Let $f ( z ) = \frac { e ^ { z } - 1 } { z ( z - 1 ) }$ be defined on the extended complex plane $\mathbb { C } \cup \{ \infty \}$. Which of the following is/are true?
(A) $z = 0 , z = 1 , z = \infty$ are poles.
(B) $z = 1$ is a simple pole.
(C) $z = 0$ is a removable singularity.
(D) $z = \infty$ is an essential singularity.

Let $f ( u ) = \tan ^ { - 1 } ( u )$, a function whose domain is the set of all real numbers and whose range is $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Let $g ( v ) = \int _ { 0 } ^ { v } f ( t ) \, d t$.
(a) $f ( 1 ) = \frac { \pi } { 4 }$.
(b) $f ( 1 ) + f ( 2 ) + f ( 3 ) = \pi$.
(c) $g$ is an increasing function on the entire real line.
(d) $g$ is an odd function, i.e., $g ( - x ) = - g ( x )$ for all real $x$.

Let $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 }$ be a smooth function whose derivative at every point is non-singular. Suppose that $f ( 0 ) = 0$ and for all $v \in \mathbb { R } ^ { 2 }$ with $| v | = 1 , | f ( v ) | \geq 1$. Let $D$ denote the open unit ball $\{ v : | v | < 1 \}$. Show that $D \subset f ( D )$. (Hint: Show that $f ( D ) \cap D$ is closed in $D$.)

Let $X$ be a topological space and $x _ { 0 } \in X$. Let $\mathcal { S } = \left\{ B \subseteq X \mid x _ { 0 } \in B \text{ and } B \text{ is connected} \right\}$. Let $$A = \bigcup _ { B \in \mathcal { S } } B .$$ Show that $A$ is closed.

Let $f : [ 1 , \infty ) \longrightarrow \mathbb { R } \backslash \{ 0 \}$ be uniformly continuous. Show that the series $\sum _ { n \geq 1 } 1 / f ( n )$ is divergent.

Show that $\int _ { 0 } ^ { \infty } x ^ { \sqrt { 10 } } e ^ { - x ^ { 1 / 100 } } d x < \infty$.

Let $E$ be a finite extension of the field $\mathbb { Q }$. We say that a homomorphism of fields $\phi : E \longrightarrow \mathbb { C }$ is real if $\phi ( E ) \subset \mathbb { R }$. Prove or disprove each of the following assertions:
(A) For each prime number $p$, the field $\mathbb { Q } \left( p ^ { 1 / 12 } \right)$ has exactly one real embedding in $\mathbb { C }$. ($p ^ { 1 / 12 }$ is the unique real number $r > 0$ such that $r ^ { 12 } = p$.)
(B) If $[ E : \mathbb { Q } ] = 11$, there exists a real embedding of $E$.
(C) If $E$ is a Galois extension of $\mathbb { Q }$ and $[ E : \mathbb { Q } ] = 11$, then every embedding $E \longrightarrow \mathbb { C }$ is a real embedding.

