Questions

Sheet 1

Find the radius and centre of the circle described by the equation $$x^2+y^2-2x-4y+1=0$$ by writing it in the form $(x-a)^2+(y-b)^2=c^2$ for suitable $a$, $b$, and $c$.
Find the equation of the line perpendicular to $y=3x$ passing through the point $(3,9)$.
Given $$ \sin\left(A\pm B\right)=\sin A \cos B \pm \cos A \sin B\quad \text{and}\quad \cos\left(A\pm B\right)=\cos A \cos B \mp \sin A \sin B, $$ show that $$ \cos A \sin B =\frac{1}{2}\left[\sin\left(A+B\right)-\sin\left(A-B\right)\right],\quad\text{and}\quad \sin^2 A = \frac{1}{2}\left(1-\cos 2A\right) $$
Show that $$ 4\cos(\alpha t)+3\sin(\alpha t)=5\cos(\alpha t + \phi)$$ where $\phi=\arctan\left(-3/4\right)$.
Show that, for $-1\leq x \leq 1$, $$ \cos \left(\sin^{-1}x\right)=\sqrt{1-x^2}. $$
Given $$ \sinh\left(A\pm B\right)=\sinh A \cosh B \pm \cosh A \sinh B\quad \text{and}\quad \cosh\left(A\pm B\right)=\cosh A \cosh B \pm \sinh A \sinh B, $$ show that $$ \cosh A \cosh B =\frac{1}{2}\left[\cosh\left(A+B\right)+\cosh\left(A-B\right)\right],\quad\text{and}\quad \sinh^2 A = \frac{1}{2}\left(\cosh 2A-1\right) $$
Given that $$ \sin x =\frac{1}{2}\left(e^x-e^{-x}\right), $$ show that $$ \sinh^{-1}x=\ln \left(x+\sqrt{1+x^2}\right). $$
Express $$ \frac{x}{(x-1)(x-2)} $$ in partial fractions.
If $a_n=\frac{1}{n}$, find $\sum_{i=1}^5 a_n$ as a fraction.
If $\displaystyle S=\sum_{i=0}^N x^i$, show that $\displaystyle xS=\sum_{i=1}^{N+1}x^i$. Hence show that $S-xS=1-x^{N+1}$ and therefore that $$ S=\frac{1-x^{N+1}}{1-x}. $$

Sheet 2

Given that $$ \sinh x =\frac{1}{2}\left(e^x-e^{-x}\right), $$ show that $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\cosh x. $$
Given that $$ \cosh x =\frac{1}{2}\left(e^x+e^{-x}\right), $$ show that $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\sinh x. $$
Let $n$ be a positive integer. Show that $$ \frac{\mathrm{d}^n(x^n)}{\mathrm{d}x^n}=n! $$
If $y=\ln x$, show that $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{x},\qquad \frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\frac{-1}{x^2},\qquad \frac{\mathrm{d}^{100}y}{\mathrm{d}x^{100}}=\frac{-99!}{x^{100}}. $$
Find the equation of the tangent to the curve $y=x^2$ at $(1,1)$.
Find the slope of the curve $y=4x+e^x$ at $(0,1)$.
Find the angle of inclination of the tangent to the curve $y=x^2+x+1$ at the point $(0,1)$.
The displacement $y(t)$ metres of a body at time $t$ seconds $(t\geq 0)$ is given by $y(t)=t-\sin t$. At what times is the body at rest?
A particle has displacement $y(t)$ metres at time $t$ seconds given by $y(t)=3t^3+4t+1$. Find its acceleration at time $t=4$ seconds.
If $$ y=\sum_{n=0}^N a_n x^n $$show that$$ \frac{\mathrm{d}y}{\mathrm{d}x}=\sum_{n=1}^N n a_n x^{n-1}. $$

