Part IA, 2009, Paper 1

Analysis I

• Paper 1, Section I, $3 \mathbf{F}$

  Analysis I | Part IA, 2009

  Determine the limits as $n \rightarrow \infty$ of the following sequences:

  (a) $a_{n}=n-\sqrt{n^{2}-n}$;

  (b) $b_{n}=\cos ^{2}\left(\pi \sqrt{n^{2}+n}\right)$.

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• Paper 1, Section I, E

  Analysis I | Part IA, 2009

  Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of complex numbers. Prove that there exists $R \in[0, \infty]$ such that the power series $\sum_{n=0}^{\infty} a_{n} z^{n}$ converges whenever $|z|<R$ and diverges whenever $|z|>R$.

  Give an example of a power series $\sum_{n=0}^{\infty} a_{n} z^{n}$ that diverges if $z=\pm 1$ and converges if $z=\pm \mathrm{i}$.

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• Paper 1, Section II, D

  Analysis I | Part IA, 2009

  State and prove Rolle's theorem.

  Let $f$ and $g$ be two continuous, real-valued functions on a closed, bounded interval $[a, b]$ that are differentiable on the open interval $(a, b)$. By considering the determinant

  $$\phi(x)=\left|\begin{array}{ccc} 1 & 1 & 0 \\ f(a) & f(b) & f(x) \\ g(a) & g(b) & g(x) \end{array}\right|=g(x)(f(b)-f(a))-f(x)(g(b)-g(a))$$

  or otherwise, show that there is a point $c \in(a, b)$ with

  $$f^{\prime}(c)(g(b)-g(a))=g^{\prime}(c)(f(b)-f(a))$$

  Suppose that $f, g:(0, \infty) \rightarrow \mathbb{R}$ are differentiable functions with $f(x) \rightarrow 0$ and $g(x) \rightarrow 0$ as $x \rightarrow 0$. Prove carefully that if the $\operatorname{limit}_{x \rightarrow 0} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\ell$ exists and is finite, then the limit $\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}$ also exists and equals $\ell$.

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• Paper 1, Section II, D

  Analysis I | Part IA, 2009

  State and prove the intermediate value theorem.

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function and let $P=(a, b)$ be a point of the plane $\mathbb{R}^{2}$. Show that the set of distances from points $(x, f(x))$ on the graph of $f$ to the point $P$ is an interval $[A, \infty)$ for some value $A \geqslant 0$.

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• Paper 1, Section II, E

  Analysis I | Part IA, 2009

  (a) What does it mean for a function $f:[a, b] \rightarrow \mathbb{R}$ to be Riemann integrable?

  (b) Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function. Suppose that for every $\delta>0$ there is a sequence

  $$0 \leqslant a_{1}<b_{1} \leqslant a_{2}<b_{2} \leqslant \ldots \leqslant a_{n}<b_{n} \leqslant 1$$

  such that for each $i$ the function $f$ is Riemann integrable on the closed interval $\left[a_{i}, b_{i}\right]$, and such that $\sum_{i=1}^{n}\left(b_{i}-a_{i}\right) \geqslant 1-\delta$. Prove that $f$ is Riemann integrable on $[0,1]$.

  (c) Let $f:[0,1] \rightarrow \mathbb{R}$ be defined as follows. We set $f(x)=1$ if $x$ has an infinite decimal expansion that consists of 2 s and $7 \mathrm{~s}$ only, and otherwise we set $f(x)=0$. Prove that $f$ is Riemann integrable and determine $\int_{0}^{1} f(x) \mathrm{d} x$.

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• Paper 1, Section II, F

  Analysis I | Part IA, 2009

  For each of the following series, determine for which real numbers $x$ it diverges, for which it converges, and for which it converges absolutely. Justify your answers briefly.

  (a) $\sum_{n \geqslant 1} \frac{3+(\sin x)^{n}}{n}(\sin x)^{n}$,

  (b) $\quad \sum_{n \geqslant 1}|\sin x|^{n} \frac{(-1)^{n}}{\sqrt{n}}$,

  (c)

  $$\sum_{n \geqslant 1} \underbrace{\sin (0.99 \sin (0.99 \ldots \sin (0.99 x) \ldots))}_{n \text { times }} .$$

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Vectors and Matrices

• Paper 1, Section I, $2 \mathrm{~B}$

  Vectors and Matrices | Part IA, 2009

  Define the Hermitian conjugate $A^{\dagger}$ of an $n \times n$ complex matrix $A$. State the conditions (i) for $A$ to be Hermitian (ii) for $A$ to be unitary.

  In the following, $A, B, C$ and $D$ are $n \times n$ complex matrices and $\mathbf{x}$ is a complex $n$-vector. A matrix $N$ is defined to be normal if $N^{\dagger} N=N N^{\dagger}$.

  (a) Let $A$ be nonsingular. Show that $B=A^{-1} A^{\dagger}$ is unitary if and only if $A$ is normal.

  (b) Let $C$ be normal. Show that $|C \mathbf{x}|=0$ if and only if $\left|C^{\dagger} \mathbf{x}\right|=0$.

  (c) Let $D$ be normal. Deduce from (b) that if $e$ is an eigenvector of $D$ with eigenvalue $\lambda$ then $e$ is also an eigenvector of $D^{\dagger}$ and find the corresponding eigenvalue.

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• Paper 1, Section I, C

  Vectors and Matrices | Part IA, 2009

  Describe geometrically the three sets of points defined by the following equations in the complex $z$ plane:

  (a) $z \bar{\alpha}+\bar{z} \alpha=0$, where $\alpha$ is non-zero;

  (b) $2|z-a|=z+\bar{z}+2 a$, where $a$ is real and non-zero;

  (c) $\log z=\mathrm{i} \log \bar{z}$.

