• # 1.I.1B

(a) Write the permutation

$(123)(234)$

as a product of disjoint cycles. Determine its order. Compute its sign.

(b) Elements $x$ and $y$ of a group $G$ are conjugate if there exists a $g \in G$ such that $g x g^{-1}=y .$

Show that if permutations $x$ and $y$ are conjugate, then they have the same sign and the same order. Is the converse true? (Justify your answer with a proof or counterexample.)

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• # 1.I.2D

Find the characteristic equation, the eigenvectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$, and the corresponding eigenvalues $\lambda_{\mathbf{a}}, \lambda_{\mathbf{b}}, \lambda_{\mathbf{c}}, \lambda_{\mathbf{d}}$ of the matrix

$A=\left(\begin{array}{cccc} i & 1 & 0 & 0 \\ 1 & i & 0 & 0 \\ 0 & 0 & i & 1 \\ 0 & 0 & -1 & i \end{array}\right)$

Show that $\{\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}\}$ spans the complex vector space $\mathbb{C}^{4}$.

Consider the four subspaces of $\mathbb{C}^{4}$ defined parametrically by

$\mathbf{z}=s \mathbf{a}, \quad \mathbf{z}=s \mathbf{b}, \quad \mathbf{z}=s \mathbf{c}, \quad \mathbf{z}=s \mathbf{d} \quad\left(\mathbf{z} \in \mathbb{C}^{4}, s \in \mathbb{C}\right)$

Show that multiplication by $A$ maps three of these subspaces onto themselves, and the remaining subspace into a smaller subspace to be specified.

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• # 1.II.5B

(a) In the standard basis of $\mathbb{R}^{2}$, write down the matrix for a rotation through an angle $\theta$ about the origin.

(b) Let $A$ be a real $3 \times 3$ matrix such that $\operatorname{det} A=1$ and $A A^{\mathrm{T}}=I$, where $A^{\mathrm{T}}$ is the transpose of $A$.

(i) Suppose that $A$ has an eigenvector $\mathbf{v}$ with eigenvalue 1 . Show that $A$ is a rotation through an angle $\theta$ about the line through the origin in the direction of $\mathbf{v}$, where $\cos \theta=\frac{1}{2}($ trace $A-1)$.

(ii) Show that $A$ must have an eigenvector $\mathbf{v}$ with eigenvalue 1 .

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• # 1.II.6A

Let $\alpha$ be a linear map

$\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} .$

Define the kernel $K$ and image $I$ of $\alpha$.

Let $\mathbf{y} \in \mathbb{R}^{3}$. Show that the equation $\alpha \mathbf{x}=\mathbf{y}$ has a solution $\mathbf{x} \in \mathbb{R}^{3}$ if and only if $\mathbf{y} \in I$

Let $\alpha$ have the matrix

$\left(\begin{array}{ccc} 1 & 1 & t \\ 0 & t & -2 b \\ 1 & t & 0 \end{array}\right)$

with respect to the standard basis, where $b \in \mathbb{R}$ and $t$ is a real variable. Find $K$ and $I$ for $\alpha$. Hence, or by evaluating the determinant, show that if $0 and $\mathbf{y} \in I$ then the equation $\alpha \mathbf{x}=\mathbf{y}$ has a unique solution $\mathbf{x} \in \mathbb{R}^{3}$ for all values of $t$.

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• # 1.II.7B

(i) State the orbit-stabilizer theorem for a group $G$ acting on a set $X$.

(ii) Denote the group of all symmetries of the cube by $G$. Using the orbit-stabilizer theorem, show that $G$ has 48 elements.

Does $G$ have any non-trivial normal subgroups?

Let $L$ denote the line between two diagonally opposite vertices of the cube, and let

$H=\{g \in G \mid g L=L\}$

be the subgroup of symmetries that preserve the line. Show that $H$ is isomorphic to the direct product $S_{3} \times C_{2}$, where $S_{3}$ is the symmetric group on 3 letters and $C_{2}$ is the cyclic group of order 2 .

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• # 1.II.8D

Let $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ and $\mathbf{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right)$ be non-zero vectors in $\mathbb{R}^{n}$. What is meant by saying that $\mathbf{x}$ and $\mathbf{y}$ are linearly independent? What is the dimension of the subspace of $\mathbb{R}^{n}$ spanned by $\mathbf{x}$ and $\mathbf{y}$ if they are (1) linearly independent, (2) linearly dependent?

Define the scalar product $\mathbf{x} \cdot \mathbf{y}$ for $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}$. Define the corresponding norm $\|\mathbf{x}\|$ of $\mathbf{x} \in \mathbb{R}^{n}$. State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality.

