Let $a_{1}, \ldots, a_{n}$ be distinct complex numbers. Show that the functions $e^{a_{1} z}, \ldots, e^{a_{n} z}$ are linearly independent over $\mathbb{C}$.

The Frattini subgroup of a finite group $G$ is the intersection of all its proper maximal subgroups. Let $p$ be a prime number. Show that the Frattini subgroup of $\mathbb{Z} / p^{n}$, $n \geq 2$, is generated by $p$.

Let $M \in M_{n}(\mathbb{C})$. Show that $M$ is diagonalizable if and only if for every polynomial $P(X) \in \mathbb{C}[X]$ such that $P(M)$ is nilpotent, $P(M) = 0$.

$X$ is said to have the universal extension property if for every normal space $Y$ and every closed subset $A \subset Y$ and every continuous function $f : A \longrightarrow X$, $f$ extends to a continuous function from $Y$ to $X$. You may assume, without proof, that $\mathbb{R}^{2}$ has the universal extension property.
(A) Prove or find a counterexample: If $X$ has the universal extension property, then $X$ is connected.
(B) Give an example (with justification) of a compact subset $X$ of $\mathbb{R}^{2}$ that does not have the universal extension property.
(C) Let $X = \{(x, \sin x) \mid x \in \mathbb{R}\}$. Then show that $X$ has the universal extension property.

Let $p$ be a prime number and $q$ a power of $p$. Let $K$ be an algebraic closure of $\mathbb{F}_{q}$. Say that a polynomial $f(X) \in K[X]$ is a $q$-polynomial if it is of the form
$$f(X) = \sum_{i=0}^{n} a_{i} X^{q^{i}}$$
Let $f(X)$ be a $q$-polynomial of degree $q^{n}$, with $a_{0} \neq 0$. Show that the set of zeros of $f(X)$ is an $n$-dimensional vector-space over $\mathbb{F}_{q}$.

Let $a_{n}$, $n \geq 1$ be a sequence of real numbers. If $a_{n} \rightarrow a$, show that
$$b_{n} = \frac{a_{1} + 2a_{2} + 3a_{3} + \cdots + na_{n}}{n^{2}} \rightarrow \frac{a}{2}.$$

$n$ and $k$ are positive integers, not necessarily distinct. You are given two stacks of cards with a number written on each card, as follows.
Stack A has $n$ cards. On each card a number in the set $\{ 1 , \ldots , k \}$ is written. Stack B has $k$ cards. On each card a number in the set $\{ 1 , \ldots , n \}$ is written. Numbers may repeat in either stack. From this, you play a game by constructing a sequence $t _ { 0 } , t _ { 1 } , t _ { 2 } , \ldots$ of integers as follows. Set $t _ { 0 } = 0$. For $j > 0$, there are two cases: If $t _ { j } \leq 0$, draw the top card of stack $A$. Set $t _ { j + 1 } = t _ { j } +$ the number written on this card. If $t _ { j } > 0$, draw the top card of stack $B$. Set $t _ { j + 1 } = t _ { j } -$ the number written on this card. In either case discard the taken card and continue. The game ends when you try to draw from an empty stack. Example: Let $n = 5 , k = 3$, stack $A = 1,3,2,3,2$ and stack $B = 2,5,1$. You can check that the game ends with the sequence $0,1 , - 1,2 , - 3 , - 1,2,1$ (and with one card from stack $A$ left unused).
(i) Prove that for every $j$ we have $- n + 1 \leq t _ { j } \leq k$.
(ii) Prove that there are at least two distinct indices $i$ and $j$ such that $t _ { i } = t _ { j }$.
(iii) Using the previous parts or otherwise, prove that there is a nonempty subset of cards in stack $A$ and another subset of cards in stack $B$ such that the sum of numbers in both the subsets is same.

Consider the improper integral $\int _ { 2 } ^ { \infty } \frac { 1 } { x ( \log x ) ^ { 2 } } d x$ and the infinite series $\sum _ { k = 2 } ^ { \infty } \frac { 1 } { k ( \log k ) ^ { 2 } }$. Which of the following is/are true?
(A) The integral converges but the series does not converge.
(B) The integral does not converge but the series converges.
(C) Both the integral and the series converge.
(D) The integral and the series both fail to converge.

