(i) Let $f$ be continuous on $[ - 1,1 ]$ and differentiable at 0. For $x \neq 0$, define a function $g$ by $g ( x ) = \frac { f ( x ) - f ( 0 ) } { x }$. Can $g ( 0 )$ be defined so that the extended function $g$ is continuous at 0?
(ii) For $f$ as in part (i), show that the following limit exists.
$$\lim _ { r \rightarrow 0 ^ { + } } \left( \int _ { - 1 } ^ { - r } \frac { f ( x ) } { x } d x + \int _ { r } ^ { 1 } \frac { f ( x ) } { x } d x \right)$$
(iii) Give an example showing that without the hypothesis of $f$ being differentiable at 0, the conclusion in (ii) need not hold.

(i) Let $f ( x ) = a _ { n } x ^ { n } + \cdots + a _ { 1 } x + a _ { 0 }$ be a polynomial, where $a _ { 0 } , \ldots , a _ { n }$ are real numbers with $a _ { n } \neq 0$. Define the ``discrete derivative of $f$'', denoted $\Delta f$, to be the function given by $\Delta f ( x ) = f ( x ) - f ( x - 1 )$. Show that $\Delta f$ is also a polynomial and find its leading term.
(ii) For integers $n \geq 0$, define polynomials $p _ { n }$ of degree $n$ as follows: $p _ { 0 } ( x ) = 1$ and for $n > 0$, let $p _ { n } ( x ) = \frac { 1 } { n ! } x ( x - 1 ) ( x - 2 ) \cdots ( x - n + 1 )$. So we have
$$p _ { 0 } ( x ) = 1 \quad , \quad p _ { 1 } ( x ) = x \quad , \quad p _ { 2 } ( x ) = \frac { x ( x - 1 ) } { 2 } \quad , \quad p _ { 3 } ( x ) = \frac { x ( x - 1 ) ( x - 2 ) } { 3 ! } \quad \ldots$$
Show that for any polynomial $f$ of degree $n$, there exist unique real numbers $b _ { 0 } , b _ { 1 } , \ldots , b _ { n }$ such that $f ( x ) = \sum _ { i = 0 } ^ { n } b _ { i } p _ { i } ( x )$.
(iii) Now suppose that $f ( x )$ is a polynomial such that for each integer $m , f ( m )$ is also an integer. Using the above parts (or otherwise), show that for such $f$, the $b _ { i }$ obtained in part (ii) are integers.

(i) Two circles $G _ { 1 } , G _ { 2 }$ intersect at points $X , Y$. Choose two other points $A , B$ on $G _ { 1 }$ as shown in the figure. The line segment from $A$ to $X$ is extended to intersect $G _ { 2 }$ at point $L$. The line segment from $L$ to $Y$ is extended to meet $G _ { 1 }$ at point $C$. Likewise the line segment from $B$ to $Y$ is extended to meet $G _ { 2 }$ at point $M$ and the segment from $M$ to $X$ is extended to meet $G _ { 1 }$ at point $D$. Show that $AB$ is parallel to $CD$.
(ii) A triangle $CDE$ is given. A point $A$ is chosen between $D$ and $E$. A point $B$ is chosen between $C$ and $E$ so that $AB$ is parallel to $CD$. Let $F$ denote the point of intersection of segments $AC$ and $BD$. Show that the line joining $E$ and $F$ bisects both segments $AB$ and segment $CD$. (Hint: You may use Ceva's theorem. Alternatively, you may additionally assume that the trapezium $ABCD$ is a cyclic quadrilateral and proceed.)
(iii) Using parts (i) and (ii) describe a procedure to do the following task: given two circles $G _ { 1 }$ and $G _ { 2 }$ intersecting at two points $X$ and $Y$ determine the center of each circle using only a straightedge. Note: Recall that a straightedge is a ruler without any markings. Given two points $A , B$, a straightedge allows one to construct the line segment joining $A , B$. Also, given any two non-parallel segments, we can use a straightedge to find the intersection point of the lines containing the two segments by extending them if necessary.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+1) = f(x)$ for all $x \in \mathbb{R}$. Which of the following statement(s) is/are true?
(A) $f$ is bounded.
(B) $f$ is bounded if it is continuous.
(C) $f$ is differentiable if it is continuous.
(D) $f$ is uniformly continuous if it is continuous.

Let $W \subset \mathbb{R}^n$ be a linear subspace of dimension at most $n-1$. Which of the following statement(s) is/are true?
(A) $W$ is nowhere dense.
(B) $W$ is closed.
(C) $\mathbb{R}^n \backslash W$ is connected.
(D) $\mathbb{R}^n \backslash W$ is not connected.

