution then it is unique.
(C) $Ax = 0$ has a unique solution.
(D) $A$ has rank $m$.

Let $A \in M_{n \times n}(\mathbb{C})$. Which of the following statement(s) is/are true?
(A) There exists $B \in M_{n \times n}(\mathbb{C})$ such that $B^2 = A$.
(B) $A$ is diagonalizable.
(C) There exists an invertible matrix $P$ such that $PAP^{-1}$ is upper-triangular.
(D) $A$ has an eigenvalue.

Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be a function. Which of the following statement(s) is/are true?
(A) Consider $f$ as a function $(f_1, f_2) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose that for $i = 1, 2$, both $\frac{\partial f_i}{\partial X}$ and $\frac{\partial f_i}{\partial Y}$ exist and are continuous. Then $f$ is entire.
(B) Assume that $f$ is entire and $|f(z)| < 1$ for all $z \in \mathbb{C}$. Then $f$ is constant.
(C) Assume that $f$ is entire and $\operatorname{Im}(f(z)) > 0$ for all $z \in \mathbb{C}$. Then $f$ is constant.

Let $\mathcal{C}(\mathbb{R})$ be the $\mathbb{R}$-vector space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Let $a_1, a_2, a_3$ be distinct real numbers. For $i = 1, 2, 3$, let $f_i \in \mathcal{C}(\mathbb{R})$ be the function $f_i(t) = e^{a_i t}$. Which of the following statement(s) is/are true?
(A) $f_1, f_2$ and $f_3$ are linearly independent.
(B) $f_1, f_2$ and $f_3$ are linearly dependent.
(C) $f_1, f_2$ and $f_3$ form a basis of $\mathcal{C}(\mathbb{R})$.

Which of the following statement(s) is/are true?
(A) The series $\sum_{n=1}^{\infty} \mathrm{e}^{-n^2}$ converges.
(B) The series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ converges.
(C) The series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ converges absolutely.
(D) The series $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2}$ converges uniformly on $\mathbb{R}$.

What is the dimension of the ring $\mathbb{Q}[x]/\left((x+1)^2\right)$ as a $\mathbb{Q}$-vector space?

Evaluate $\lim_{n \rightarrow \infty} \left[\frac{\pi \sum_{i=1}^{n} \sin\left(\frac{i\pi}{n}\right)}{n}\right]$.

Show that the set of rank two matrices in $M_{2 \times 3}(\mathbb{R})$ is open.

(A) Let $F$ be a finite field extension of $\mathbb{Q}$. Show that any field homomorphism $\phi : F \rightarrow F$ is an isomorphism. (Note that $\phi(1) = 1$ by definition.)
(B) Let $F$ be a finite field whose characteristic is not 2. Let $F^\times$ denote the multiplicative group of nonzero elements of $F$. An element $a \in F^\times$ is called a square if there exists $x \in F^\times$ such that $x^2 = a$. Show that exactly half the elements of $F^\times$ are squares.

Let $n \in \mathbb{N}$. Show that the determinant map $\det : M_{n \times n}(\mathbb{R}) \rightarrow \mathbb{R}$ is infinitely differentiable and compute the total derivative $d(\operatorname{det})$ at every point $A \in M_{n \times n}(\mathbb{R})$. Find a necessary and sufficient condition on the rank of $A$ for $d(\operatorname{det}) = 0$ at $A$.

Let $a_i, i \in \mathbb{R}$ be non-negative real numbers such that $\sup\left\{\sum_{i \in F} a_i \mid F \subseteq \mathbb{R} \text{ a finite subset}\right\}$ is finite. Show that $a_i = 0$ except for countably many $i \in \mathbb{R}$. Give an example to show that 'countably' cannot be replaced by 'finite'. (Hint: consider $F_n := \left\{i \left\lvert\, a_i \geq \frac{1}{n}\right.\right\}$.)

