Part IA, 2013, Paper 1

Analysis I

• Paper 1, Section I, D

  Analysis I | Part IA, 2013

  Show that $\exp (x) \geqslant 1+x$ for $x \geqslant 0$.

  Let $\left(a_{j}\right)$ be a sequence of positive real numbers. Show that for every $n$,

  $$\sum_{1}^{n} a_{j} \leqslant \prod_{1}^{n}\left(1+a_{j}\right) \leqslant \exp \left(\sum_{1}^{n} a_{j}\right)$$

  Deduce that $\prod_{1}^{n}\left(1+a_{j}\right)$ tends to a limit as $n \rightarrow \infty$ if and only if $\sum_{1}^{n} a_{j}$ does.

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• Paper 1, Section I, F

  Analysis I | Part IA, 2013

  (a) Suppose $b_{n} \geqslant b_{n+1} \geqslant 0$ for $n \geqslant 1$ and $b_{n} \rightarrow 0$. Show that $\sum_{n=1}^{\infty}(-1)^{n-1} b_{n}$ converges.

  (b) Does the series $\sum_{n=2}^{\infty} \frac{1}{n \log n}$ converge or diverge? Explain your answer.

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• Paper 1, Section II, D

  Analysis I | Part IA, 2013

  (a) Determine the radius of convergence of each of the following power series:

  $$\sum_{n \geqslant 1} \frac{x^{n}}{n !}, \quad \sum_{n \geqslant 1} n ! x^{n}, \quad \sum_{n \geqslant 1}(n !)^{2} x^{n^{2}}$$

  (b) State Taylor's theorem.

  Show that

  $$(1+x)^{1 / 2}=1+\sum_{n \geqslant 1} c_{n} x^{n}$$

  for all $x \in(0,1)$, where

  $$c_{n}=\frac{\frac{1}{2}\left(\frac{1}{2}-1\right) \ldots\left(\frac{1}{2}-n+1\right)}{n !}$$

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• Paper 1, Section II, E

  Analysis I | Part IA, 2013

  (i) State (without proof) Rolle's Theorem.

  (ii) State and prove the Mean Value Theorem.

  (iii) Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous, and differentiable on $(a, b)$ with $g^{\prime}(x) \neq 0$ for all $x \in(a, b)$. Show that there exists $\xi \in(a, b)$ such that

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

  Deduce that if moreover $f(a)=g(a)=0$, and the limit

  $$\ell=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}$$

  exists, then

  $$\frac{f(x)}{g(x)} \rightarrow \ell \text { as } x \rightarrow a$$

  (iv) Deduce that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable then for any $a \in \mathbb{R}$

  $$f^{\prime \prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} .$$

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• Paper 1, Section II, E

  Analysis I | Part IA, 2013

  (a) Let $f:[a, b] \rightarrow \mathbb{R}$. Suppose that for every sequence $\left(x_{n}\right)$ in $[a, b]$ with limit $y \in[a, b]$, the sequence $\left(f\left(x_{n}\right)\right)$ converges to $f(y)$. Show that $f$ is continuous at $y$.

  (b) State the Intermediate Value Theorem.

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a function with $f(a)=c<f(b)=d$. We say $f$ is injective if for all $x, y \in[a, b]$ with $x \neq y$, we have $f(x) \neq f(y)$. We say $f$ is strictly increasing if for all $x, y$ with $x<y$, we have $f(x)<f(y)$.

  (i) Suppose $f$ is strictly increasing. Show that it is injective, and that if $f(x)<f(y)$ then $x<y .$

  (ii) Suppose $f$ is continuous and injective. Show that if $a<x<b$ then $c<f(x)<d$. Deduce that $f$ is strictly increasing.

  (iii) Suppose $f$ is strictly increasing, and that for every $y \in[c, d]$ there exists $x \in[a, b]$ with $f(x)=y$. Show that $f$ is continuous at $b$. Deduce that $f$ is continuous on $[a, b]$.

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• Paper 1, Section II, F

  Analysis I | Part IA, 2013

  Fix a closed interval $[a, b]$. For a bounded function $f$ on $[a, b]$ and a dissection $\mathcal{D}$ of $[a, b]$, how are the lower sum $s(f, \mathcal{D})$ and upper sum $S(f, \mathcal{D})$ defined? Show that $s(f, \mathcal{D}) \leqslant S(f, \mathcal{D})$.

