Suppose $\varphi = (\varphi_2, \ldots, \varphi_n) : \mathbb{R}^n \rightarrow \mathbb{R}^{n-1}$ is a $C^2$ function, i.e. all second order partial derivatives of the $\varphi_i$ exist and are continuous. Show that the symbolic determinant $$\left| \begin{array}{cccc} \frac{\partial}{\partial x_1} & \frac{\partial \varphi_2}{\partial x_1} & \ldots & \frac{\partial \varphi_n}{\partial x_1} \\ \vdots & \vdots & & \vdots \\ \frac{\partial}{\partial x_n} & \frac{\partial \varphi_2}{\partial x_n} & \ldots & \frac{\partial \varphi_n}{\partial x_n} \end{array} \right|$$ vanishes identically.

Evaluate:
(a) $\lim _ { x \rightarrow 1 } \frac { n - \sum _ { k = 1 } ^ { n } x ^ { k } } { 1 - x }$
(b) $\lim _ { x \rightarrow 0 } \frac { e ^ { - 1 / x } } { x }$

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.

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 $U$.

Consider the following subsets of $\mathbb { R } ^ { 2 } : X _ { 1 } = \left\{ \left. \left( x , \sin \frac { 1 } { x } \right) \right\rvert \, 0 < x < 1 \right\} , X _ { 2 } = [ 0,1 ] \times \{ 0 \}$, and $X _ { 3 } = \{ ( 0,1 ) \}$. Then,
(a) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is a connected set;
(b) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is a path-connected set;
(c) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is not path-connected, but $X _ { 1 } \cup X _ { 2 }$ is path-connected;
(d) $X _ { 1 } \cup X _ { 2 }$ is not path-connected, but every open neighbourhood of a point in this set contains a smaller open neighbourhood which is path-connected.

For a set $A \subset \mathbb { R }$, denote by $\operatorname { Cl } ( A )$ the closure of $A$, and by $\operatorname { Int } ( A )$ the interior of $A$. There is a set $A \subset \mathbb { R }$ such that
(a) $A , Cl ( A )$, and $\operatorname { Int } ( A )$ are pairwise distinct;
(b) $A , Cl ( A ) , \operatorname { Int } ( A )$, and $\operatorname { Cl } ( \operatorname { Int } ( A ) )$ are pairwise distinct;
(c) $A , \operatorname { Cl } ( A ) , \operatorname { Int } ( A )$, and $\operatorname { Int } ( \operatorname { Cl } ( A ) )$ are pairwise distinct;
(d) $A , Cl ( A ) , \operatorname { Int } ( A ) , \operatorname { Int } ( Cl ( A ) )$, and $\operatorname { Cl } ( \operatorname { Int } ( A ) )$ are pairwise distinct.

Let $f , g : [ 0,1 ] \rightarrow \mathbb { R }$ be given by $$\begin{gathered} f ( x ) : = \begin{cases} x ^ { 2 } & \text { if } x \text { is rational, } \\ 0 & \text { if } x \text { is irrational; } \end{cases} \\ g ( x ) : = \begin{cases} 1 / q & \text { if } x = \frac { p } { q } \text { is rational, with } \operatorname { gcd } ( p , q ) = 1 , \\ 0 & \text { if } x \text { is irrational. } \end{cases} \end{gathered}$$ Then,
(a) $g$ is Riemann integrable, but not $f$;
(b) both $f$ and $g$ are Riemann integrable;
(c) the Riemann integral $\int _ { 0 } ^ { 1 } f ( x ) d x = 0$;
(d) the Riemann integral $\int _ { 0 } ^ { 1 } g ( x ) d x = 0$.

Let $C$ be the ellipse $24 x ^ { 2 } + x y + 5 y ^ { 2 } + 3 x + 2 y + 1 = 0$. Then, the line integral $\oint \left( x ^ { 2 } y \, d y + x y ^ { 2 } \, d x \right)$
(a) lies in $( 0,1 )$;
(b) is 1;
(c) is either 1 or $-1$ depending on whether $C$ is traversed clockwise or counterclockwise;
(d) is 0.