Part IA, 2011, Paper 1

Analysis I

• Paper 1, Section I, $3 F$

  Analysis I | Part IA, 2011

  (a) State, without proof, the Bolzano-Weierstrass Theorem.

  (b) Give an example of a sequence that does not have a convergent subsequence.

  (c) Give an example of an unbounded sequence having a convergent subsequence.

  (d) Let $a_{n}=1+(-1)^{\lfloor n / 2\rfloor}(1+1 / n)$, where $\lfloor x\rfloor$ denotes the integer part of $x$. Find all values $c$ such that the sequence $\left\{a_{n}\right\}$ has a subsequence converging to $c$. For each such value, provide a subsequence converging to it.

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• Paper 1, Section I, D

  Analysis I | Part IA, 2011

  Find the radius of convergence of each of the following power series. (i) $\sum_{n \geqslant 1} n^{2} z^{n}$ (ii) $\sum_{n \geqslant 1} n^{n^{1 / 3}} z^{n}$

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• Paper 1, Section II, D

  Analysis I | Part IA, 2011

  State and prove the Fundamental Theorem of Calculus.

  Let $f:[0,1] \rightarrow \mathbb{R}$ be integrable, and set $F(x)=\int_{0}^{x} f(t) \mathrm{d} t$ for $0<x<1$. Must $F$ be differentiable?

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable, and set $g(x)=f^{\prime}(x)$ for $0 \leqslant x \leqslant 1$. Must the Riemann integral of $g$ from 0 to 1 exist?

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• Paper 1, Section II, E

  Analysis I | Part IA, 2011

  For each of the following two functions $f: \mathbb{R} \rightarrow \mathbb{R}$, determine the set of points at which $f$ is continuous, and also the set of points at which $f$ is differentiable.

  $$\begin{aligned} &\text { (i) } f(x)= \begin{cases}x & \text { if } x \in \mathbb{Q} \\ -x & \text { if } x \notin \mathbb{Q}\end{cases} \\ &\text { (ii) } f(x)= \begin{cases}x \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases} \end{aligned}$$

  By modifying the function in (i), or otherwise, find a function (not necessarily continuous) $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is differentiable at 0 and nowhere else.

  Find a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is not differentiable at the points $1 / 2,1 / 3,1 / 4, \ldots$, but is differentiable at all other points.

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• Paper 1, Section II, E

  Analysis I | Part IA, 2011

  State and prove the Intermediate Value Theorem.

  A fixed point of a function $f: X \rightarrow X$ is an $x \in X$ with $f(x)=x$. Prove that every continuous function $f:[0,1] \rightarrow[0,1]$ has a fixed point.

  Answer the following questions with justification.

  (i) Does every continuous function $f:(0,1) \rightarrow(0,1)$ have a fixed point?

  (ii) Does every continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ have a fixed point?

  (iii) Does every function $f:[0,1] \rightarrow[0,1]$ (not necessarily continuous) have a fixed point?

  (iv) Let $f:[0,1] \rightarrow[0,1]$ be a continuous function with $f(0)=1$ and $f(1)=0$. Can $f$ have exactly two fixed points?

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• Paper 1, Section II, F

  Analysis I | Part IA, 2011

  (a) State, without proof, the ratio test for the series $\sum_{n \geqslant 1} a_{n}$, where $a_{n}>0$. Give examples, without proof, to show that, when $a_{n+1}<a_{n}$ and $a_{n+1} / a_{n} \rightarrow 1$, the series may converge or diverge.

  (b) Prove that $\sum_{k=1}^{n-1} \frac{1}{k} \geqslant \log n$.

  (c) Now suppose that $a_{n}>0$ and that, for $n$ large enough, $\frac{a_{n+1}}{a_{n}} \leqslant 1-\frac{c}{n}$ where $c>1$. Prove that the series $\sum_{n \geqslant 1} a_{n}$ converges.

  [You may find it helpful to prove the inequality $\log (1-x)<-x$ for $0<x<1$.]

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Vectors and Matrices

• Paper 1, Section I, $1 \mathrm{C}$

  Vectors and Matrices | Part IA, 2011

  For $z, a \in \mathbb{C}$ define the principal value of $\log z$ and hence of $z^{a}$. Hence find all solutions to (i) $z^{\mathrm{i}}=1$ (ii) $z^{\mathrm{i}}+\bar{z}^{\mathrm{i}}=2 \mathrm{i}$,

  and sketch the curve $\left|z^{\mathrm{i}+1}\right|=1$.

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• Paper 1, Section I, A

  Vectors and Matrices | Part IA, 2011

  The matrix

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

  represents a linear map $\Phi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ with respect to the bases

  $$B=\left\{\left(\begin{array}{l} 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 1 \\ -1 \end{array}\right)\right\} \quad, \quad C=\left\{\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right)\right\}$$

  Find the matrix $A^{\prime}$ that represents $\Phi$ with respect to the bases

  $$B^{\prime}=\left\{\left(\begin{array}{l} 0 \\ 2 \end{array}\right),\left(\begin{array}{l} 2 \\ 0 \end{array}\right)\right\} \quad, \quad C^{\prime}=\left\{\left(\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right)\right\}$$

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• Paper 1, Section II, $5 \mathrm{C}$

  Vectors and Matrices | Part IA, 2011

  Explain why each of the equations

  $$\begin{aligned} &\mathbf{x}=\mathbf{a}+\lambda \mathbf{b} \\ &\mathbf{x} \times \mathbf{c}=\mathbf{d} \end{aligned}$$

  describes a straight line, where $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are constant vectors in $\mathbb{R}^{3}, \mathbf{b}$ and $\mathbf{c}$ are non-zero, $\mathbf{c} \cdot \mathbf{d}=0$ and $\lambda$ is a real parameter. Describe the geometrical relationship of a, $\mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ to the relevant line, assuming that $\mathbf{d} \neq \mathbf{0}$.