Let $G$ be a finite group. An element $a \in G$ is called a square if there exists $x \in G$ such that $x^2 = a$. Which of the following statement(s) is/are true?
(A) If $a, b \in G$ are not squares, $ab$ is a square.
(B) Suppose that $G$ is cyclic. Then if $a, b \in G$ are not squares, $ab$ is a square.
(C) $G$ has a normal subgroup.
(D) If every proper subgroup of $G$ is cyclic then $G$ is cyclic.

Let $A \in M_{m \times n}(\mathbb{R})$ and let $b_0 \in \mathbb{R}^m$. Suppose the system of equations $Ax = b_0$ has a unique solution. Which of the following statement(s) is/are true?
(A) $Ax = b$ has a solution for every $b \in \mathbb{R}^m$.
(B) If $Ax = b$ has a solution then it is unique.
(C) $Ax = 0$ has a unique solution.
(D) $A$ has rank $m$.

Let $A \in M_{n \times n}(\mathbb{C})$. Which of the following statement(s) is/are true?
(A) There exists $B \in M_{n \times n}(\mathbb{C})$ such that $B^2 = A$.
(B) $A$ is diagonalizable.
(C) There exists an invertible matrix $P$ such that $PAP^{-1}$ is upper-triangular.
(D) $A$ has an eigenvalue.

Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be a function. Which of the following statement(s) is/are true?
(A) Consider $f$ as a function $(f_1, f_2) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose that for $i = 1, 2$, both $\frac{\partial f_i}{\partial X}$ and $\frac{\partial f_i}{\partial Y}$ exist and are continuous. Then $f$ is entire.
(B) Assume that $f$ is entire and $|f(z)| < 1$ for all $z \in \mathbb{C}$. Then $f$ is constant.
(C) Assume that $f$ is entire and $\operatorname{Im}(f(z)) > 0$ for all $z \in \mathbb{C}$. Then $f$ is constant.

Let $\mathcal{C}(\mathbb{R})$ be the $\mathbb{R}$-vector space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Let $a_1, a_2, a_3$ be distinct real numbers. For $i = 1, 2, 3$, let $f_i \in \mathcal{C}(\mathbb{R})$ be the function $f_i(t) = e^{a_i t}$. Which of the following statement(s) is/are true?
(A) $f_1, f_2$ and $f_3$ are linearly independent.
(B) $f_1, f_2$ and $f_3$ are linearly dependent.
(C) $f_1, f_2$ and $f_3$ form a basis of $\mathcal{C}(\mathbb{R})$.

Which of the following statement(s) is/are true?
(A) The series $\sum_{n=1}^{\infty} \mathrm{e}^{-n^2}$ converges.
(B) The series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ converges.
(C) The series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ converges absolutely.
(D) The series $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2}$ converges uniformly on $\mathbb{R}$.

What is the dimension of the ring $\mathbb{Q}[x]/\left((x+1)^2\right)$ as a $\mathbb{Q}$-vector space?

Evaluate $\lim_{n \rightarrow \infty} \left[\frac{\pi \sum_{i=1}^{n} \sin\left(\frac{i\pi}{n}\right)}{n}\right]$.

Show that the set of rank two matrices in $M_{2 \times 3}(\mathbb{R})$ is open.

(A) Let $F$ be a finite field extension of $\mathbb{Q}$. Show that any field homomorphism $\phi : F \rightarrow F$ is an isomorphism. (Note that $\phi(1) = 1$ by definition.)
(B) Let $F$ be a finite field whose characteristic is not 2. Let $F^\times$ denote the multiplicative group of nonzero elements of $F$. An element $a \in F^\times$ is called a square if there exists $x \in F^\times$ such that $x^2 = a$. Show that exactly half the elements of $F^\times$ are squares.

Let $n \in \mathbb{N}$. Show that the determinant map $\det : M_{n \times n}(\mathbb{R}) \rightarrow \mathbb{R}$ is infinitely differentiable and compute the total derivative $d(\operatorname{det})$ at every point $A \in M_{n \times n}(\mathbb{R})$. Find a necessary and sufficient condition on the rank of $A$ for $d(\operatorname{det}) = 0$ at $A$.

Let $a_i, i \in \mathbb{R}$ be non-negative real numbers such that $\sup\left\{\sum_{i \in F} a_i \mid F \subseteq \mathbb{R} \text{ a finite subset}\right\}$ is finite. Show that $a_i = 0$ except for countably many $i \in \mathbb{R}$. Give an example to show that 'countably' cannot be replaced by 'finite'. (Hint: consider $F_n := \left\{i \left\lvert\, a_i \geq \frac{1}{n}\right.\right\}$.)