By means of a sketch, give a geometric interpretation of the scalar product $\mathbf{x} \cdot \mathbf{y}$ in the case $n=3$, relating the value of $\mathbf{x} \cdot \mathbf{y}$ to the angle $\alpha$ between the directions of $\mathbf{x}$ and $\mathbf{y}$.

What is a unit vector? Let $\mathbf{u}, \mathbf{v}, \mathbf{w}$ be unit vectors in $\mathbb{R}^{3}$. Let

$S=\mathbf{u} \cdot \mathbf{v}+\mathbf{v} \cdot \mathbf{w}+\mathbf{w} \cdot \mathbf{u}$

Show that

(i) for any fixed, linearly independent $\mathbf{u}$ and $\mathbf{v}$, the minimum of $S$ over $\mathbf{w}$ is attained when $\mathbf{w}=\lambda(\mathbf{u}+\mathbf{v})$ for some $\lambda \in \mathbb{R}$;

(ii) $\lambda \leqslant-\frac{1}{2}$ in all cases;

(iii) $\lambda=-1$ and $S=-3 / 2$ in the case where $\mathbf{u} \cdot \mathbf{v}=\cos (2 \pi / 3)$.

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• # 1.I.3B

Define what it means for a function of a real variable to be differentiable at $x \in \mathbb{R}$.

Prove that if a function is differentiable at $x \in \mathbb{R}$, then it is continuous there.

Show directly from the definition that the function

$f(x)= \begin{cases}x^{2} \sin (1 / x) & x \neq 0 \\ 0 & x=0\end{cases}$

is differentiable at 0 with derivative 0 .

Show that the derivative $f^{\prime}(x)$ is not continuous at 0 .

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• # 1.I.4C

Explain what is meant by the radius of convergence of a power series.

Find the radius of convergence $R$ of each of the following power series: (i) $\sum_{n=1}^{\infty} n^{-2} z^{n}$, (ii) $\sum_{n=1}^{\infty}\left(n+\frac{1}{2^{n}}\right) z^{n}$.

In each case, determine whether the series converges on the circle $|z|=R$.

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• # 1.II.10F

State without proof the Integral Comparison Test for the convergence of a series $\sum_{n=1}^{\infty} a_{n}$ of non-negative terms.

Determine for which positive real numbers $\alpha$ the series $\sum_{n=1}^{\infty} n^{-\alpha}$ converges.

In each of the following cases determine whether the series is convergent or divergent: (i) $\sum_{n=3}^{\infty} \frac{1}{n \log n}$, (ii) $\sum_{n=3}^{\infty} \frac{1}{(n \log n)(\log \log n)^{2}}$, (iii) $\sum_{n=3}^{\infty} \frac{1}{n^{(1+1 / n) \log n}}$.

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• # 1.II.11B

Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous. Define the integral $\int_{a}^{b} f(x) d x$. (You are not asked to prove existence.)

Suppose that $m, M$ are real numbers such that $m \leqslant f(x) \leqslant M$ for all $x \in[a, b]$. Stating clearly any properties of the integral that you require, show that

$m(b-a) \leqslant \int_{a}^{b} f(x) d x \leqslant M(b-a) .$

The function $g:[a, b] \rightarrow \mathbb{R}$ is continuous and non-negative. Show that

$m \int_{a}^{b} g(x) d x \leqslant \int_{a}^{b} f(x) g(x) d x \leqslant M \int_{a}^{b} g(x) d x$

Now let $f$ be continuous on $[0,1]$. By suitable choice of $g$ show that

$\lim _{n \rightarrow \infty} \int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x=f(0),$

and by making an appropriate change of variable, or otherwise, show that

$\lim _{n \rightarrow \infty} \int_{0}^{1} n f(x) e^{-n x} d x=f(0) .$

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• # 1.II.12C

State carefully the formula for integration by parts for functions of a real variable.

Let $f:(-1,1) \rightarrow \mathbb{R}$ be infinitely differentiable. Prove that for all $n \geqslant 1$ and all $t \in(-1,1)$,

$f(t)=f(0)+f^{\prime}(0) t+\frac{1}{2 !} f^{\prime \prime}(0) t^{2}+\ldots+\frac{1}{(n-1) !} f^{(n-1)}(0) t^{n-1}+\frac{1}{(n-1) !} \int_{0}^{t} f^{(n)}(x)(t-x)^{n-1} d x .$

By considering the function $f(x)=\log (1-x)$ at $x=1 / 2$, or otherwise, prove that the series

$\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}$

converges to $\log 2$.

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• # 1.II.9F

Prove the Axiom of Archimedes.

Let $x$ be a real number in $[0,1]$, and let $m, n$ be positive integers. Show that the limit

$\lim _{m \rightarrow \infty}\left[\lim _{n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right]$

exists, and that its value depends on whether $x$ is rational or irrational.

[You may assume standard properties of the cosine function provided they are clearly stated.]

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