Part IA, 2016, Paper 1

Analysis I

• Paper 1, Section I, D

  Analysis I | Part IA, 2016

  What does it mean to say that a sequence of real numbers $\left(x_{n}\right)$ converges to $x$ ? Suppose that $\left(x_{n}\right)$ converges to $x$. Show that the sequence $\left(y_{n}\right)$ given by

  $$y_{n}=\frac{1}{n} \sum_{i=1}^{n} x_{i}$$

  also converges to $x$.

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• Paper 1, Section I, F

  Analysis I | Part IA, 2016

  Let $a_{n}$ be the number of pairs of integers $(x, y) \in \mathbb{Z}^{2}$ such that $x^{2}+y^{2} \leqslant n^{2}$. What is the radius of convergence of the series $\sum_{n=0}^{\infty} a_{n} z^{n}$ ? [You may use the comparison test, provided you state it clearly.]

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• Paper 1, Section II, 12F

  Analysis I | Part IA, 2016

  Let $f:[0,1] \rightarrow \mathbb{R}$ satisfy $|f(x)-f(y)| \leqslant|x-y|$ for all $x, y \in[0,1]$.

  Show that $f$ is continuous and that for all $\varepsilon>0$, there exists a piecewise constant function $g$ such that

  $$\sup _{x \in[0,1]}|f(x)-g(x)| \leqslant \varepsilon .$$

  For all integers $n \geqslant 1$, let $u_{n}=\int_{0}^{1} f(t) \cos (n t) d t$. Show that the sequence $\left(u_{n}\right)$ converges to 0 .

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• Paper 1, Section II, D

  Analysis I | Part IA, 2016

  If $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are sequences converging to $x$ and $y$ respectively, show that the sequence $\left(x_{n}+y_{n}\right)$ converges to $x+y$.

  If $x_{n} \neq 0$ for all $n$ and $x \neq 0$, show that the sequence $\left(\frac{1}{x_{n}}\right)$ converges to $\frac{1}{x}$.

  (a) Find $\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+n}-n\right)$.

  (b) Determine whether $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n}}$ converges.

  Justify your answers.

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• Paper 1, Section II, E

  Analysis I | Part IA, 2016

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$. We say that $x \in \mathbb{R}$ is a real root of $f$ if $f(x)=0$. Show that if $f$ is differentiable and has $k$ distinct real roots, then $f^{\prime}$ has at least $k-1$ real roots. [Rolle's theorem may be used, provided you state it clearly.]

  Let $p(x)=\sum_{i=1}^{n} a_{i} x^{d_{i}}$ be a polynomial in $x$, where all $a_{i} \neq 0$ and $d_{i+1}>d_{i}$. (In other words, the $a_{i}$ are the nonzero coefficients of the polynomial, arranged in order of increasing power of $x$.) The number of sign changes in the coefficients of $p$ is the number of $i$ for which $a_{i} a_{i+1}<0$. For example, the polynomial $x^{5}-x^{3}-x^{2}+1$ has 2 sign changes. Show by induction on $n$ that the number of positive real roots of $p$ is less than or equal to the number of sign changes in its coefficients.

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• Paper 1, Section II, E

  Analysis I | Part IA, 2016

  State the Bolzano-Weierstrass theorem. Use it to show that a continuous function $f:[a, b] \rightarrow \mathbb{R}$ attains a global maximum; that is, there is a real number $c \in[a, b]$ such that $f(c) \geqslant f(x)$ for all $x \in[a, b]$.

  A function $f$ is said to attain a local maximum at $c \in \mathbb{R}$ if there is some $\varepsilon>0$ such that $f(c) \geqslant f(x)$ whenever $|x-c|<\varepsilon$. Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable, and that $f^{\prime \prime}(x)<0$ for all $x \in \mathbb{R}$. Show that there is at most one $c \in \mathbb{R}$ at which $f$ attains a local maximum.

  If there is a constant $K<0$ such that $f^{\prime \prime}(x)<K$ for all $x \in \mathbb{R}$, show that $f$ attains a global maximum. [Hint: if $g^{\prime}(x)<0$ for all $x \in \mathbb{R}$, then $g$ is decreasing.]

  Must $f: \mathbb{R} \rightarrow \mathbb{R}$ attain a global maximum if we merely require $f^{\prime \prime}(x)<0$ for all $x \in \mathbb{R} ?$ Justify your answer.

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Vectors and Matrices

• Paper 1, Section I, A

  Vectors and Matrices | Part IA, 2016

  Let $z \in \mathbb{C}$ be a solution of

  $$z^{2}+b z+1=0$$

  where $b \in \mathbb{R}$ and $|b| \leqslant 2$. For which values of $b$ do the following hold?

  (i) $\left|e^{z}\right|<1$.

  (ii) $\left|e^{i z}\right|=1$.

  (iii) $\operatorname{Im}(\cosh z)=0$.

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• Paper 1, Section I, C

  Vectors and Matrices | Part IA, 2016

  Write down the general form of a $2 \times 2$ rotation matrix. Let $R$ be a real $2 \times 2$ matrix with positive determinant such that $|R \mathbf{x}|=|\mathbf{x}|$ for all $\mathbf{x} \in \mathbb{R}^{2}$. Show that $R$ is a rotation matrix.

  Let

  $$J=\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)$$

  Show that any real $2 \times 2$ matrix $A$ which satisfies $A J=J A$ can be written as $A=\lambda R$, where $\lambda$ is a real number and $R$ is a rotation matrix.

