Part IA, 2018, Paper 1

Analysis I

• Paper 1, Section I, $4 \mathrm{D}$

  Analysis I | Part IA, 2018

  Define the radius of convergence $R$ of a complex power series $\sum a_{n} z^{n}$. Prove that $\sum a_{n} z^{n}$ converges whenever $|z|<R$ and diverges whenever $|z|>R$.

  If $\left|a_{n}\right| \leqslant\left|b_{n}\right|$ for all $n$ does it follow that the radius of convergence of $\sum a_{n} z^{n}$ is at least that of $\sum b_{n} z^{n}$ ? Justify your answer.

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• Paper 1, Section I, E

  Analysis I | Part IA, 2018

  Prove that an increasing sequence in $\mathbb{R}$ that is bounded above converges.

  Let $f: \mathbb{R} \rightarrow(0, \infty)$ be a decreasing function. Let $x_{1}=1$ and $x_{n+1}=x_{n}+f\left(x_{n}\right)$. Prove that $x_{n} \rightarrow \infty$ as $n \rightarrow \infty$.

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• Paper 1, Section II, $10 F$

  Analysis I | Part IA, 2018

  (a) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable at $x_{0} \in \mathbb{R}$. Show that $f$ is continuous at $x_{0}$.

  (b) State the Mean Value Theorem. Prove the following inequalities:

  $$\left|\cos \left(e^{-x}\right)-\cos \left(e^{-y}\right)\right| \leqslant|x-y| \quad \text { for } x, y \geqslant 0$$

  and

  $$\log (1+x) \leqslant \frac{x}{\sqrt{1+x}} \text { for } x \geqslant 0 .$$

  (c) Determine at which points the following functions from $\mathbb{R}$ to $\mathbb{R}$ are differentiable, and find their derivatives at the points at which they are differentiable:

  $$f(x)=\left\{\begin{array}{ll} |x|^{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0, \end{array} \quad g(x)=\cos (|x|), \quad h(x)=x|x|\right.$$

  (d) Determine the points at which the following function from $\mathbb{R}$ to $\mathbb{R}$ is continuous:

  $$f(x)= \begin{cases}0 & \text { if } x \notin \mathbb{Q} \text { or } x=0 \\ 1 / q & \text { if } x=p / q \text { where } p \in \mathbb{Z} \backslash\{0\} \text { and } q \in \mathbb{N} \text { are relatively prime. }\end{cases}$$

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• Paper 1, Section II, D

  Analysis I | Part IA, 2018

  (a) Let $q_{1}, q_{2}, \ldots$ be a fixed enumeration of the rationals in $[0,1]$. For positive reals $a_{1}, a_{2}, \ldots$, define a function $f$ from $[0,1]$ to $\mathbb{R}$ by setting $f\left(q_{n}\right)=a_{n}$ for each $n$ and $f(x)=0$ for $x$ irrational. Prove that if $a_{n} \rightarrow 0$ then $f$ is Riemann integrable. If $a_{n} \rightarrow 0$, can $f$ be Riemann integrable? Justify your answer.

  (b) State and prove the Fundamental Theorem of Calculus.

  Let $f$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}$, and set $g(x)=f^{\prime}(x)$ for $0 \leqslant x \leqslant 1$. Must $g$ be Riemann integrable on $[0,1]$ ?

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• Paper 1, Section II, E

  Analysis I | Part IA, 2018

  State and prove the Comparison Test for real series.

  Assume $0 \leqslant x_{n}<1$ for all $n \in \mathbb{N}$. Show that if $\sum x_{n}$ converges, then so do $\sum x_{n}^{2}$ and $\sum \frac{x_{n}}{1-x_{n}}$. In each case, does the converse hold? Justify your answers.

  Let $\left(x_{n}\right)$ be a decreasing sequence of positive reals. Show that if $\sum x_{n}$ converges, then $n x_{n} \rightarrow 0$ as $n \rightarrow \infty$. Does the converse hold? If $\sum x_{n}$ converges, must it be the case that $(n \log n) x_{n} \rightarrow 0$ as $n \rightarrow \infty$ ? Justify your answers.