Let $G$ be a finite group of order $2n$ for some integer $n$. Consider the map $\phi : G \rightarrow G$ given by $\phi(a) = a^2$. Show that $\phi$ is not surjective.

Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be an entire function.
(A) Construct a sequence $\{z_n\}$ in $\mathbb{C}$ such that $|z_n| \rightarrow \infty$ and $e^{z_n} \rightarrow 1$.
(B) Show that the function $g(z) = f\left(e^z\right)$ is not a polynomial.

For $F = \mathbb{R}$ and $F = \mathbb{C}$, let $O_n(F) = \left\{A \in M_{n \times n}(F) \mid AA^t = I_n\right\}$.
(A) Show that $O_n(\mathbb{R})$ is compact.
(B) Is $O_n(\mathbb{R})$ connected? Justify.
(C) Is $O_n(\mathbb{C})$ compact? Justify.

Let $\Omega$ be a region in $\mathbb{C}$. Let $\{a_n\}$ be a sequence of nonzero elements in $\Omega$ such that $a_n \rightarrow 0$ as $n \rightarrow \infty$. Let $\{b_n\}$ be a sequence of complex numbers such that $\lim_{n \rightarrow \infty} \frac{b_n}{a_n^k} = 0$ for every nonnegative integer $k$. Suppose that $f : \Omega \rightarrow \mathbb{C}$ is an entire function such that $f(a_n) = b_n$ for all $n$. Show that $b_n = 0$ for every $n$.

Let $G$ be a finite group of order $n$ and let $H$ be a subgroup of $G$ of order $m$. Assume that $\left(\frac{n}{m}\right)! < 2n$. Show that $G$ is not simple, that is: $G$ has a nontrivial proper normal subgroup. (Hint: Think along the lines of Cayley's theorem.)

Let $$\mathcal{C}_0(\mathbb{R}) = \left\{f : \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ is continuous}, \lim_{x \rightarrow \infty} |f(x)| = 0 \text{ and } \lim_{x \rightarrow -\infty} |f(x)| = 0\right\},$$ and $\mathcal{C}_0^\infty(\mathbb{R}) = \left\{f \in \mathcal{C}_0(\mathbb{R}) \mid f \text{ is infinitely differentiable}\right\}$. Let $\phi \in \mathcal{C}_0(\mathbb{R})$. For $f \in \mathcal{C}_0(\mathbb{R})$, define $\phi^*(f) = f \circ \phi$.
(A) Show that $\phi^*(f) \in \mathcal{C}_0(\mathbb{R})$ if $f \in \mathcal{C}_0(\mathbb{R})$.
(B) If $\phi^*\left(\mathcal{C}_0^\infty(\mathbb{R})\right) \subseteq \mathcal{C}_0^\infty(\mathbb{R})$, then show that $\phi$ is infinitely differentiable.

Carefully solve the following series of questions. If you cannot solve an earlier part, you may still assume the result in it to solve a later part.
(a) For any polynomial $p(t)$, the limit $\lim_{t \rightarrow \infty} \frac{p(t)}{e^t}$ is independent of the polynomial $p$. Justify this statement and find the value of the limit.
(b) Consider the function defined by
$$\begin{aligned} q(x) &= e^{-1/x} \text{ when } x > 0 \\ &= 0 \text{ when } x = 0 \\ &= e^{1/x} \text{ when } x < 0 \end{aligned}$$
Show that $q'(0)$ exists and find its value. Why is it enough to calculate the relevant limit from only one side?
(c) Now for any positive integer $n$, show that $q^{(n)}(0)$ exists and find its value. Here $q(x)$ is the function in part (b) and $q^{(n)}(0)$ denotes its $n$-th derivative at $x = 0$.

You are given the following: a circle, one of its diameters $AB$ and a point $X$.
(a) Using only a straight-edge, show in the given figure how to draw a line perpendicular to $AB$ passing through $X$. No credit will be given without full justification. (Recall that a straight-edge is a ruler without any markings. Given two points, a straight-edge can be used to draw the line passing through the given points.)
(b) Do NOT draw any of your work for this part in the given figure. Reconsider your procedure to see if it can be made to work if the point $X$ is in some other position, e.g., when it is inside the circle or to the ``left/right'' of the circle. Clearly specify all positions of the point $X$ for which your procedure in part (a), or a small extension/variation of it, can be used to obtain the perpendicular to $AB$ through $X$. Justify your answer.