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• Paper 1, Section II, $8 \mathbf{C}$

  Vectors and Matrices | Part IA, 2016

  (a) Show that the equations

  $$\begin{array}{r} 1+s+t=a \\ 1-s+t=b \\ 1-2 t=c \end{array}$$

  determine $s$ and $t$ uniquely if and only if $a+b+c=3$.

  Write the following system of equations

  $$\begin{aligned} &5 x+2 y-z=1+s+t \\ &2 x+5 y-z=1-s+t \\ &-x-y+8 z=1-2 t \end{aligned}$$

  in matrix form $A \mathbf{x}=\mathbf{b}$. Use Gaussian elimination to solve the system for $x, y$, and $z$. State a relationship between the rank and the kernel of a matrix. What is the rank and what is the kernel of $A$ ?

  For which values of $x, y$, and $z$ is it possible to solve the above system for $s$ and $t$ ?

  (b) Define a unitary $n \times n$ matrix. Let $A$ be a real symmetric $n \times n$ matrix, and let $I$ be the $n \times n$ identity matrix. Show that $|(A+i I) \mathbf{x}|^{2}=|A \mathbf{x}|^{2}+|\mathbf{x}|^{2}$ for arbitrary $\mathbf{x} \in \mathbb{C}^{n}$, where $|\mathbf{x}|^{2}=\sum_{j=1}^{n}\left|x_{j}\right|^{2}$. Find a similar expression for $|(A-i I) \mathbf{x}|^{2}$. Prove that $(A-i I)(A+i I)^{-1}$ is well-defined and is a unitary matrix.

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• Paper 1, Section II, $\mathbf{6 B}$

  Vectors and Matrices | Part IA, 2016

  The $n \times n$ real symmetric matrix $M$ has eigenvectors of unit length $\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}$, with corresponding eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$, where $\lambda_{1}>\lambda_{2}>\cdots>\lambda_{n}$. Prove that the eigenvalues are real and that $\mathbf{e}_{a} \cdot \mathbf{e}_{b}=\delta_{a b}$.

  Let $\mathbf{x}$ be any (real) unit vector. Show that

  $$\mathbf{x}^{\mathrm{T}} M \mathrm{x} \leqslant \lambda_{1}$$

  What can be said about $\mathbf{x}$ if $\mathbf{x}^{\mathrm{T}} M \mathbf{x}=\lambda_{1} ?$

  Let $S$ be the set of all (real) unit vectors of the form

  $$\mathbf{x}=\left(0, x_{2}, \ldots, x_{n}\right)$$

  Show that $\alpha_{1} \mathbf{e}_{1}+\alpha_{2} \mathbf{e}_{2} \in S$ for some $\alpha_{1}, \alpha_{2} \in \mathbb{R}$. Deduce that

  $$\underset{\mathbf{x} \in S}{\operatorname{Max}} \mathbf{x}^{\mathrm{T}} M \mathbf{x} \geqslant \lambda_{2}$$

  The $(n-1) \times(n-1)$ matrix $A$ is obtained by removing the first row and the first column of $M$. Let $\mu$ be the greatest eigenvalue of $A$. Show that

  $$\lambda_{1} \geqslant \mu \geqslant \lambda_{2}$$

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• Paper 1, Section II, A

  Vectors and Matrices | Part IA, 2016

  (a) Use suffix notation to prove that

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

  (b) Show that the equation of the plane through three non-colinear points with position vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ is

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

  where $\mathbf{r}$ is the position vector of a point in this plane.

  Find a unit vector normal to the plane in the case $\mathbf{a}=(2,0,1), \mathbf{b}=(0,4,0)$ and $\mathbf{c}=(1,-1,2)$.

  (c) Let $\mathbf{r}$ be the position vector of a point in a given plane. The plane is a distance $d$ from the origin and has unit normal vector $\mathbf{n}$, where $\mathbf{n} \cdot \mathbf{r} \geqslant 0$. Write down the equation of this plane.

  This plane intersects the sphere with centre at $\mathbf{p}$ and radius $q$ in a circle with centre at $\mathbf{m}$ and radius $\rho$. Show that

  $$\mathbf{m}-\mathbf{p}=\gamma \mathbf{n}$$

  Find $\gamma$ in terms of $q$ and $\rho$. Hence find $\rho$ in terms of $\mathbf{n}, d, \mathbf{p}$ and $q$.

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• Paper 1, Section II, B

  Vectors and Matrices | Part IA, 2016

  What does it mean to say that a matrix can be diagonalised? Given that the $n \times n$ real matrix $M$ has $n$ eigenvectors satisfying $\mathbf{e}_{a} \cdot \mathbf{e}_{b}=\delta_{a b}$, explain how to obtain the diagonal form $\Lambda$ of $M$. Prove that $\Lambda$ is indeed diagonal. Obtain, with proof, an expression for the trace of $M$ in terms of its eigenvalues.

  The elements of $M$ are given by

  $$M_{i j}= \begin{cases}0 & \text { for } i=j \\ 1 & \text { for } i \neq j\end{cases}$$

  Determine the elements of $M^{2}$ and hence show that, if $\lambda$ is an eigenvalue of $M$, then

  $$\lambda^{2}=(n-1)+(n-2) \lambda$$

  Assuming that $M$ can be diagonalised, give its diagonal form.

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