Part IA, 2012, Paper 1

# Part IA, 2012, Paper 1

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Paper 1, Section I, $4 \mathbf{F}$

commentLet $f, g:[0,1] \rightarrow \mathbb{R}$ be continuous functions with $g(x) \geqslant 0$ for $x \in[0,1]$. Show that

$\int_{0}^{1} f(x) g(x) d x \leqslant M \int_{0}^{1} g(x) d x$

where $M=\sup \{|f(x)|: x \in[0,1]\}$.

Prove there exists $\alpha \in[0,1]$ such that

$\int_{0}^{1} f(x) g(x) d x=f(\alpha) \int_{0}^{1} g(x) d x$

[Standard results about continuous functions and their integrals may be used without proof, if clearly stated.]

Paper 1, Section I, E

commentWhat does it mean to say that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous at $x_{0} \in \mathbb{R}$ ?

Give an example of a continuous function $f:(0,1] \rightarrow \mathbb{R}$ which is bounded but attains neither its upper bound nor its lower bound.

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and non-negative, and satisfies $f(x) \rightarrow 0$ as $x \rightarrow \infty$ and $f(x) \rightarrow 0$ as $x \rightarrow-\infty$. Show that $f$ is bounded above and attains its upper bound.

[Standard results about continuous functions on closed bounded intervals may be used without proof if clearly stated.]

Paper 1, Section II, D

commentLet $f$ be a continuous function from $(0,1)$ to $(0,1)$ such that $f(x)<x$ for every $0<x<1$. We write $f^{n}$ for the $n$-fold composition of $f$ with itself (so for example $\left.f^{2}(x)=f(f(x))\right)$.

(i) Prove that for every $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$.

(ii) Must it be the case that for every $\epsilon>0$ there exists $n$ with the property that $f^{n}(x)<\epsilon$ for all $0<x<1$ ? Justify your answer.

Now suppose that we remove the condition that $f$ be continuous.

(iii) Give an example to show that it need not be the case that for every $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$.

(iv) Must it be the case that for some $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$ ? Justify your answer.

Paper 1, Section II, D

commentLet $\left(a_{n}\right)$ be a sequence of reals.

(i) Show that if the sequence $\left(a_{n+1}-a_{n}\right)$ is convergent then so is the sequence $\left(\frac{a_{n}}{n}\right)$.

(ii) Give an example to show the sequence $\left(\frac{a_{n}}{n}\right)$ being convergent does not imply that the sequence $\left(a_{n+1}-a_{n}\right)$ is convergent.

(iii) If $a_{n+k}-a_{n} \rightarrow 0$ as $n \rightarrow \infty$ for each positive integer $k$, does it follow that $\left(a_{n}\right)$ is convergent? Justify your answer.

(iv) If $a_{n+f(n)}-a_{n} \rightarrow 0$ as $n \rightarrow \infty$ for every function $f$ from the positive integers to the positive integers, does it follow that $\left(a_{n}\right)$ is convergent? Justify your answer.

Paper 1, Section II, E

comment(a) What does it mean to say that the sequence $\left(x_{n}\right)$ of real numbers converges to $\ell \in \mathbb{R} ?$

Suppose that $\left(y_{n}^{(1)}\right),\left(y_{n}^{(2)}\right), \ldots,\left(y_{n}^{(k)}\right)$ are sequences of real numbers converging to the same limit $\ell$. Let $\left(x_{n}\right)$ be a sequence such that for every $n$,

$x_{n} \in\left\{y_{n}^{(1)}, y_{n}^{(2)}, \ldots, y_{n}^{(k)}\right\}$

Show that $\left(x_{n}\right)$ also converges to $\ell$.

Find a collection of sequences $\left(y_{n}^{(j)}\right), j=1,2, \ldots$ such that for every $j,\left(y_{n}^{(j)}\right) \rightarrow \ell$ but the sequence $\left(x_{n}\right)$ defined by $x_{n}=y_{n}^{(n)}$ diverges.

(b) Let $a, b$ be real numbers with $0<a<b$. Sequences $\left(a_{n}\right),\left(b_{n}\right)$ are defined by $a_{1}=a, b_{1}=b$ and

$\text { for all } n \geqslant 1, \quad a_{n+1}=\sqrt{a_{n} b_{n}}, \quad b_{n+1}=\frac{a_{n}+b_{n}}{2} \text {. }$

Show that $\left(a_{n}\right)$ and $\left(b_{n}\right)$ converge to the same limit.

Paper 1, Section II, F

comment(a) (i) State the ratio test for the convergence of a real series with positive terms.

(ii) Define the radius of convergence of a real power series $\sum_{n=0}^{\infty} a_{n} x^{n}$.

(iii) Prove that the real power series $f(x)=\sum_{n} a_{n} x^{n}$ and $g(x)=\sum_{n}(n+1) a_{n+1} x^{n}$ have equal radii of convergence.

(iv) State the relationship between $f(x)$ and $g(x)$ within their interval of convergence.

(b) (i) Prove that the real series

$f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}, \quad g(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}$

have radius of convergence $\infty$.

(ii) Show that they are differentiable on the real line $\mathbb{R}$, with $f^{\prime}=-g$ and $g^{\prime}=f$, and deduce that $f(x)^{2}+g(x)^{2}=1$.

[You may use, without proof, general theorems about differentiating within the interval of convergence, provided that you give a clear statement of any such theorem.]