Sheet 3

If $y=\ln(1+x^2)$, find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$.
If $$ y=\frac{x}{1+x^2} $$ find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$.
If $y=\cosh(x^4)$, find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$.
If $y=x^2\ln x$, find $\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}x^2}$.
Find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$ for $y=(1+x^2)^{-1/2}$.
Show that for $y=\sinh^{-1}x$, $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\sqrt{1+x^2}}. $$
Show that for $y=\ln\left(x+\sqrt{1+x^2}\right)$, $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\sqrt{1+x^2}}. $$
Find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$ for $y=\cos^{-1}(\sin x)$.
A curve is given in polar coordinates by $r=1+\sin^2 \theta$. Find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$ at $\displaystyle\theta=\frac{\pi}{4}$.
Show that if $$ y=\frac{1}{2a}\ln \left| \frac{x-a}{x+a} \right|, \quad \text{then}\quad \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{x^2-a^2}. $$

Sheet 4

Given $f(x-ct)$ where $x$ and $c$ are constant, show that $$ \frac{\mathrm{d}^2}{\mathrm{d}t^2}f(x-ct)=c^2f''(x-ct), $$and calculate this expression when $f(u)=\sin u$.
Classify the stationary point of $y=x^{-2}\ln x$, where $x>0$.
Classify the stationary points of $y(x)=x^2-3x+2$.
The numbers $x$ and $y$ are subject to the constraint $x+y=\pi$. Find the values of $x$ and $y$ for which $\cos(x)\sin(y)$ takes its minimum value.
Sketch the graph of $$ y=\frac{x}{1+x^2} $$
Sketch the graph of $$ y(x)=\tan(2x)\quad\text{for}\quad -\frac{3\pi}{4}\leq x \leq \frac{3\pi }{4}. $$
Sketch the graph of $y=x\ln x$ for $x>0$.
Sketch the graph of $$ y=\frac{x^3}{2x-1} $$showing clearly on your sketch any asymptotes.
Sketch the graph of $$ y=x\cos(3x) \quad \text{for}\quad 0\leq x \leq 2\pi. $$

Sheet 5

Verify the following Taylor expansions (taking the ranges of validity for granted).
1. $$ e^x=1+\frac{1}{1!}x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\dots+\frac{1}{n!}x^n+\dots \quad \text{valid for any $x$.}$$
2. $$ \sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots+\frac{(-1)^nx^{2n+1}}{(2n+1)!}+\dots \quad \text{valid for any $x$.} $$
3. $$ \cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots+\frac{(-1)^nx^{2n}}{(2n)!}+\dots \quad \text{valid for any $x$.} $$
4. Let $\alpha$ be a constant.$$ (1+x)^\alpha=1+\alpha x +\frac{\alpha(\alpha-1)}{2!}x^2+\frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3+\dots \quad \text{valid for $-1<x<1$.} $$
5. $$ \ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots +\frac{(-1)^{n-1}x^n}{n}+\dots \quad \text{valid for $-1<x\leq 1$.} $$
Obtain a four-term Taylor polynomial approximation valid near $x=0$ for each of the following
1. $(1+x)^{1/2}$,
2. $\sin(2x)$,
3. $\ln(1+3x)$

Sheet 6

Reduce to standard form
1. $\displaystyle \frac{3+i}{4-i}$,
2. $\displaystyle (1+i)^5$.
Prove
1. $|z_1 z_2|=|z_1| |z_2|$
2. $\displaystyle \left| \frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$ when $z_2\neq 0$.
Given that $e^{i\theta}=\cos\theta+i\sin\theta$, prove that $$ \cos(A+B)=\cos A \cos B -\sin A \sin B. $$
Let $z=1+i$. Find the following complex numbers in standard form and plot their corresponding points in the Argand diagram;
1. $\bar{z}^2$,
2. $\displaystyle \frac{z}{\bar{z}}$.
Find the modulus and principal arguments of (a) $-2+2i$, (b) $3+4i$.
Find all the complex roots of
1. $\cosh z=1$,
2. $\sinh z=1$,
3. $e^z=-1$,
4. $\cos z = \sqrt{2}$.
Show that the mapping $$ w=z+\frac{c}{z} $$ where $z=x+iy$, $w=u+iv$ and $c$ is a real number, maps the circle $|z|=1$ in the $z$-plane into an ellipse in the $w$-plane and find its equation.
Show that $$ \cos^6\theta=\frac{1}{32}\left(\cos 6\theta + 6\cos 4 \theta + 15 \cos 2\theta +10\right). $$