  Suppose $\mathcal{D}^{\prime}$ is a dissection of $[a, b]$ such that $\mathcal{D} \subseteq \mathcal{D}^{\prime}$. Show that

  $$s(f, \mathcal{D}) \leqslant s\left(f, \mathcal{D}^{\prime}\right) \text { and } S\left(f, \mathcal{D}^{\prime}\right) \leqslant S(f, \mathcal{D})$$

  By using the above inequalities or otherwise, show that if $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are two dissections of $[a, b]$ then

  $$s\left(f, \mathcal{D}_{1}\right) \leqslant S\left(f, \mathcal{D}_{2}\right)$$

  For a function $f$ and dissection $\mathcal{D}=\left\{x_{0}, \ldots, x_{n}\right\}$ let

  $$p(f, \mathcal{D})=\prod_{k=1}^{n}\left[1+\left(x_{k}-x_{k-1}\right) \inf _{x \in\left[x_{k-1}, x_{k}\right]} f(x)\right]$$

  If $f$ is non-negative and Riemann integrable, show that

  $$p(f, \mathcal{D}) \leqslant e^{\int_{a}^{b} f(x) d x} .$$

  [You may use without proof the inequality $e^{t} \geqslant t+1$ for all $t$.]

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Vectors and Matrices

• Paper 1, Section I, $1 \mathrm{C}$

  Vectors and Matrices | Part IA, 2013

  (a) State de Moivre's theorem and use it to derive a formula for the roots of order $n$ of a complex number $z=a+i b$. Using this formula compute the cube roots of $z=-8$.

  (b) Consider the equation $|z+3 i|=3|z|$ for $z \in \mathbb{C}$. Give a geometric description of the set $S$ of solutions and sketch $S$ as a subset of the complex plane.

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• Paper 1, Section I, A

  Vectors and Matrices | Part IA, 2013

  Let $A$ be a real $3 \times 3$ matrix.

  (i) For $B=R_{1} A$ with

  $$R_{1}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta_{1} & -\sin \theta_{1} \\ 0 & \sin \theta_{1} & \cos \theta_{1} \end{array}\right)$$

  find an angle $\theta_{1}$ so that the element $b_{31}=0$, where $b_{i j}$ denotes the $i j^{\text {th }}$entry of the matrix $B$.

  (ii) For $C=R_{2} B$ with $b_{31}=0$ and

  $$R_{2}=\left(\begin{array}{ccc} \cos \theta_{2} & -\sin \theta_{2} & 0 \\ \sin \theta_{2} & \cos \theta_{2} & 0 \\ 0 & 0 & 1 \end{array}\right)$$

  show that $c_{31}=0$ and find an angle $\theta_{2}$ so that $c_{21}=0$.

  (iii) For $D=R_{3} C$ with $c_{31}=c_{21}=0$ and

  $$R_{3}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta_{3} & -\sin \theta_{3} \\ 0 & \sin \theta_{3} & \cos \theta_{3} \end{array}\right)$$

  show that $d_{31}=d_{21}=0$ and find an angle $\theta_{3}$ so that $d_{32}=0$.

  (iv) Deduce that any real $3 \times 3$ matrix can be written as a product of an orthogonal matrix and an upper triangular matrix.

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• Paper 1, Section II, $5 \mathrm{C}$

  Vectors and Matrices | Part IA, 2013

  Let $\mathbf{x}$ and $\mathbf{y}$ be non-zero vectors in $\mathbb{R}^{n}$. What is meant by saying that $\mathbf{x}$ and $\mathbf{y}$ are linearly independent? What is the dimension of the subspace of $\mathbb{R}^{n}$ spanned by $\mathbf{x}$ and $\mathbf{y}$ if they are (1) linearly independent, (2) linearly dependent?

  Define the scalar product $\mathbf{x} \cdot \mathbf{y}$ for $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}$. Define the corresponding norm $\|\mathbf{x}\|$ of $\mathbf{x} \in \mathbb{R}^{n}$. State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality. Under what condition does equality hold in the Cauchy-Schwarz inequality?

  Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ be unit vectors in $\mathbb{R}^{3}$. Let

  $$S=\mathbf{x} \cdot \mathbf{y}+\mathbf{y} \cdot \mathbf{z}+\mathbf{z} \cdot \mathbf{x}$$

  Show that for any fixed, linearly independent vectors $\mathbf{x}$ and $\mathbf{y}$, the minimum of $S$ over $\mathbf{z}$ is attained when $\mathbf{z}=\lambda(\mathbf{x}+\mathbf{y})$ for some $\lambda \in \mathbb{R}$, and that for this value of $\lambda$ we have

  (i) $\lambda \leqslant-\frac{1}{2}$ (for any choice of $\mathbf{x}$ and $\left.\mathbf{y}\right)$;

  (ii) $\lambda=-1$ and $S=-\frac{3}{2}$ in the case where $\mathbf{x} \cdot \mathbf{y}=\cos \frac{2 \pi}{3}$.

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

  Vectors and Matrices | Part IA, 2013

  Define the kernel and the image of a linear map $\alpha$ from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$.