Let $G$ be a finite group of order $2n$ for some integer $n$. Consider the map $\phi : G \rightarrow G$ given by $\phi(a) = a^2$. Show that $\phi$ is not surjective.

Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be an entire function.
(A) Construct a sequence $\{z_n\}$ in $\mathbb{C}$ such that $|z_n| \rightarrow \infty$ and $e^{z_n} \rightarrow 1$.
(B) Show that the function $g(z) = f\left(e^z\right)$ is not a polynomial.

For $F = \mathbb{R}$ and $F = \mathbb{C}$, let $O_n(F) = \left\{A \in M_{n \times n}(F) \mid AA^t = I_n\right\}$.
(A) Show that $O_n(\mathbb{R})$ is compact.
(B) Is $O_n(\mathbb{R})$ connected? Justify.
(C) Is $O_n(\mathbb{C})$ compact? Justify.

Let $\Omega$ be a region in $\mathbb{C}$. Let $\{a_n\}$ be a sequence of nonzero elements in $\Omega$ such that $a_n \rightarrow 0$ as $n \rightarrow \infty$. Let $\{b_n\}$ be a sequence of complex numbers such that $\lim_{n \rightarrow \infty} \frac{b_n}{a_n^k} = 0$ for every nonnegative integer $k$. Suppose that $f : \Omega \rightarrow \mathbb{C}$ is an entire function such that $f(a_n) = b_n$ for all $n$. Show that $b_n = 0$ for every $n$.

Let $G$ be a finite group of order $n$ and let $H$ be a subgroup of $G$ of order $m$. Assume that $\left(\frac{n}{m}\right)! < 2n$. Show that $G$ is not simple, that is: $G$ has a nontrivial proper normal subgroup. (Hint: Think along the lines of Cayley's theorem.)

Let $$\mathcal{C}_0(\mathbb{R}) = \left\{f : \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ is continuous}, \lim_{x \rightarrow \infty} |f(x)| = 0 \text{ and } \lim_{x \rightarrow -\infty} |f(x)| = 0\right\},$$ and $\mathcal{C}_0^\infty(\mathbb{R}) = \left\{f \in \mathcal{C}_0(\mathbb{R}) \mid f \text{ is infinitely differentiable}\right\}$. Let $\phi \in \mathcal{C}_0(\mathbb{R})$. For $f \in \mathcal{C}_0(\mathbb{R})$, define $\phi^*(f) = f \circ \phi$.
(A) Show that $\phi^*(f) \in \mathcal{C}_0(\mathbb{R})$ if $f \in \mathcal{C}_0(\mathbb{R})$.
(B) If $\phi^*\left(\mathcal{C}_0^\infty(\mathbb{R})\right) \subseteq \mathcal{C}_0^\infty(\mathbb{R})$, then show that $\phi$ is infinitely differentiable.

Carefully solve the following series of questions. If you cannot solve an earlier part, you may still assume the result in it to solve a later part.
(a) For any polynomial $p(t)$, the limit $\lim_{t \rightarrow \infty} \frac{p(t)}{e^t}$ is independent of the polynomial $p$. Justify this statement and find the value of the limit.
(b) Consider the function defined by
$$\begin{aligned} q(x) &= e^{-1/x} \text{ when } x > 0 \\ &= 0 \text{ when } x = 0 \\ &= e^{1/x} \text{ when } x < 0 \end{aligned}$$
Show that $q'(0)$ exists and find its value. Why is it enough to calculate the relevant limit from only one side?
(c) Now for any positive integer $n$, show that $q^{(n)}(0)$ exists and find its value. Here $q(x)$ is the function in part (b) and $q^{(n)}(0)$ denotes its $n$-th derivative at $x = 0$.

Let $p$, $q$ and $r$ be real numbers with $p^2 + q^2 + r^2 = 1$.
(a) Prove the inequality $3p^2 q + 3p^2 r + 2q^3 + 2r^3 \leq 2$.
(b) Also find the smallest possible value of $3p^2 q + 3p^2 r + 2q^3 + 2r^3$. Specify exactly when the smallest and the largest possible value is achieved.

