• Paper 1, Section I, $1 A$

(i) The spherical polar unit basis vectors $\mathbf{e}_{r}, \mathbf{e}_{\phi}$ and $\mathbf{e}_{\theta}$ in $\mathbb{R}^{3}$ are given in terms of the Cartesian unit basis vectors $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ by

\begin{aligned} &\mathbf{e}_{r}=\mathbf{i} \cos \phi \sin \theta+\mathbf{j} \sin \phi \sin \theta+\mathbf{k} \cos \theta \\ &\mathbf{e}_{\theta}=\mathbf{i} \cos \phi \cos \theta+\mathbf{j} \sin \phi \cos \theta-\mathbf{k} \sin \theta \\ &\mathbf{e}_{\phi}=-\mathbf{i} \sin \phi+\mathbf{j} \cos \phi \end{aligned}

Express $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ in terms of $\mathbf{e}_{r}, \mathbf{e}_{\phi}$ and $\mathbf{e}_{\theta}$.

(ii) Use suffix notation to prove the following identity for the vectors $\mathbf{A}, \mathbf{B}$, and $\mathbf{C}$ in $\mathbb{R}^{3}$ :

$(\mathbf{A} \times \mathbf{B}) \times(\mathbf{A} \times \mathbf{C})=(\mathbf{A} \cdot \mathbf{B} \times \mathbf{C}) \mathbf{A}$

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• Paper 1, Section I, B

For the equations

$\begin{gathered} p x+y+z=1 \\ x+2 y+4 z=t \\ x+4 y+10 z=t^{2} \end{gathered}$

find the values of $p$ and $t$ for which

(i) there is a unique solution;

(ii) there are infinitely many solutions;

(iii) there is no solution.

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• Paper 1, Section II, A

(i) Show that any line in the complex plane $\mathbb{C}$ can be represented in the form

$\bar{c} z+c \bar{z}+r=0,$

where $c \in \mathbb{C}$ and $r \in \mathbb{R}$.

(ii) If $z$ and $u$ are two complex numbers for which

$\left|\frac{z+u}{z+\bar{u}}\right|=1$

show that either $z$ or $u$ is real.

(iii) Show that any Möbius transformation

$w=\frac{a z+b}{c z+d} \quad(b c-a d \neq 0)$

that maps the real axis $z=\bar{z}$ into the unit circle $|w|=1$ can be expressed in the form

$w=\lambda \frac{z+k}{z+\bar{k}}$

where $\lambda, k \in \mathbb{C}$ and $|\lambda|=1$.

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• Paper 1, Section II, C

Let $\mathbf{x}$ and $\mathbf{y}$ be non-zero vectors in a real vector space with scalar product denoted by $\mathbf{x} \cdot \mathbf{y}$. Prove that $(\mathbf{x} \cdot \mathbf{y})^{2} \leqslant(\mathbf{x} \cdot \mathbf{x})(\mathbf{y} \cdot \mathbf{y})$, and prove also that $(\mathbf{x} \cdot \mathbf{y})^{\mathbf{2}}=(\mathbf{x} \cdot \mathbf{x})(\mathbf{y} \cdot \mathbf{y})$ if and only if $\mathbf{x}=\lambda \mathbf{y}$ for some scalar $\lambda$.

(i) By considering suitable vectors in $\mathbb{R}^{3}$, or otherwise, prove that the inequality $x^{2}+y^{2}+z^{2} \geqslant y z+z x+x y$ holds for any real numbers $x, y$ and $z$.

(ii) By considering suitable vectors in $\mathbb{R}^{4}$, or otherwise, show that only one choice of real numbers $x, y, z$ satisfies $3\left(x^{2}+y^{2}+z^{2}+4\right)-2(y z+z x+x y)-4(x+y+z)=0$, and find these numbers.

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• Paper 1, Section II, C

Let $\mathcal{M}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be the linear map defined by

$\mathbf{x} \mapsto \mathbf{x}^{\prime}=a \mathbf{x}+b(\mathbf{n} \times \mathbf{x})$

where $a$ and $b$ are positive scalar constants, and $\mathbf{n}$ is a unit vector.

(i) By considering the effect of $\mathcal{M}$ on $\mathbf{n}$ and on a vector orthogonal to $\mathbf{n}$, describe geometrically the action of $\mathcal{M}$.

(ii) Express the map $\mathcal{M}$ as a matrix $M$ using suffix notation. Find $a, b$ and $\mathbf{n}$ in the case

$M=\left(\begin{array}{rrr} 2 & -2 & 2 \\ 2 & 2 & -1 \\ -2 & 1 & 2 \end{array}\right)$

(iii) Find, in the general case, the inverse map (i.e. express $\mathbf{x}$ in terms of $\mathbf{x}^{\prime}$ in vector form).