  Show that the solutions of (2) satisfy an equation of the form (1), defining $\mathbf{a}, \mathbf{b}$ and $\lambda(\mathbf{x})$ in terms of $\mathbf{c}$ and $\mathbf{d}$ such that $\mathbf{a} \cdot \mathbf{b}=0$ and $|\mathbf{b}|=|\mathbf{c}|$. Deduce that the conditions on $\mathbf{c}$ and $\mathbf{d}$ are sufficient for (2) to have solutions.

  For each of the lines described by (1) and (2), find the point $\mathbf{x}$ that is closest to a given fixed point $\mathbf{y}$.

  Find the line of intersection of the two planes $\mathbf{x} \cdot \mathbf{m}=\mu$ and $\mathbf{x} \cdot \mathbf{n}=\nu$, where $\mathbf{m}$ and $\mathbf{n}$ are constant unit vectors, $\mathbf{m} \times \mathbf{n} \neq \mathbf{0}$ and $\mu$ and $\nu$ are constants. Express your answer in each of the forms (1) and (2), giving both $\mathbf{a}$ and $\mathbf{d}$ as linear combinations of $\mathbf{m}$ and $\mathbf{n}$.

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• Paper 1, Section II, $6 \mathrm{~A}$

  Vectors and Matrices | Part IA, 2011

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

  (a) Find the inverse map $\Phi^{-1}$, when it exists, and determine the values of $\alpha$ for which it does.

  (b) When $\Phi$ is not invertible, find its image and kernel, and explain geometrically why these subspaces are perpendicular.

  (c) Let $\mathbf{y}=\Phi(\mathbf{x})$. 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 $x_{i}=B_{i j} y_{j}$.

  (d) Now let $A$ be as defined in (c) for the case $\mathbf{n}=\frac{1}{\sqrt{3}}(1,1,1)$, and let

  $$C=\frac{1}{3}\left(\begin{array}{rrr} 2 & 2 & -1 \\ -1 & 2 & 2 \\ 2 & -1 & 2 \end{array}\right)$$

  By analysing a suitable determinant, for all values of $\alpha$ find all vectors $\mathbf{x}$ such that $A \mathbf{x}=C \mathbf{x}$. Explain your results by interpreting $A$ and $C$ geometrically.

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• Paper 1, Section II, B

  Vectors and Matrices | Part IA, 2011

  (a) Let $M$ be a real symmetric $n \times n$ matrix. Prove the following.

  (i) Each eigenvalue of $M$ is real.

  (ii) Each eigenvector can be chosen to be real.

  (iii) Eigenvectors with different eigenvalues are orthogonal.

  (b) Let $A$ be a real antisymmetric $n \times n$ matrix. Prove that each eigenvalue of $A^{2}$ is real and is less than or equal to zero.

  If $-\lambda^{2}$ and $-\mu^{2}$ are distinct, non-zero eigenvalues of $A^{2}$, show that there exist orthonormal vectors $\mathbf{u}, \mathbf{u}^{\prime}, \mathbf{w}, \mathbf{w}^{\prime}$ with

  $$\begin{array}{rlr} A \mathbf{u}=\lambda \mathbf{u}^{\prime}, & A \mathbf{w}=\mu \mathbf{w}^{\prime} \\ A \mathbf{u}^{\prime}=-\lambda \mathbf{u}, & A \mathbf{w}^{\prime}=-\mu \mathbf{w} \end{array}$$

  Part IA, 2011 List of Questions

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• Paper 1, Section II, B

  Vectors and Matrices | Part IA, 2011

  (a) Find the eigenvalues and eigenvectors of the matrix

  $$M=\left(\begin{array}{rrr} 2 & 0 & 1 \\ 1 & 1 & 1 \\ 2 & -2 & 3 \end{array}\right)$$

  (b) Under what conditions on the $3 \times 3$ matrix $A$ and the vector $\mathbf{b}$ in $\mathbb{R}^{3}$ does the equation

  $$A \mathbf{x}=\mathbf{b}$$

  have 0,1 , or infinitely many solutions for the vector $\mathbf{x}$ in $\mathbb{R}^{3}$ ? Give clear, concise arguments to support your answer, explaining why just these three possibilities are allowed.

  (c) Using the results of $(\mathrm{a})$, or otherwise, find all solutions to $(*)$ when

  $$A=M-\lambda I \quad \text { and } \quad \mathbf{b}=\left(\begin{array}{l} 4 \\ 3 \\ 2 \end{array}\right)$$

  in each of the cases $\lambda=0,1,2$.

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