Let $A \in M _ { 2 } ( \mathbb { R } )$ be a nonzero matrix. Pick the correct statement(s) from below.
(A) If $A ^ { 2 } = 0$, then $\left( I _ { 2 } - A \right) ^ { 5 } = 0$.
(B) If $A ^ { 2 } = 0$, then ( $I _ { 2 } - A$ ) is invertible.
(C) If $A ^ { 3 } = 0$, then $A ^ { 2 } = 0$.
(D) If $A ^ { 2 } = A ^ { 3 } \neq 0$, then $A$ is invertible.

Let $f : [ 0,1 ] \longrightarrow [ 0,1 ]$ be a continuous function. Which of the following is/are true?
(A) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $g ( 0 ) = 0$ and $g ( 1 ) = 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.
(B) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $g ( 0 ) < 0$ and $g ( 1 ) > 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.
(C) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $0 < g ( 0 ) < 1$ and $0 < g ( 1 ) < 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.
(D) For every continuous $g : [ 0,1 ] \longrightarrow [ 0,1 ]$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.

Let $I , J$ be nonempty open intervals in $\mathbb { R }$. Let $f : I \longrightarrow J$ and $g : J \longrightarrow \mathbb { R }$ be functions. Let $h : I \longrightarrow \mathbb { R }$ be the composite function $g \circ f$. Pick the correct statement(s) from below.
(A) If $f , g$ are continuous, then $h$ is continuous.
(B) If $f , g$ are uniformly continuous, then $h$ is uniformly continuous.
(C) If $h$ is continuous, then $f$ is continuous.
(D) If $h$ is continuous, then $g$ is continuous.

Let $A , B$ be non-empty subsets of $\mathbb { R } ^ { 2 }$. Pick the correct statement(s) from below:
(A) If $A$ is compact, $B$ is open and $A \cup B$ is compact, then $A \cap B \neq \varnothing$.
(B) If $A$ and $B$ are path-connected and $A \cap B \neq \varnothing$ then $A \cup B$ is path-connected.
(C) If $A$ and $B$ are connected and open and $A \cap B \neq \varnothing$, then $A \cap B$ is connected.
(D) If $A$ is countable with $| A | \geq 2$, then $A$ is not connected.

Pick the correct statement(s) from below.
(A) $X = \prod _ { n = 1 } ^ { \infty } X _ { n }$ where $X _ { n } = \left\{ 1,2 , \ldots , 2 ^ { n } \right\}$ for $n \geq 1$ is not compact in the product topology.
(B) $Y = \prod _ { n = 1 } ^ { \infty } Y _ { n }$ where $Y _ { n } = \left[ 0,2 ^ { n } \right] \subseteq \mathbb { R }$ for $n \geq 1$ is path-connected in the product topology.
(C) $Z = \prod _ { n = 1 } ^ { \infty } Z _ { n }$ where $Z _ { n } = \left( 0 , \frac { 1 } { n } \right) \subseteq \mathbb { R }$ for $n \geq 1$ is compact in the product topology.
(D) $P = \prod _ { n = 1 } ^ { \infty } P _ { n }$ where $P _ { n } = \{ 0,1 \}$ for $n \geq 1$ (with product topology) is homeomorphic to $( 0,1 )$.

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.

For $A \in M _ { 3 } ( \mathbb { C } )$, let $W _ { A } = \left\{ B \in M _ { 3 } ( \mathbb { C } ) \mid A B = B A \right\}$. Which of the following is/are true?
(A) For all diagonal $A \in M _ { 3 } ( \mathbb { C } )$, $W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } \geq 3$.
(B) For all $A \in M _ { 3 } ( \mathbb { C } ) , W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } > 3$.
(C) There exists $A \in M _ { 3 } ( \mathbb { C } )$ such that $W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } = 3$.
(D) If $A \in M _ { 3 } ( \mathbb { C } )$ is diagonalizable, then every element of $W _ { A }$ is diagonalizable.