Paper 1, Section I, $1 \mathrm{C}$

comment(a) Let $R$ be the set of all $z \in \mathbb{C}$ with real part 1 . Draw a picture of $R$ and the image of $R$ under the map $z \mapsto e^{z}$ in the complex plane.

(b) For each of the following equations, find all complex numbers $z$ which satisfy it:

(i) $e^{z}=e$,

(ii) $(\log z)^{2}=-\frac{\pi^{2}}{4}$.

(c) Prove that there is no complex number $z$ satisfying $|z|-z=i$.

Paper 1, Section I, A

commentDefine what is meant by the terms rotation, reflection, dilation and shear. Give examples of real $2 \times 2$ matrices representing each of these.

Consider the three $2 \times 2$ matrices

$A=\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array}\right), \quad B=\frac{1}{\sqrt{2}}\left(\begin{array}{ll} 1 & 1 \\ 1 & 3 \end{array}\right) \quad \text { and } \quad C=A B$

Identify the three matrices in terms of your definitions above.

Paper 1, Section II, $5 \mathrm{C}$

commentThe equation of a plane $\Pi$ in $\mathbb{R}^{3}$ is

$\mathbf{x} \cdot \mathbf{n}=d$

where $d$ is a constant scalar and $\mathbf{n}$ is a unit vector normal to $\Pi$. What is the distance of the plane from the origin $O$ ?

A sphere $S$ with centre $\mathbf{p}$ and radius $r$ satisfies the equation

$|\mathbf{x}-\mathbf{p}|^{2}=r^{2}$

Show that the intersection of $\Pi$ and $S$ contains exactly one point if $|\mathbf{p} \cdot \mathbf{n}-d|=r$.

The tetrahedron $O A B C$ is defined by the vectors $\mathbf{a}=\overrightarrow{O A}, \mathbf{b}=\overrightarrow{O B}$, and $\mathbf{c}=\overrightarrow{O C}$with $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})>0$. What does the condition $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})>0$ imply about the set of vectors $\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$ ? A sphere $T_{r}$ with radius $r>0$ lies inside the tetrahedron and intersects each of the three faces $O A B, O B C$, and $O C A$ in exactly one point. Show that the centre $P$ of $T_{r}$ satisfies

$\overrightarrow{O P}=r \frac{|\mathbf{b} \times \mathbf{c}| \mathbf{a}+|\mathbf{c} \times \mathbf{a}| \mathbf{b}+|\mathbf{a} \times \mathbf{b}| \mathbf{c}}{\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})}$

Given that the vector $\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}$ is orthogonal to the plane $\Psi$ of the face $A B C$, obtain an equation for $\Psi$. What is the distance of $\Psi$ from the origin?

Paper 1, Section II, 7B

comment(a) Consider the matrix

$M=\left(\begin{array}{rrr} 2 & 1 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 4 \end{array}\right)$

Determine whether or not $M$ is diagonalisable.

(b) Prove that if $A$ and $B$ are similar matrices then $A$ and $B$ have the same eigenvalues with the same corresponding algebraic multiplicities.

Is the converse true? Give either a proof (if true) or a counterexample with a brief reason (if false).

(c) State the Cayley-Hamilton theorem for a complex matrix $A$ and prove it in the case when $A$ is a $2 \times 2$ diagonalisable matrix.

Suppose that an $n \times n$ matrix $B$ has $B^{k}=\mathbf{0}$ for some $k>n$ (where $\mathbf{0}$ denotes the zero matrix). Show that $B^{n}=\mathbf{0}$.

Paper 1, Section II, A

commentExplain why the number of solutions $\mathbf{x}$ of the simultaneous linear equations $A \mathbf{x}=\mathbf{b}$ is 0,1 or infinity, where $A$ is a real $3 \times 3$ matrix and $\mathbf{x}$ and $\mathbf{b}$ are vectors in $\mathbb{R}^{3}$. State necessary and sufficient conditions on $A$ and $\mathrm{b}$ for each of these possibilities to hold.

Let $A$ and $B$ be real $3 \times 3$ matrices. Give necessary and sufficient conditions on $A$ for there to exist a unique real $3 \times 3$ matrix $X$ satisfying $A X=B$.

Find $X$ when

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

Paper 1, Section II, B

comment(a) (i) Find the eigenvalues and eigenvectors of the matrix

$A=\left(\begin{array}{lll} 3 & 1 & 1 \\ 1 & 2 & 0 \\ 1 & 0 & 2 \end{array}\right)$

(ii) Show that the quadric $\mathcal{Q}$ in $\mathbb{R}^{3}$ defined by

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

is an ellipsoid. Find the matrix $B$ of a linear transformation of $\mathbb{R}^{3}$ that will map $\mathcal{Q}$ onto the unit sphere $x^{2}+y^{2}+z^{2}=1$.

(b) Let $P$ be a real orthogonal matrix. Prove that:

(i) as a mapping of vectors, $P$ preserves inner products;

(ii) if $\lambda$ is an eigenvalue of $P$ then $|\lambda|=1$ and $\lambda^{*}$ is also an eigenvalue of $P$.

Now let $Q$ be a real orthogonal $3 \times 3$ matrix having $\lambda=1$ as an eigenvalue of algebraic multiplicity 2. Give a geometrical description of the action of $Q$ on $\mathbb{R}^{3}$, giving a reason for your answer. [You may assume that orthogonal matrices are always diagonalisable.]