Let $f : ( 0 , \infty ) \longrightarrow \mathbb { R }$ be a continuous function. Then $f$ maps any Cauchy sequence to a Cauchy sequence.

Let $\left\{ f _ { n } : \mathbb { R } \longrightarrow \mathbb { R } \right\}$ be a sequence of continuous functions. Let $x _ { n } \longrightarrow x$ be a convergent sequence of reals. If $f _ { n } \longrightarrow f$ uniformly then $f _ { n } \left( x _ { n } \right) \longrightarrow f ( x )$.

Let $K \subset \mathbb { R } ^ { n }$ such that every real valued continuous function on $K$ is bounded. Then $K$ is compact (i.e closed and bounded).

If $A \subset \mathbb { R } ^ { 2 }$ is a countable set, then $\mathbb { R } ^ { 2 } \backslash A$ is connected.

The set $A = \left\{ ( z , w ) \in \mathbb { C } ^ { 2 } \mid z ^ { 2 } + w ^ { 2 } = 1 \right\}$ is bounded in $\mathbb { C } ^ { 2 }$.

Let $f , g : \mathbb { C } \longrightarrow \mathbb { C }$ be complex analytic, and let $h : [ 0,1 ] \longrightarrow \mathbb { C }$ be a non-constant continuous map. Suppose $f ( z ) = g ( z )$ for every $z \in \operatorname { Im } h$, then $f = g$. (Here $\operatorname { Im } h$ denotes the image of the function $h$.)

The matrix $\left( \begin{array} { c c } \pi & \pi \\ 0 & \frac { 22 } { 7 } \end{array} \right)$ is diagonalizable over $\mathbb { C }$.

There are no infinite group with subgroups of index 5.

Every finite group of odd order is isomorphic to a subgroup of $A _ { n }$, the group of all even permutations.

Two abelian groups of the same order are isomorphic.

There is a non-constant continuous function $f : \mathbb { R } \rightarrow \mathbb { R }$ whose image is contained in $\mathbb { Q }$.

Suppose $f : \mathbb { R } \mapsto \mathbb { R } ^ { n }$ be a differentiable mapping satisfying $\| f ( t ) \| = 1$ for all $t \in \mathbb { R }$. Show that $\left\langle f ^ { \prime } ( t ) , f ( t ) \right\rangle = 0$ for all $t \in \mathbb { R }$. (Here $\|$.$\|$ denotes standard norm or length of a vector in $\mathbb { R } ^ { n }$, and $\langle . , .\rangle$ denotes the standard inner product (or scalar product) in $\mathbb { R } ^ { n }$.)

Let $A , B \subset \mathbb { R } ^ { n }$ and define $A + B = \{ a + b ; a \in A , b \in B \}$. If $A$ and $B$ are open, is $A + B$ open? If $A$ and $B$ are closed, is $A + B$ closed? Justify your answers.

Let $f : X \mapsto Y$ be continuous map onto $Y$, and let $X$ be compact. Also $g : Y \mapsto Z$ is such that $g \circ f$ is continuous. Show $g$ is continuous.

Let $A$ be a $n \times m$ matrix with real entries, and let $B = A A ^ { t }$ and let $\alpha$ be the supremum of $x ^ { t } B x$ where supremum is taken over all vectors $x \in \mathbb { R } ^ { n }$ with norm less than or equal to 1. Consider $$C _ { k } = I + \sum _ { j = 1 } ^ { k } B ^ { j }$$ Show that the sequence of matrices $C _ { k }$ converges if and only if $\alpha < 1$.

Show that a power series $\sum _ { n \geq 0 } a _ { n } z ^ { n }$ where $a _ { n } \rightarrow 0$ as $n \rightarrow \infty$ cannot have a pole on the unit circle. Is the statement true with the hypothesis that $\left( a _ { n } \right)$ is a bounded sequence?

Show that a biholomorphic map of the unit ball onto itself which fixes the origin is necessarily a rotation.

(i) Let $G = G L \left( 2 , \mathbb { F } _ { p } \right)$. Prove that there is a Sylow $p$-subgroup $H$ of $G$ whose normalizer $N _ { G } ( H )$ is the group of all upper triangular matrices in $G$.
(ii) Hence prove that the number of Sylow subgroups of $G$ is $1 + p$.

Calculate the minimal polynomial of $\sqrt { 2 } e ^ { \frac { 2 \pi i } { 3 } }$ over $\mathbb { Q }$.

Let $G$ be a group $\mathbb { F }$ a field and $n$ a positive integer. A linear action of $G$ on $\mathbb { F } ^ { n }$ is a map $\alpha : G \times \mathbb { F } ^ { n } \rightarrow \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = \rho ( g ) v$ for some group homomorphism $\rho : G \rightarrow \mathrm { GL } _ { n } ( \mathbb { F } )$. Show that for every finite group $G$, there is an $n$ such that there is a linear action $\alpha$ of $G$ on $\mathbb { F } ^ { n }$ and such that there is a nonzero vector $v \in \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = v$ for all $g \in G$.

Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite-dimensional vector space over $F$, then $R$ is a field.

Let $G$ be a group. The following statements hold.
(a) If $G$ has nontrivial centre $C$, then $G / C$ has trivial centre.
(b) If $G \neq 1$, there exists a nontrivial homomorphism $h : \mathbb { Z } \rightarrow G$.
(c) If $| G | = p ^ { 3 }$, for $p$ a prime, then $G$ is abelian.
(d) If $G$ is nonabelian, then it has a nontrivial automorphism.

Let $C [ 0,1 ]$ be the space of continuous real-valued functions on the interval $[ 0,1 ]$. This is a ring under point-wise addition and multiplication. The following are true.
(a) For any $x \in [ 0,1 ]$, the ideal $M ( x ) = \{ f \in C [ 0,1 ] \mid f ( x ) = 0 \}$ is maximal.
(b) $C [ 0,1 ]$ is an integral domain.
(c) The group of units of $C [ 0,1 ]$ is cyclic.
(d) The linear functions form a vector-space basis of $C [ 0,1 ]$ over $\mathbb { R }$.

Let $h : \mathbb { C } \rightarrow \mathbb { C }$ be an analytic function such that $h ( 0 ) = 0 ; h \left( \frac { 1 } { 2 } \right) = 5$, and $| h ( z ) | < 10$ for $| z | < 1$. Then,
(a) the set $\{ z : | h ( z ) | = 5 \}$ is unbounded by the Maximum Principle;
(b) the set $\left\{ z : \left| h ^ { \prime } ( z ) \right| = 5 \right\}$ is a circle of strictly positive radius;
(c) $h ( 1 ) = 10$;
(d) regardless of what $h ^ { \prime }$ is, $h ^ { \prime \prime } \equiv 0$.

Suppose that $f ( z )$ is analytic, and satisfies the condition $\left| f ( z ) ^ { 2 } - 1 \right| = | f ( z ) - 1 | \cdot | f ( z ) + 1 | < 1$ on a non-empty connected open set $U$. Then,
(a) $f$ is constant.
(b) The imaginary part of $f , \operatorname { Im } ( f )$, is positive on $U$.
(c) The real part of $f , \operatorname { Re } ( f )$, is non-zero on $U$.
(d) $\operatorname { Re } ( f )$ is of fixed sign on $U$.