  Let $\left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{m}\right\}$ be a basis of $\mathbb{R}^{m}$ and $\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \ldots, \mathbf{f}_{n}\right\}$ a basis of $\mathbb{R}^{n}$. Explain how to represent $\alpha$ by a matrix $A$ relative to the given bases.

  A second set of bases $\left\{\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \ldots, \mathbf{e}_{m}^{\prime}\right\}$ and $\left\{\mathbf{f}_{1}^{\prime}, \mathbf{f}_{2}^{\prime}, \ldots, \mathbf{f}_{n}^{\prime}\right\}$ is now used to represent $\alpha$ by a matrix $A^{\prime}$. Relate the elements of $A^{\prime}$ to the elements of $A$.

  Let $\beta$ be a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{3}$ defined by

  $$\beta\left(\begin{array}{l} 1 \\ 1 \end{array}\right)=\left(\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right), \quad \beta\left(\begin{array}{c} 1 \\ -1 \end{array}\right)=\left(\begin{array}{l} 6 \\ 4 \\ 2 \end{array}\right)$$

  Either find one or more $\mathbf{x}$ in $\mathbb{R}^{2}$ such that

  $$\beta \mathbf{x}=\left(\begin{array}{c} 1 \\ -2 \\ 1 \end{array}\right)$$

  or explain why one cannot be found.

  Let $\gamma$ be a linear map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ defined by

  $$\gamma\left(\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right)=\left(\begin{array}{l} 1 \\ 3 \end{array}\right), \quad \gamma\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right)=\left(\begin{array}{c} -2 \\ 1 \end{array}\right), \quad \gamma\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right)=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)$$

  Find the kernel of $\gamma$.

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• Paper 1, Section II, B

  Vectors and Matrices | Part IA, 2013

  (a) Let $A$ and $A^{\prime}$ be the matrices of a linear map $L$ on $\mathbb{C}^{2}$ relative to bases $\mathcal{B}$ and $\mathcal{B}^{\prime}$ respectively. In this question you may assume without proof that $A$ and $A^{\prime}$ are similar.

  (i) State how the matrix $A$ of $L$ relative to the basis $\mathcal{B}=\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\}$ is constructed from $L$ and $\mathcal{B}$. Also state how $A$ may be used to compute $L \mathbf{v}$ for any $\mathbf{v} \in \mathbb{C}^{2}$.

  (ii) Show that $A$ and $A^{\prime}$ have the same characteristic equation.

  (iii) Show that for any $k \neq 0$ the matrices

  $$\left(\begin{array}{ll} a & c \\ b & d \end{array}\right) \text { and }\left(\begin{array}{cc} a & c / k \\ b k & d \end{array}\right)$$

  are similar. [Hint: if $\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\}$ is a basis then so is $\left\{k \mathbf{e}_{1}, \mathbf{e}_{2}\right\}$.]

  (b) Using the results of (a), or otherwise, prove that any $2 \times 2$ complex matrix $M$ with equal eigenvalues is similar to one of

  $$\left(\begin{array}{ll} a & 0 \\ 0 & a \end{array}\right) \text { and }\left(\begin{array}{ll} a & 1 \\ 0 & a \end{array}\right) \text { with } a \in \mathbb{C} .$$

  (c) Consider the matrix

  $$B(r)=\frac{1}{2}\left(\begin{array}{ccc} 1+r & 1-r & 1 \\ 1-r & 1+r & -1 \\ -1 & 1 & 2 r \end{array}\right)$$

  Show that there is a real value $r_{0}>0$ such that $B\left(r_{0}\right)$ is an orthogonal matrix. Show that $B\left(r_{0}\right)$ is a rotation and find the axis and angle of the rotation.

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• Paper 1, Section II, B

  Vectors and Matrices | Part IA, 2013

  (a) Let $\lambda_{1}, \ldots, \lambda_{d}$ be distinct eigenvalues of an $n \times n$ matrix $A$, with corresponding eigenvectors $\mathbf{v}_{1}, \ldots, \mathbf{v}_{d}$. Prove that the set $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{d}\right\}$ is linearly independent.

  (b) Consider the quadric surface $Q$ in $\mathbb{R}^{3}$ defined by

  $$2 x^{2}-4 x y+5 y^{2}-z^{2}+6 \sqrt{5} y=0 .$$

  Find the position of the origin $\tilde{O}$ and orthonormal coordinate basis vectors $\tilde{\mathbf{e}}_{1}, \tilde{\mathbf{e}}_{2}$ and $\tilde{\mathbf{e}}_{3}$, for a coordinate system $(\tilde{x}, \tilde{y}, \tilde{z})$ in which $Q$ takes the form

  $$\alpha \tilde{x}^{2}+\beta \tilde{y}^{2}+\gamma \tilde{z}^{2}=1 .$$

  Also determine the values of $\alpha, \beta$ and $\gamma$, and describe the surface geometrically.

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