There is $2 \times 2$ real matrix with characteristic polynomial $x ^ { 2 } + 1$.

There are at least three non-isomorphic rings with 4 elements.

The group $( \mathbb { Q } , + )$ is a finitely generated abelian group.

$\mathbb { Q } ( \sqrt { 7 } )$ and $\mathbb { Q } ( \sqrt { 17 } )$ are isomorphic as fields.

A vector space of dimension $\geq 2$ can be expressed as a union of two proper subspaces.

There is a bijective analytic function from the complex plane to the upper half-plane.

There is a non-constant bounded analytic function on $\mathbb { C } \backslash \{ 0 \}$.

(a) Consider the ring $R$ of polynomials in $n$ variables with integer coefficients. Prove that the polynomial $f \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { n } \right) = x _ { 1 } x _ { 2 } \cdots x _ { n }$ has $2 ^ { n + 1 } - 2$ non-constant polynomials in $R$ dividing it.
(b) Let $p _ { 1 } , p _ { 2 } , \ldots , p _ { n }$ be distinct prime numbers. Then show that the number $N = p _ { 1 } p _ { 2 } ^ { 2 } p _ { 3 } ^ { 3 } \cdots p _ { n } ^ { n }$ has $( n + 1 ) !$ positive divisors.

Let $f ( x ) = \left( x ^ { 2 } - 2 \right) \left( x ^ { 2 } - 3 \right) \left( x ^ { 2 } - 6 \right)$. For every prime number $p$, show that $f ( x ) \equiv 0 ( \bmod p )$ has a solution in $\mathbb { Z }$.

Let $\mathbf { S }$ denote the group of all those permutations of the English alphabet that fix the letters T,E,N,D,U,L,K,A and R. Other letters may or may not be fixed. Show that $\mathbf { S }$ has elements $\sigma , \tau$ of order 36 and 39 respectively, but does not have any element of order 37 or 38.

Show that there are at least two non-isomorphic groups of order 198. Show that in all those groups the number of elements of order 11 is the same.

Suppose $f , g , h$ are functions from the set of positive real numbers into itself satisfying $f ( x ) g ( y ) = h \left( \sqrt { x ^ { 2 } + y ^ { 2 } } \right)$ for all $x , y \in ( 0 , \infty )$. Show that the three functions $f ( x ) / g ( x ) , g ( x ) / h ( x )$, and $h ( x ) / f ( x )$ are all constant.

Let $a , b > 0$.
(a) Prove that $\lim _ { n \rightarrow \infty } \left( a ^ { n } + b ^ { n } \right) ^ { 1 / n } = \max \{ a , b \}$.
(b) Define a sequence by $x _ { 1 } = a , x _ { 2 } = b$ and $x _ { n } = \frac { 1 } { 2 } \left( x _ { n - 1 } + x _ { n - 2 } \right)$ for $n > 2$. Show that $\left\{ x _ { n } \right\}$ is a convergent sequence.

Let $f : \mathbb { C } \rightarrow \mathbb { C }$ be an entire function with the following property: In the power series expansion around any $a \in \mathbb { C }$, given as $f ( z ) = \sum _ { n = o } ^ { \infty } c _ { n } ( a ) ( z - a ) ^ { n }$, the coefficient $c _ { n } ( a )$ is zero for some $n$ ( with $n$ depending on $a$). Show that $f ( z )$ is in fact a polynomial.

(a) Show that in a Hausdorff topological space any compact set is closed.
(b) If $\left( X , d _ { 1 } \right)$ and $\left( Y , d _ { 2 } \right)$ are two metric spaces that are homeomorphic then does completeness of $\left( X , d _ { 1 } \right)$ imply the completeness of $\left( Y , d _ { 2 } \right)$? Give reasons for your answer.

Fix an integer $n > 1$. Show that there is a real $n \times n$ diagonal matrix $D$ such that the condition $A D = D A$ is valid only for a diagonal matrix $A$.

Given positive real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { 2011 }$ whose product $a _ { 1 } a _ { 2 } \cdots a _ { 2011 }$ is 1 , what can you say about their sum $S = a _ { 1 } + a _ { 2 } + \cdots + a _ { 2011 }$ ?
(A) $S$ can be any positive number.
(B) $1 \leq S \leq 2011$.
(C) $2011 \leq S$ and $S$ is unbounded above.
(D) $2011 \leq S$ and $S$ is bounded above.

Show that there are infinitely many perfect squares that can be written as a sum of six consecutive natural numbers. Find the smallest such square.

Show that there is no solid figure with exactly 11 faces such that each face is a polygon having an odd number of sides.

A real-valued function $f$ defined on a closed interval $[ a , b ]$ has the properties that $f ( a ) = f ( b ) = 0$ and $f ( x ) = f ^ { \prime } ( x ) + f ^ { \prime \prime } ( x )$ for all $x$ in $[ a , b ]$. Show that $f ( x ) = 0$ for all $x$ in $[ a , b ]$.

Show that there are exactly 16 pairs of integers $( x , y )$ such that $11 x + 8 y + 17 = x y$. You need not list the solutions.

A function $g$ from a set X to itself satisfies $g ^ { m } = g ^ { n }$ for positive integers $m$ and $n$ with $m > n$. Here $g ^ { n }$ stands for $g \circ g \circ \cdots \circ g$ ( $n$ times). Show that $g$ is one-to-one if and only if $g$ is onto.

In a quadrilateral ABCD , angles at vertices B and D are right angles. AM and CN are respectively altitudes of the triangles ABD and CBD. Show that $BN = DM$.

The function $f : \mathbb { R } ^ { n } \rightarrow \mathbb { R }$, defined as $f \left( x _ { 1 } , \cdots , x _ { n } \right) = \operatorname { Max } \left\{ \left| x _ { i } \right| \right\} , i = 1 , \cdots , n$, is uniformly continuous.

Let $x _ { n }$ be a sequence with the following property: Every subsequence of $x _ { n }$ has a further subsequence which converges to $x$. Then the sequence $x _ { n }$ converges to $x$.