A continuous map $f : A \longrightarrow B$ between two metric spaces $( A , d _ { A } )$, $( B , d _ { B } )$ is said to be a bilipschitz map if there exists a real number $\lambda > 0$ such that $( 1 / \lambda ) d _ { A } \left( a _ { 0 } , a _ { 1 } \right) \leq d _ { B } \left( f \left( a _ { 0 } \right) , f \left( a _ { 1 } \right) \right) \leq \lambda d _ { A } \left( a _ { 0 } , a _ { 1 } \right)$ for each $a _ { 0 } , a _ { 1 } \in A$.
Let $X = \mathbb { R } ^ { 2 } \backslash \{ 0 \}$ and $Y = \mathbb { S } ^ { 1 } \times \mathbb { R } = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Let $d _ { X }$ (respectively, $d _ { Y }$ ) be the euclidean metric on $X$ induced from $\mathbb { R } ^ { 2 }$ (respectively, on $Y$ induced from $\mathbb { R } ^ { 3 }$ ). Let $f : X \longrightarrow Y$ be a bilipschitz map.
(A) Let $R > 0$ and $C _ { R } \subseteq X$ the circle of radius $R$ with centre at 0. Let $\bar { f } : X \longrightarrow \mathbb { R }$ be the composite of $f$ and the projection from $Y = \mathbb { S } ^ { 1 } \times \mathbb { R }$ to its second factor $\mathbb { R }$. Let $L _ { R }$ be the length of the interval $\bar { f } \left( C _ { R } \right) \subseteq \mathbb { R }$. Let $a , b \in X$ be such that $\bar { f } ( b ) = \bar { f } ( a ) + L _ { R }$. Show that $d _ { X } ( a , b ) \geq ( 2 R - 2 \lambda ) / \lambda ^ { 2 }$.
(B) Let $C _ { 1 }$ and $C _ { 2 }$ be the two arcs of $C _ { R }$, joining $a$ and $b$. Show that there exists $x _ { 1 } \in C _ { 1 }$ and $x _ { 2 } \in C _ { 2 }$ such that $\bar { f } \left( x _ { 1 } \right) = \bar { f } \left( x _ { 2 } \right) = \frac { f ( a ) + f ( b ) } { 2 }$. Show that $d _ { Y } \left( f \left( x _ { 1 } \right) , f \left( x _ { 2 } \right) \right) \leq 2$.
(C) Show that for all sufficiently large $R , d _ { Y } \left( f \left( x _ { 1 } \right) , f \left( x _ { 2 } \right) \right) > 2$. (Hint: Let $a _ { i } \in C _ { i }$ be such that $d _ { X } \left( a , a _ { i } \right) = R / \lambda ^ { 2 }$; show that $d _ { X } \left( x _ { 1 } , x _ { 2 } \right) \geq d _ { X } \left( a _ { 1 } , a _ { 2 } \right)$.)
(D) What is your conclusion about $f$?

Let $R$ be an integral domain containing $\mathbb { C }$ such that it is a finite-dimensional $\mathbb { C }$-vector-space. Pick the correct statement(s) from below.
(A) For every $a \in R$, the set $\left\{ 1 , a , a ^ { 2 } , \ldots \right\}$ is linearly dependent over $\mathbb { C }$.
(B) $R$ is a field.
(C) $R = \mathbb { C }$.
(D) The transcendence degree of $R$ over $\mathbb { C }$ is 1 .

Let $R$ be a euclidean domain that is not a field. Let $d : R \backslash \{ 0 \} \longrightarrow \mathbb { N }$ be the euclidean size (degree) function. Write $R ^ { \times }$for the invertible elements of $R$. Pick the correct statements from below.
(A) $R = R ^ { \times } \cup \{ 0 \}$.
(B) There exists $a \in R \backslash \left( R ^ { \times } \cup \{ 0 \} \right)$ such that $d ( a ) = \inf \left\{ d ( r ) \mid r \in R \backslash \left( R ^ { \times } \cup \{ 0 \} \right) \right\}$.
(C) With $a$ defined as above, for all $r \in R$, there exists $u \in R ^ { \times } \cup \{ 0 \}$ such that $a$ divides ( $r - u$ ).
(D) With $a$ defined as above, the ideal generated by $a$ is a maximal ideal.

Let $X$ be a compact topological space. Let $f : X \longrightarrow \mathbb { R }$ be a function satisfying $f ^ { - 1 } ( [ n , \infty ) )$ is closed for all $n \in \mathbb { N }$. Pick the correct statements from below.
(A) $f$ is continuous.
(B) $f ( U )$ is open for each open subset $U$ of $X$.
(C) $f ( U )$ is closed for each closed subset $U$ of $X$.
(D) $f$ is bounded above.

Let $f : [ 0,1 ] \longrightarrow \mathbb { R }$ be a continuous function and $E \subseteq [ 0,1 ]$. Which of the following are true?
(A) If $E$ is closed, then $f ( E )$ is closed.
(B) If $E$ is open, then $f ( E )$ is open.
(C) If $E$ is a countable union of closed sets, then $f ( E )$ is a countable union of closed sets.
(D) If $f$ injective and $E$ is a countable intersection of open sets, then $f ( E )$ is a countable intersection of open sets.

