Part IA, 2005, Paper 1

# Part IA, 2005, Paper 1

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1.I.1C

commentConvert the following expressions from suffix notation (assuming the summation convention in three dimensions) into standard notation using vectors and/or matrices, where possible, identifying the one expression that is incorrectly formed:

(i) $\delta_{i j}$,

(ii) $\delta_{i i} \delta_{i j}$,

(iii) $\delta_{l l} a_{i} b_{j} C_{i j} d_{k}-C_{i k} d_{i}$,

(iv) $\epsilon_{i j k} a_{k} b_{j}$,

(v) $\epsilon_{i j k} a_{j} a_{k}$.

Write the vector triple product $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ in suffix notation and derive an equivalent expression that utilises scalar products. Express the result both in suffix notation and in standard vector notation. Hence or otherwise determine $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ when $\mathbf{a}$ and $\mathbf{b}$ are orthogonal and $\mathbf{c}=\mathbf{a}+\mathbf{b}+\mathbf{a} \times \mathbf{b}$.

1.I.2B

commentLet $\mathbf{n} \in \mathbb{R}^{3}$ be a unit vector. Consider the operation

$\mathbf{x} \mapsto \mathbf{n} \times \mathbf{x}$

Write this in matrix form, i.e., find a $3 \times 3$ matrix $\mathbf{A}$ such that $\mathbf{A} \mathbf{x}=\mathbf{n} \times \mathbf{x}$ for all $\mathbf{x}$, and compute the eigenvalues of $\mathbf{A}$. In the case when $\mathbf{n}=(0,0,1)$, compute $\mathbf{A}^{2}$ and its eigenvalues and eigenvectors.

1.II.5C

commentGive the real and imaginary parts of each of the following functions of $z=x+i y$, with $x, y$ real, (i) $e^{z}$, (ii) $\cos z$, (iii) $\log z$, (iv) $\frac{1}{z}+\frac{1}{\bar{z}}$, (v) $z^{3}+3 z^{2} \bar{z}+3 z \bar{z}^{2}+\bar{z}^{3}-\bar{z}$,

where $\bar{z}$ is the complex conjugate of $z$.

An ant lives in the complex region $R$ given by $|z-1| \leq 1$. Food is found at $z$ such that

$(\log z)^{2}=-\frac{\pi^{2}}{16} .$

Drink is found at $z$ such that

$\frac{z+\frac{1}{2} \bar{z}}{\left(z-\frac{1}{2} \bar{z}\right)^{2}}=3, \quad z \neq 0$

Identify the places within $R$ where the ant will find the food or drink.

1.II.6B

commentLet $\mathbf{A}$ be a real $3 \times 3$ matrix. Define the rank of $\mathbf{A}$. Describe the space of solutions of the equation

$\tag{†} \mathbf{A x}=\mathbf{b},$

organizing your discussion with reference to the rank of $\mathbf{A}$.

Write down the equation of the tangent plane at $(0,1,1)$ on the sphere $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=2$ and the equation of a general line in $\mathbb{R}^{3}$ passing through the origin $(0,0,0)$.

Express the problem of finding points on the intersection of the tangent plane and the line in the form $(†)$. Find, and give geometrical interpretations of, the solutions.

1.II.7A

commentConsider two vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^{n}$. Show that a may be written as the sum of two vectors: one parallel (or anti-parallel) to $\mathbf{b}$ and the other perpendicular to $\mathbf{b}$. By setting the former equal to $\cos \theta|\mathbf{a}| \hat{\mathbf{b}}$, where $\hat{\mathbf{b}}$ is a unit vector along $\mathbf{b}$, show that

$\cos \theta=\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$

Explain why this is a sensible definition of the angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$.

Consider the $2^{n}$ vertices of a cube of side 2 in $\mathbb{R}^{n}$, centered on the origin. Each vertex is joined by a straight line through the origin to another vertex: the lines are the $2^{n-1}$ diagonals of the cube. Show that no two diagonals can be perpendicular if $n$ is odd.

For $n=4$, what is the greatest number of mutually perpendicular diagonals? List all the possible angles between the diagonals.