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• Paper 1, Section II, C

(i) Describe geometrically the following surfaces in three-dimensional space:

(a) $\mathbf{r} \cdot \mathbf{u}=\alpha|\mathbf{r}|$, where $0<|\alpha|<1$

(b) $|\mathbf{r}-(\mathbf{r} \cdot \mathbf{u}) \mathbf{u}|=\beta$, where $\beta>0$.

Here $\alpha$ and $\beta$ are fixed scalars and $\mathbf{u}$ is a fixed unit vector. You should identify the meaning of $\alpha, \beta$ and $\mathbf{u}$ for these surfaces.

(ii) The plane $\mathbf{n} \cdot \mathbf{r}=p$, where $\mathbf{n}$ is a fixed unit vector, and the sphere with centre $\mathbf{c}$ and radius $a$ intersect in a circle with centre $\mathbf{b}$ and radius $\rho$.

(a) Show that $\mathbf{b}-\mathbf{c}=\lambda \mathbf{n}$, where you should give $\lambda$ in terms of $a$ and $\rho$.

(b) Find $\rho$ in terms of $\mathbf{c}, \mathbf{n}, a$ and $p$.

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• Paper 1, Section I, $3 F$

Prove that, for positive real numbers $a$ and $b$,

$2 \sqrt{a b} \leqslant a+b$

For positive real numbers $a_{1}, a_{2}, \ldots$, prove that the convergence of

$\sum_{n=1}^{\infty} a_{n}$

implies the convergence of

$\sum_{n=1}^{\infty} \frac{\sqrt{a_{n}}}{n}$

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• Paper 1, Section I, D

Let $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a complex power series. Show that there exists $R \in[0, \infty]$ such that $\sum_{n=0}^{\infty} a_{n} z^{n}$ converges whenever $|z| and diverges whenever $|z|>R$.

Find the value of $R$ for the power series

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

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• Paper 1, Section II, D

Explain carefully what it means to say that a bounded function $f:[0,1] \rightarrow \mathbb{R}$ is Riemann integrable.

Prove that every continuous function $f:[0,1] \rightarrow \mathbb{R}$ is Riemann integrable.

For each of the following functions from $[0,1]$ to $\mathbb{R}$, determine with proof whether or not it is Riemann integrable:

(i) the function $f(x)=x \sin \frac{1}{x}$ for $x \neq 0$, with $f(0)=0$;

(ii) the function $g(x)=\sin \frac{1}{x}$ for $x \neq 0$, with $g(0)=0$.

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• Paper 1, Section II, E

Let $a be real numbers, and let $f:[a, b] \rightarrow \mathbb{R}$ be continuous. Show that $f$ is bounded on $[a, b]$, and that there exist $c, d \in[a, b]$ such that for all $x \in[a, b]$, $f(c) \leqslant f(x) \leqslant f(d)$.

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that

$\lim _{x \rightarrow+\infty} g(x)=\lim _{x \rightarrow-\infty} g(x)=0$

Show that $g$ is bounded. Show also that, if $a$ and $c$ are real numbers with $0, then there exists $x \in \mathbb{R}$ with $g(x)=c$.

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• Paper 1, Section II, E

State and prove the Mean Value Theorem.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that, for every $x \in \mathbb{R}, f^{\prime \prime}(x)$ exists and is non-negative.

(i) Show that if $x \leqslant y$ then $f^{\prime}(x) \leqslant f^{\prime}(y)$.

(ii) Let $\lambda \in(0,1)$ and $a. Show that there exist $x$ and $y$ such that

$f(\lambda a+(1-\lambda) b)=f(a)+(1-\lambda)(b-a) f^{\prime}(x)=f(b)-\lambda(b-a) f^{\prime}(y)$

and that

$f(\lambda a+(1-\lambda) b) \leqslant \lambda f(a)+(1-\lambda) f(b) .$

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• Paper 1, Section II, F

Let $a_{1}=\sqrt{2}$, and consider the sequence of positive real numbers defined by

$a_{n+1}=\sqrt{2+\sqrt{a}_{n}}, \quad n=1,2,3, \ldots$

Show that $a_{n} \leqslant 2$ for all $n$. Prove that the sequence $a_{1}, a_{2}, \ldots$ converges to a limit.

Suppose instead that $a_{1}=4$. Prove that again the sequence $a_{1}, a_{2}, \ldots$ converges to a limit.

Prove that the limits obtained in the two cases are equal.

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