Let $A$ be the ring of all entire functions under point-wise addition and multiplication. Then which of the following are true?
(A) $A$ does not have non-zero nilpotent elements.
(B) In the group of the units of $A$ (under multiplication), every element other than 1 has infinite order.
(C) For every $f \in A$, there is a sequence of polynomials which converges to $f$ uniformly on compact sets.
(D) The ideal generated by $z$ and $\sin z$ is principal.

Let $p , q$ be distinct prime numbers and let $\zeta _ { p } , \zeta _ { q }$ denote (any) primitive $p$-th and $q$-th roots of unity, respectively. Choose all the correct statements.
(A) $\zeta _ { 13 } \notin \mathbb { Q } \left( \zeta _ { 31 } \right)$.
(B) If $p$ divides $q - 1$, then $\zeta _ { p } \in \mathbb { Q } \left( \zeta _ { q } \right)$.
(C) If $\zeta _ { p } \in \mathbb { Q } \left( \zeta _ { q } \right)$, then $p - 1$ divides $q - 1$.
(D) If there exists a field homomorphism $\mathbb { Q } \left( \zeta _ { p } \right) \longrightarrow \mathbb { Q } \left( \zeta _ { q } \right)$, then $p - 1$ divides $q - 1$.

Let $f , g : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R }$ be functions. Let $F = ( f , g ) : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 }$. Assume that $F$ is infinitely differentiable and that $F ( 0,0 ) = ( 0,0 )$. Suppose further that the function $f g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$ is everywhere non-negative. Then
(A) $f _ { x } ( 0,0 ) = 0 , f _ { y } ( 0,0 ) = 0$.
(B) $g _ { x } ( 0,0 ) = 0 , g _ { y } ( 0,0 ) = 0$.
(C) The image of $F$ is not dense in $\mathbb { R } ^ { 2 }$.
(D) $\operatorname { det } J ( 0,0 ) = 0$ where $J$ is the matrix of first partial derivatives (i.e., the jacobian matrix).

Let $f$ be a non-constant entire function with $f ( 0 ) = 0$. Let $u$ and $v$ be the real and imaginary parts of $f$ respectively. Let $R > 0$ and
$$B = \sup \{ u ( z ) : | z | = R \}$$
(A) (2 marks) Show that $B > 0$.
(B) (2 marks) Consider the function
$$F ( z ) : = \frac { f ( z ) } { z ( 2 B - f ( z ) ) }$$
Show that $F$ is analytic on the open ball with radius $R$ and continuous on the boundary $\{ z : | z | = R \}$.
(C) (3 marks) Show that $\sup \{ | F ( z ) | : | z | = R \} \leq \frac { 1 } { R }$.
(D) (3 marks) Show that
$$\sup \left\{ | f ( z ) | : | z | = \frac { R } { 2 } \right\} \leq 2 B$$

A list of $k$ elements, possibly with repeats, is given. The goal is to find if there is a majority element. This is defined to be an element $x$ such that the number of times $x$ occurs in the list is strictly greater than $\frac{k}{2}$. (Note that there need not be such an element, but if it is there, it must be unique.) A celebrated efficient way to do this task uses two functions $f$ and $m$ with domain $\{1, 2, \ldots, k\}$. The functions are defined inductively as follows.
Define $f(1) =$ first element of the list, $m(1) = 1$. Assuming $f$ and $m$ are defined for all inputs from 1 to $i$, define $$f(i+1) = \begin{cases} f(i) & \text{if } m(i) > 0 \\ (i+1)^{\text{th}} \text{ element of the list} & \text{if } m(i) = 0 \end{cases}$$ $$m(i+1) = \begin{cases} m(i) - 1 & \text{if } m(i) > 0 \text{ and } (i+1)^{\text{th}} \text{ element of the list is other than } f(i) \\ m(i) + 1 & \text{otherwise} \end{cases}$$
(a) For the example of length 15 given below, write a sequence of 15 letters showing the values of $f(i)$ and a sequence of 15 numbers directly underneath showing the values of $m(i)$ for $i = 1, 2, \ldots, 15$.
$$\text{a a b a b c c b b b a b b c b}$$
(b) Prove that in general the list can be divided into two disjoint parts A and B such that

• Part A contains $m(k)$ elements of the list each of which is $f(k)$.
• Part B contains the remaining $k - m(k)$ elements of the list and B can be written as disjoint union of pairs such that the two elements in each pair are distinct.