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• Paper 1, Section II, F

  Analysis I | Part IA, 2018

  (a) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function, and let $x \in \mathbb{R}$. Define what it means for $f$ to be continuous at $x$. Show that $f$ is continuous at $x$ if and only if $f\left(x_{n}\right) \rightarrow f(x)$ for every sequence $\left(x_{n}\right)$ with $x_{n} \rightarrow x$.

  (b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a non-constant polynomial. Show that its image $\{f(x): x \in \mathbb{R}\}$ is either the real line $\mathbb{R}$, the interval $[a, \infty)$ for some $a \in \mathbb{R}$, or the interval $(-\infty, a]$ for some $a \in \mathbb{R}$.

  (c) Let $\alpha>1$, let $f:(0, \infty) \rightarrow \mathbb{R}$ be continuous, and assume that $f(x)=f\left(x^{\alpha}\right)$ holds for all $x>0$. Show that $f$ must be constant.

  Is this also true when the condition that $f$ be continuous is dropped?

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Vectors and Matrices

• Paper 1, Section I, A

  Vectors and Matrices | Part IA, 2018

  The map $\boldsymbol{\Phi}(\mathbf{x})=\alpha(\mathbf{n} \cdot \mathbf{x}) \mathbf{n}-\mathbf{n} \times(\mathbf{n} \times \mathbf{x})$ is defined for $\mathbf{x} \in \mathbb{R}^{3}$, where $\mathbf{n}$ is a unit vector in $\mathbb{R}^{3}$ and $\alpha$ is a real constant.

  (i) Find the values of $\alpha$ for which the inverse map $\Phi^{-1}$ exists, as well as the inverse map itself in these cases.

  (ii) When $\boldsymbol{\Phi}$ is not invertible, find its image and kernel. What is the value of the rank and the value of the nullity of $\Phi$ ?

  (iii) Let $\mathbf{y}=\mathbf{\Phi}(\mathbf{x})$. Find the components $A_{i j}$ of the matrix $A$ such that $y_{i}=A_{i j} x_{j}$. When $\Phi$ is invertible, find the components of the matrix $B$ such that $x_{i}=B_{i j} y_{j}$.

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• Paper 1, Section I, C

  Vectors and Matrices | Part IA, 2018

  For $z, w \in \mathbb{C}$ define the principal value of $z^{w}$. State de Moivre's theorem.

  Hence solve the equations (i) $z^{6}=\sqrt{3}+i$, (ii) $z^{1 / 6}=\sqrt{3}+i$, (iii) $i^{z}=\sqrt{3}+i$ (iv) $\left(e^{5 i \pi / 2}\right)^{z}=\sqrt{3}+i$

  [In each expression, the principal value is to be taken.]

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• Paper 1, Section II, $5 \mathbf{V}$

  Vectors and Matrices | Part IA, 2018

  Let $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}$ be non-zero real vectors. Define the inner product $\mathbf{x} \cdot \mathbf{y}$ in terms of the components $x_{i}$ and $y_{i}$, and define the norm $|\mathbf{x}|$. Prove that $\mathbf{x} \cdot \mathbf{y} \leqslant|\mathbf{x}||\mathbf{y}|$. When does equality hold? Express the angle between $\mathbf{x}$ and $\mathbf{y}$ in terms of their inner product.

  Use suffix notation to expand $(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{b} \times \mathbf{c})$.

  Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be given unit vectors in $\mathbb{R}^{3}$, and let $\mathbf{m}=(\mathbf{a} \times \mathbf{b})+(\mathbf{b} \times \mathbf{c})+(\mathbf{c} \times \mathbf{a})$. Obtain expressions for the angle between $\mathbf{m}$ and each of $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, in terms of $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $|\mathbf{m}|$. Calculate $|\mathbf{m}|$ for the particular case when the angles between $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are all equal to $\theta$, and check your result for an example with $\theta=0$ and an example with $\theta=\pi / 2$.