Which of the following topological spaces is/are connected?
(A) $\mathrm{GL}_{1}(\mathbb{R})$
(B) $\mathrm{GL}_{1}(\mathbb{C})$
(C) $\mathrm{GL}_{2}(\mathbb{R})$
(D) $\left\{\left[\begin{array}{cc}x & -y \\ y & x\end{array}\right] : x, y \in \mathbb{R}, x^{2}+y^{2}=1\right\}$

Consider $f : \{z \in \mathbb{C} : |z| > 1\} \longrightarrow \mathbb{C},\ f(z) = \frac{1}{z}$. Choose the correct statement(s):
(A) There are infinitely many entire functions $g$ such that $g(z) = f(z)$ for every $z \in \mathbb{C}$ with $|z| > 1$.
(B) There does not exist an entire function $g$ such that $g(z) = f(z)$ for every $z \in \mathbb{C}$ with $|z| > 1$.
(C) $g : \mathbb{C} \longrightarrow \mathbb{C}$ with $$g(z) = \begin{cases} 1 - \frac{1}{2}z^{2}, & |z| \leq 1 \\ \frac{1}{z}, & |z| > 1 \end{cases}$$ is an entire function such that $g(z) = f(z)$ for every $z \in \mathbb{C}$ with $|z| > 1$.

Consider all finite letter-strings formed by using only two letters A and B. We consider the usual dictionary order on these strings.
Formal rule: To compare two strings $w_1$ and $w_2$, read them from left to right. We say ``$w_1$ is smaller than $w_2$'' or ``$w_1 < w_2$'' if the first letter in which $w_1$ and $w_2$ differ is A in $w_1$ and B in $w_2$ (for example, $\mathrm{ABAA} < \mathrm{ABB}$ by looking at the third letters) or if $w_2$ is obtained by appending some letters at the end of $w_1$ (e.g. $\mathrm{AB} < \mathrm{ABAA}$).
For each of the statements below, state if it is true or false. Write your answers as a sequence of three letters (T for True and F for False) in correct order.
(a) Let $w$ be an arbitrary string. There exists a unique string $y$ satisfying both the following properties: (i) $w < y$ and (ii) there is no string $x$ with $w < x < y$.
(b) It is possible to give an infinite decreasing sequence of strings, i.e. a sequence $w_1, w_2, \ldots$, such that $w_{i+1} < w_i$ for each positive integer $i$.
(c) Fewer than 50 strings are smaller than ABBABABB.

Choose the correct statement(s):
(A) There is a continuous surjective function from $[0,1)$ to $\mathbb{R}$;
(B) $\mathbb{R}$ and $[0,1)$ are homeomorphic to each other;
(C) There is a bijective function from $[0,1)$ to $\mathbb{R}$;
(D) Bounded subspaces of $\mathbb{R}$ cannot be homeomorphic to $\mathbb{R}$.

Which of the following complex numbers has/have a prime number as the degree of its minimal polynomial over $\mathbb{Q}$?
(A) $\zeta_{7}$, a primitive 7th root of unity;
(B) $\sqrt{2}+\sqrt{3}$;
(C) $\sqrt{-1}$;
(D) $\sqrt[3]{2}$.

Consider the polynomial $p(x) = (x + a_1)(x + a_2) \cdots (x + a_{10})$ where $a_i$ is a real number for each $i = 1, \ldots, 10$. Suppose all of the eleven coefficients of $p(x)$ are positive. For each of the following statements, decide if it is true or false. Write your answers as a sequence of four letters (T/F) in correct order.
(i) The polynomial $p(x)$ must have a global minimum.
(ii) Each $a_i$ must be positive.
(iii) All real roots of $p'(x)$ must be negative.
(iv) All roots of $p'(x)$ must be real.