(c) If there is a majority element, show that it must be $f(k)$. You may assume part (b) even if you did not do it.
(d) Assuming $f(k)$ is the majority element, answer the following two questions. Show by examples that the number of occurrences of $f(k)$ in the list does not determine the value of $m(k)$. Can the value of $m(k)$ be anything in $\{0, \ldots, k\}$? Find constraints if any on the possible values of $m(k)$.
(e) Now assume instead that an element occurs exactly $\frac{k}{2}$ times in the list. Is it necessary that $f(k)$ is such an element?

Let $f ( z ) = z ^ { 7 } - 4 z ^ { 3 } - 11$. Pick the correct statement(s) from below.
(A) $f ( z )$ has at least 1 zero in the open set $\{ | z | > 2 \}$.
(B) $f ( z )$ has at least 5 zeroes in the annular region $\{ 1 < | z | < 2 \}$.
(C) $f ( z )$ has exactly 6 zeroes in the annular region $\{ 1 < | z | < 2 \}$.
(D) $f ( z )$ has exactly 1 zero in the closed disc $\{ | z | \leq 1 \}$.

A region in $\mathbb { C }$ is a non-empty open connected set. Select all the statement(s) that are true.
(A) Let $f$ be a function on a region $\Omega$ such that the integral of $f$ along the boundary of any closed triangle in $\Omega$ is zero. Then $f$ is analytic on $\Omega$.
(B) There exist a region $\Omega$ containing the real interval $( 0,1 )$ and a non-zero analytic function $f : \Omega \rightarrow \mathbb { C }$ such that $f \left( \frac { 1 } { n } \right) = 0$ for all positive integers $n$.
(C) Let $f$ be an analytic function on $\mathbb { C } \backslash \{ 0 \}$ with an essential singularity at $z = 0$. Then $\lim _ { z \rightarrow 0 } | f ( z ) | = \infty$.
(D) Every bounded analytic function on $\mathbb { C } \backslash \{ 0 \}$ is constant.

Let $u$ and $v$ be real-valued functions on $\mathbb { R } ^ { 2 }$ defined as follows:
$$\begin{aligned} & u ( x , y ) = \begin{cases} \frac { x ^ { 3 } - 3 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\ 0 & \text { otherwise } \end{cases} \\ & v ( x , y ) = \begin{cases} \frac { y ^ { 3 } - 3 y x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$
Let $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 }$ be the function $f ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. Pick the correct statement(s) from below.
(A) $\frac { \partial u } { \partial x } , \frac { \partial u } { \partial y } , \frac { \partial v } { \partial x }$ and $\frac { \partial v } { \partial y }$ exist at $( 0,0 )$.
(B) $\frac { \partial u } { \partial x }$ is continuous at $( 0,0 )$.
(C) For every fixed $( a , b ) \neq ( 0,0 ) \in \mathbb { R } ^ { 2 }$, the function $t \mapsto f ( t a , t b )$ is a differentiable function (of $t$ ).
(D) $f$ is differentiable at $( 0,0 )$.