  Consider three planes in $\mathbb{R}^{3}$ passing through the points $\mathbf{p}, \mathbf{q}$ and $\mathbf{r}$, respectively, with unit normals $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, respectively. State a condition that must be satisfied for the three planes to intersect at a single point, and find the intersection point.

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• Paper 1, Section II, A

  Vectors and Matrices | Part IA, 2018

  What is the definition of an orthogonal matrix $M$ ?

  Write down a $2 \times 2$ matrix $R$ representing the rotation of a 2-dimensional vector $(x, y)$ by an angle $\theta$ around the origin. Show that $R$ is indeed orthogonal.

  Take a matrix

  $$A=\left(\begin{array}{ll} a & b \\ b & c \end{array}\right)$$

  where $a, b, c$ are real. Suppose that the $2 \times 2$ matrix $B=R A R^{T}$ is diagonal. Determine all possible values of $\theta$.

  Show that the diagonal entries of $B$ are the eigenvalues of $A$ and express them in terms of the determinant and trace of $A$.

  Using the above results, or otherwise, find the elements of the matrix

  $$\left(\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right)^{2 N}$$

  as a function of $N$, where $N$ is a natural number.

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• Paper 1, Section II, B

  Vectors and Matrices | Part IA, 2018

  Let $A$ be a real symmetric $n \times n$ matrix.

  (a) Prove the following:

  (i) Each eigenvalue of $A$ is real and there is a corresponding real eigenvector.

  (ii) Eigenvectors corresponding to different eigenvalues are orthogonal.

  (iii) If there are $n$ distinct eigenvalues then the matrix is diagonalisable.

  Assuming that $A$ has $n$ distinct eigenvalues, explain briefly how to choose (up to an arbitrary scalar factor) the vector $v$ such that $\frac{v^{T} A v}{v^{T} v}$ is maximised.

  (b) A scalar $\lambda$ and a non-zero vector $v$ such that

  $$A v=\lambda B v$$

  are called, for a specified $n \times n$ matrix $B$, respectively a generalised eigenvalue and a generalised eigenvector of $A$.

  Assume the matrix $B$ is real, symmetric and positive definite (i.e. $\left(u^{*}\right)^{T} B u>0$ for all non-zero complex vectors $u$ ).

  Prove the following:

  (i) If $\lambda$ is a generalised eigenvalue of $A$ then it is a root of $\operatorname{det}(A-\lambda B)=0$.

  (ii) Each generalised eigenvalue of $A$ is real and there is a corresponding real generalised eigenvector.

  (iii) Two generalised eigenvectors $u, v$, corresponding to different generalised eigenvalues, are orthogonal in the sense that $u^{T} B v=0$.

  (c) Find, up to an arbitrary scalar factor, the vector $v$ such that the value of $F(v)=\frac{v^{T} A v}{v^{T} B v}$ is maximised, and the corresponding value of $F(v)$, where

  $$A=\left(\begin{array}{ccc} 4 & 2 & 0 \\ 2 & 3 & 0 \\ 0 & 0 & 10 \end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 3 \end{array}\right)$$

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• Paper 1, Section II, B

  Vectors and Matrices | Part IA, 2018

  (a) Consider the matrix

  $$R=\left(\begin{array}{ccc} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right)$$

  representing a rotation about the $z$-axis through an angle $\theta$.

  Show that $R$ has three eigenvalues in $\mathbb{C}$ each with modulus 1 , of which one is real and two are complex (in general), and give the relation of the real eigenvector and the two complex eigenvalues to the properties of the rotation.

  Now consider the rotation composed of a rotation by angle $\pi / 2$ about the $z$-axis followed by a rotation by angle $\pi / 2$ about the $x$-axis. Determine the rotation axis $\mathbf{n}$ and the magnitude of the angle of rotation $\phi$.

  (b) A surface in $\mathbb{R}^{3}$ is given by

  $$7 x^{2}+4 x y+3 y^{2}+2 x z+3 z^{2}=1 .$$

  By considering a suitable eigenvalue problem, show that the surface is an ellipsoid, find the lengths of its semi-axes and find the position of the two points on the surface that are closest to the origin.

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