Let $K$ be a field of order 243 and let $F$ be a subfield of $K$ of order 3. Pick the correct statement(s) from below.
(A) There exists $\alpha \in K$ such that $K = F ( \alpha )$.
(B) The polynomial $x ^ { 242 } = 1$ has exactly 242 solutions in $K$.
(C) The polynomial $x ^ { 26 } = 1$ has exactly 26 roots in $K$.
(D) Let $f ( x ) \in F [ x ]$ be an irreducible polynomial of degree 5. Then $f ( x )$ has a root in $K$.

Let $G$ be a finite group and $X$ the set of all abelian subgroups $H$ of $G$ such that $H$ is a maximal subgroup of $G$ (under inclusion) and is not normal in $G$. Let $M$ and $N$ be distinct elements of $X$. Show the following:
(A) The subgroup of $G$ generated by $M$ and $N$ is contained in the centralizer of $M \cap N$ in $G$.
(B) $M \cap N$ is the centre of $G$.

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

Consider the following statement: Let $F$ be a field and $R = F [ X ]$ the polynomial ring over $F$ in one variable. Let $I _ { 1 }$ and $I _ { 2 }$ be maximal ideals of $R$ such that the fields $R / I _ { 1 } \simeq R / I _ { 2 } \neq F$. Then $I _ { 1 } = I _ { 2 }$.
Prove or find a counterexample to the following claims:
(A) The above statement holds if $F$ is a finite field.
(B) The above statement holds if $F = \mathbb { R }$.

Let $\mathrm { O } ( 2 , \mathbb { R } )$ be the subgroup of $\mathrm { GL } ( 2 , \mathbb { R } )$ consisting of orthogonal matrices, i.e., matrices $A$ satisfying $A ^ { \operatorname { tr } } A = I$. Let $\mathrm { B } _ { + } ( 2 , \mathbb { R } )$ be the subgroup of $\mathrm { GL } ( 2 , \mathbb { R } )$ consisting of upper triangular matrices with positive entries on the diagonal.
(A) Let $A \in \mathrm { GL } ( 2 , \mathbb { R } )$. Show that there exist $A _ { o } \in \mathrm { O } ( 2 , \mathbb { R } )$ and $A _ { b } \in \mathrm { B } _ { + } ( 2 , \mathbb { R } )$ such that $A = A _ { o } A _ { b }$. (Hint: use appropriate elementary column operations.)
(B) Show that the map $$\phi : \mathrm { O } ( 2 , \mathbb { R } ) \times \mathrm { B } _ { + } ( 2 , \mathbb { R } ) \longrightarrow \mathrm { GL } ( 2 , \mathbb { R } ) \quad \left( A ^ { \prime } , A ^ { \prime \prime } \right) \mapsto A ^ { \prime } A ^ { \prime \prime }$$ is injective.
(C) Show that $\mathrm { GL } ( 2 , \mathbb { R } )$ is homeomorphic to $\mathrm { O } ( 2 , \mathbb { R } ) \times \mathrm { B } _ { + } ( 2 , \mathbb { R } )$. (Hint: first show that the map $A \mapsto A _ { b }$ is continuous.)

Let $F$ be a field of characteristic $p > 0$ and $V$ a finite-dimensional $F$-vector-space. Let $\phi \in \mathrm { GL } ( V )$ be an element of order $p ^ { 3 }$. Show that there exists a basis of $V$ with respect to which $\phi$ is given by an upper-triangular matrix with 1's on the diagonal.

Let $\zeta _ { 5 } \in \mathbb { C }$ be a primitive 5th root of unity; let $\sqrt [ 5 ] { 2 }$ denote a real 5th root of 2, and let $l$ denote a square root of $-1$. Let $K = \mathbb { Q } \left( \zeta _ { 5 } , \sqrt [ 5 ] { 2 } \right)$.
(A) Find the degree $[ K : \mathbb { Q } ]$ of the field $K$ over $\mathbb { Q }$.
(B) Determine if $l \in \mathbb { Q } \left( \zeta _ { 5 } \right)$. (Hint: You may use, without proof, the following fact: if $\zeta _ { 20 } \in \mathbb { C }$ is a primitive 20th root of unity, then $\left[ \mathbb { Q } \left( \zeta _ { 20 } \right) : \mathbb { Q } \right] > 4$.)
(C) Determine if $l \in K$.