Let $X$ be a subset of $\mathbb { R } ^ { 3 }$. We say that $X$ has property $S$ if it contains at least two elements and every Cauchy sequence in $X$ has a limit point in $X$. Pick the correct statement(s) from below.
(A) If $X$ has property $S$ then it must be compact.
(B) If $X$ has property $S$ then it must be closed.
(C) Suppose that $X$ has property $S$ and it further satisfies the following condition: if $\left( a _ { 1 } , b _ { 1 } , c _ { 1 } \right) , \left( a _ { 2 } , b _ { 2 } , c _ { 2 } \right) \in X$, then $\left( a _ { 1 } + a _ { 2 } , b _ { 1 } + b _ { 2 } , c _ { 1 } + c _ { 2 } \right) \in X$. Then $X$ is dense in $\mathbb { R } ^ { 3 }$.
(D) Suppose that $X$ has property $S$ and it further satisfies the following condition: if $( a , b , c ) \in X$, then $\left( \frac { a } { 2 } , \frac { b } { 2 } , \frac { c } { 2 } \right) \in X$. Then $X$ is dense in $\mathbb { R } ^ { 3 }$.

Which of the following statement(s) are true?
(A) If $F _ { 1 } , F _ { 2 }$ are finite field extensions of $\mathbb { Q }$ such that $\left[ F _ { 1 } : \mathbb { Q } \right] = \left[ F _ { 2 } : \mathbb { Q } \right]$, then $F _ { 1 } , F _ { 2 }$ are isomorphic as fields.
(B) If $F _ { 1 } , F _ { 2 }$ are finite field extensions of $\mathbb { R }$ such that $\left[ F _ { 1 } : \mathbb { R } \right] = \left[ F _ { 2 } : \mathbb { R } \right]$, then $F _ { 1 } , F _ { 2 }$ are isomorphic as fields.
(C) Let $\mathbb { F }$ be a finite field. If $F _ { 1 } , F _ { 2 }$ be finite field extensions of $\mathbb { F }$ such that $\left[ F _ { 1 } : \mathbb { F } \right] = \left[ F _ { 2 } : \mathbb { F } \right]$, then $F _ { 1 } , F _ { 2 }$ are isomorphic as fields.
(D) Let $\omega \in \mathbb { C }$ be a primitive cube root of unity and let $\sqrt [ 3 ] { 2 } \in \mathbb { R }$ be a cube root of 2. Let $K = \mathbb { Q } ( \omega , \sqrt [ 3 ] { 2 } )$. If $F _ { 1 } , F _ { 2 }$ are subfields of $K$ such that $\left[ K : F _ { 1 } \right] = \left[ K : F _ { 2 } \right] = 2$, then $F _ { 1 } = F _ { 2 }$.

Let $X = \{ \alpha \in \mathbb { C } \mid \alpha$ satisfies a monic polynomial over $\mathbb { Q } \}$. (I.e., $X$ is the algebraic closure of $\mathbb { Q }$ in $\mathbb { C }$.) Endow $X$ with the subspace topology of the euclidean metric topology from $\mathbb { C }$. Pick the correct statement(s) from below.
(A) $X$ is closed.
(B) $X$ is complete.
(C) $X$ is unbounded.
(D) $X$ is connected.

Let $n \geq 3$ be an integer. Write $D _ { 2 n }$ for the dihedral group with $2 n$ elements. Show that the automorphism group of $D _ { 2 n }$ has at most $n \varphi ( n )$ elements. (Here $\varphi ( n )$ is the number of positive integers that are relatively prime to $n$.)

Let $\operatorname { SL } ( 2 , \mathbb { R } )$ be the group of $2 \times 2$ matrices with real entries and determinant 1, endowed with the subspace topology of $\mathbb { R } ^ { 4 }$. Consider the continuous map $f : \operatorname { SL } ( 2 , \mathbb { R } ) \longrightarrow \mathbb { C }$ given by
$$f \left( \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right] \right) = \frac { a i + b } { c i + d }$$
(A) (4 marks) Show that $f$ maps SL$( 2 , \mathbb { R } )$ onto the upper half plane $H = \{ z : \operatorname { Im } ( z ) > 0 \}$.
(B) (6 marks) Assume the following two facts:
(i) For all $M , N \in \mathrm { SL } ( 2 , \mathbb { R } ) , f ( M ) = f ( N )$ if and only if $M ^ { - 1 } N$ is an orthogonal matrix.
(ii) The map $f$ is an open map.
Now show that for every compact $K \subseteq H , f ^ { - 1 } ( K )$ is compact.