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• Paper 1, Section II, $5 \mathrm{C}$

  Vectors and Matrices | Part IA, 2009

  Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be unit vectors. By using suffix notation, prove that

  $$(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{a} \times \mathbf{c})=\mathbf{b} \cdot \mathbf{c}-(\mathbf{a} \cdot \mathbf{b})(\mathbf{a} \cdot \mathbf{c})$$

  and

  $$(\mathbf{a} \times \mathbf{b}) \times(\mathbf{a} \times \mathbf{c})=[\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})] \mathbf{a}$$

  The three distinct points $A, B, C$ with position vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ lie on the surface of the unit sphere centred on the origin $O$. The spherical distance between the points $A$ and $B$, denoted $\delta(A, B)$, is the length of the (shorter) arc of the circle with centre $O$ passing through $A$ and $B$. Show that

  $$\cos \delta(A, B)=\mathbf{a} \cdot \mathbf{b}$$

  A spherical triangle with vertices $A, B, C$ is a region on the sphere bounded by the three circular arcs $A B, B C, C A$. The interior angles of a spherical triangle at the vertices $A, B, C$ are denoted $\alpha, \beta, \gamma$, respectively.

  By considering the normals to the planes $O A B$ and $O A C$, or otherwise, show that

  $$\cos \alpha=\frac{(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{a} \times \mathbf{c})}{|\mathbf{a} \times \mathbf{b} \| \mathbf{a} \times \mathbf{c}|}$$

  Using identities (1) and (2), prove that

  $$\cos \delta(B, C)=\cos \delta(A, B) \cos \delta(A, C)+\sin \delta(A, B) \sin \delta(A, C) \cos \alpha$$

  and

  $$\frac{\sin \alpha}{\sin \delta(B, C)}=\frac{\sin \beta}{\sin \delta(A, C)}=\frac{\sin \gamma}{\sin \delta(A, B)}$$

  For an equilateral spherical triangle show that $\alpha>\pi / 3$.

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• Paper 1, Section II, $7 \mathrm{~A}$

  Vectors and Matrices | Part IA, 2009

  Let $A$ be an $n \times n$ Hermitian matrix. Show that all the eigenvalues of $A$ are real.

  Suppose now that $A$ has $n$ distinct eigenvalues.

  (a) Show that the eigenvectors of $A$ are orthogonal.

  (b) Define the characteristic polynomial $P_{A}(t)$ of $A$. Let

  $$P_{A}(t)=\sum_{r=0}^{n} a_{r} t^{r}$$

  Prove the matrix identity

  $$\sum_{r=0}^{n} a_{r} A^{r}=0$$

  (c) What is the range of possible values of

  $$\frac{\mathbf{x}^{\dagger} A \mathbf{x}}{\mathbf{x}^{\dagger} \mathbf{x}}$$

  for non-zero vectors $\mathbf{x} \in \mathbb{C}^{n} ?$ Justify your answer.

  (d) For any (not necessarily symmetric) real $2 \times 2$ matrix $B$ with real eigenvalues, let $\lambda_{\max }(B)$ denote its maximum eigenvalue. Is it possible to find a constant $C$ such that

  $$\frac{\mathbf{x}^{\dagger} B \mathbf{x}}{\mathbf{x}^{\dagger} \mathbf{x}} \leqslant C \lambda_{\max }(B)$$

  for all non-zero vectors $\mathbf{x} \in \mathbb{R}^{2}$ and all such matrices $B$ ? Justify your answer.

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• Paper 1, Section II, A

  Vectors and Matrices | Part IA, 2009

  (a) Explain what is meant by saying that a $2 \times 2$ real transformation matrix

  $$\begin{aligned} &A=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \text { preserves the scalar product with respect to the Euclidean metric } \\ &I=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right) \text { on } \mathbb{R}^{2} . \end{aligned}$$

  Derive a description of all such matrices that uses a single real parameter together with choices of $\operatorname{sign}(\pm 1)$. Show that these matrices form a group.

  (b) Explain what is meant by saying that a $2 \times 2$ real transformation matrix $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ preserves the scalar product with respect to the Minkowski metric $J=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right)$ on $\mathbb{R}^{2}$

  Consider now the set of such matrices with $a>0$. Derive a description of all matrices in this set that uses a single real parameter together with choices of sign $(\pm 1)$. Show that these matrices form a group.

  (c) What is the intersection of these two groups?

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• Paper 1, Section II, B

  Vectors and Matrices | Part IA, 2009

  Explain why the number of solutions $\mathbf{x} \in \mathbb{R}^{3}$ of the matrix equation $A \mathbf{x}=\mathbf{c}$ is 0,1 or infinity, where $A$ is a real $3 \times 3$ matrix and $\mathbf{c} \in \mathbb{R}^{3}$. State conditions on $A$ and $\mathbf{c}$ that distinguish between these possibilities, and state the relationship that holds between any two solutions when there are infinitely many.

  Consider the case

  $$A=\left(\begin{array}{lll} a & a & b \\ b & a & a \\ a & b & a \end{array}\right) \quad \text { and } \mathbf{c}=\left(\begin{array}{l} 1 \\ c \\ 1 \end{array}\right)$$

  Use row and column operations to find and factorize the determinant of $A$.

  Find the kernel and image of the linear map represented by $A$ for all values of $a$ and $b$. Find the general solution to $A \mathbf{x}=\mathbf{c}$ for all values of $a, b$ and $c$ for which a solution exists.

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