1.II.8A

commentGiven a non-zero vector $v_{i}$, any $3 \times 3$ symmetric matrix $T_{i j}$ can be expressed as

$T_{i j}=A \delta_{i j}+B v_{i} v_{j}+\left(C_{i} v_{j}+C_{j} v_{i}\right)+D_{i j}$

for some numbers $A$ and $B$, some vector $C_{i}$ and a symmetric matrix $D_{i j}$, where

$C_{i} v_{i}=0, \quad D_{i i}=0, \quad D_{i j} v_{j}=0,$

and the summation convention is implicit.

Show that the above statement is true by finding $A, B, C_{i}$ and $D_{i j}$ explicitly in terms of $T_{i j}$ and $v_{j}$, or otherwise. Explain why $A, B, C_{i}$ and $D_{i j}$ together provide a space of the correct dimension to parameterise an arbitrary symmetric $3 \times 3$ matrix $T_{i j}$.

1.I.3F

commentDefine the supremum or least upper bound of a non-empty set of real numbers.

Let $A$ denote a non-empty set of real numbers which has a supremum but no maximum. Show that for every $\epsilon>0$ there are infinitely many elements of $A$ contained in the open interval

$(\sup A-\epsilon, \sup A) .$

Give an example of a non-empty set of real numbers which has a supremum and maximum and for which the above conclusion does not hold.

1.I.4D

commentLet $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series in the complex plane with radius of convergence $R$. Show that $\left|a_{n} z^{n}\right|$ is unbounded in $n$ for any $z$ with $|z|>R$. State clearly any results on absolute convergence that are used.

For every $R \in[0, \infty]$, show that there exists a power series $\sum_{n=0}^{\infty} a_{n} z^{n}$ with radius of convergence $R$.

1.II.10D

commentExplain what it means for a bounded function $f:[a, b] \rightarrow \mathbb{R}$ to be Riemann integrable.

Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a strictly decreasing continuous function. Show that for each $x \in(0, \infty)$, there exists a unique point $g(x) \in(0, x)$ such that

$\frac{1}{x} \int_{0}^{x} f(t) d t=f(g(x)) .$

Find $g(x)$ if $f(x)=e^{-x}$.

Suppose now that $f$ is differentiable and $f^{\prime}(x)<0$ for all $x \in(0, \infty)$. Prove that $g$ is differentiable at all $x \in(0, \infty)$ and $g^{\prime}(x)>0$ for all $x \in(0, \infty)$, stating clearly any results on the inverse of $f$ you use.

1.II.11E

commentProve that if $f$ is a continuous function on the interval $[a, b]$ with $f(a)<0<f(b)$ then $f(c)=0$ for some $c \in(a, b)$.

Let $g$ be a continuous function on $[0,1]$ satisfying $g(0)=g(1)$. By considering the function $f(x)=g\left(x+\frac{1}{2}\right)-g(x)$ on $\left[0, \frac{1}{2}\right]$, show that $g\left(c+\frac{1}{2}\right)=g(c)$ for some $c \in\left[0, \frac{1}{2}\right]$. Show, more generally, that for any positive integer $n$ there exists a point $c_{n} \in\left[0, \frac{n-1}{n}\right]$ for which $g\left(c_{n}+\frac{1}{n}\right)=g\left(c_{n}\right)$.

1.II.12E

commentState and prove Rolle's Theorem.

Prove that if the real polynomial $p$ of degree $n$ has all its roots real (though not necessarily distinct), then so does its derivative $p^{\prime}$. Give an example of a cubic polynomial $p$ for which the converse fails.

1.II.9F

commentExamine each of the following series and determine whether or not they converge.

Give reasons in each case.

$(i)$

$(i i)$

$\sum_{n=1}^{\infty} \frac{1}{n^{2}+(-1)^{n+1} 2 n+1}$

(iii)

$\sum_{n=1}^{\infty} \frac{n^{3}+(-1)^{n} 8 n^{2}+1}{n^{4}+(-1)^{n+1} n^{2}}$

$(i v)$

$\sum_{n=1}^{\infty} \frac{n^{3}}{e^{e^{n}}}$