Part IA, 2012, Paper 1

Analysis I

• Paper 1, Section I, $4 \mathbf{F}$

  Analysis I | Part IA, 2012

  Let $f, g:[0,1] \rightarrow \mathbb{R}$ 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.]

  comment

• Paper 1, Section I, E

  Analysis I | Part IA, 2012

  What 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.]

  comment

• Paper 1, Section II, D

  Analysis I | Part IA, 2012

  Let $f$ 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.

  comment

• Paper 1, Section II, D

  Analysis I | Part IA, 2012

  Let $\left(a_{n}\right)$ 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.

  comment

• Paper 1, Section II, E

  Analysis I | Part IA, 2012

  (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.

  comment

• Paper 1, Section II, F

  Analysis I | Part IA, 2012

  (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.]

  comment

Vectors and Matrices

• Paper 1, Section I, $1 \mathrm{C}$

  Vectors and Matrices | Part IA, 2012

  (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 $|z|-z=i$.

  comment

• Paper 1, Section I, A

  Vectors and Matrices | Part IA, 2012

  Define 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.

  comment

• Paper 1, Section II, $5 \mathrm{C}$

  Vectors and Matrices | Part IA, 2012

  The 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?

  comment

• Paper 1, Section II, 7B

  Vectors and Matrices | Part IA, 2012

  (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 $B^{n}=\mathbf{0}$.

  comment

• Paper 1, Section II, A

  Vectors and Matrices | Part IA, 2012

  Explain 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

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

  comment

• Paper 1, Section II, B

  Vectors and Matrices | Part IA, 2012

  (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.]

  comment