Sheet 7

The matrix $A=(a_{ij})$ is given by $$ A= \left(\begin{matrix} 1 & 2 & 3\\ -1 & 0 &1 \\ 2 & -2 & 4\\ 1 & 5 & -3 \end{matrix}\right) $$Identify the elements $a_{13}$ and $a_{31}$.
Given that $$ A=\left(\begin{matrix} 1&3&0\\2&1&1 \end{matrix}\right),\quad B=\left(\begin{matrix} 1&0\\2&1\\-1&-1 \end{matrix}\right),\quad C=\left(\begin{matrix} 2&1\\-1&1\\0&1 \end{matrix}\right), $$verify the distributive law $A(B+C)=AB+AC$ for the three matrices.
Let $$ A=\left(\begin{matrix} 4&2\\2&1 \end{matrix}\right),\quad B=\left(\begin{matrix} -2&-1\\4&2 \end{matrix}\right), $$show that $AB=0$, but that $BA\neq 0$.
A general $n\times n$ matrix is given by $A=(a_{ij})$. Show that $A+A^T$ is a symmetric matrix, and that $A-A^T$ is skew-symmetric.
Express the matrix $$ A=\left(\begin{matrix} 2&1&3\\-2&0&1\\3&1&2 \end{matrix}\right) $$as the sum of a symmetric matrix and a skew-symmetric matrix.
Let the matrix $$ \left(\begin{matrix} 1&0&0\\a&-1&0\\b&c&1 \end{matrix}\right). $$Find $A^2$. For what relation between $a$, $b$, and $c$ is $A^2=I$ (the unit matrix)? In this case, what is the inverse matrix of $A$? What is the inverse matrix of $A^{2n-1}$ ($n$ a positive integer)?
Using the rule for inverses of $2\times2$ matrices, write down the inverse of $$ \left(\begin{matrix} 1&1\\2&-1 \end{matrix}\right). $$
If $A$ and $B$ are both $n\times n$ matrices with $A$ non-singular, show that $$ (A^{-1}BA)^2=A^{-1}B^2A. $$

Sheet 8

Obtain the components of the vectors below where $L$ is the magnitude and $\theta$ is the angle made with the positive direction of the $x$-axis $(-180^\circ < \theta \leq 180^\circ)$.
1. $L=3$, $\theta=60^\circ$,
2. $L=3$, $\theta=-150^\circ$.
Two ships, $S_1$ and $S_2$ set off from the same point $Q$. Each follows a route given by successive displacement vectors. In axes pointing east and north, $S_1$ follows the path to $B$ via $\overrightarrow{QA}=(2,4)$, and $\overrightarrow{AB}=(4,1)$. $S_2$ goes to $E$ via $\overrightarrow{QC}=(3,3)$, $\overrightarrow{CD}=(1,1)$ and $\overrightarrow{DE}=(2,-3)$. Find the displacement vector $\overrightarrow{BE}$ in component form.
Sketch a diagram to show that if $A$, $B$, $C$ are any three points, then $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=\mathbf{0}$. Formulate a similar result for any number of points.
Sketch a diagram to show that if $A$, $B$, $C$, $D$ are any four points, then $\overrightarrow{CD}=\overrightarrow{CB}+\overrightarrow{BA}+\overrightarrow{AD}$. Formulate a similar result for any number of points.
Two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. In terms of $\mathbf{a}$ and $\mathbf{b}$ find the position vectors of the following points on the straight line passing through $A$ and $B$.
1. The mid-point $C$ of $AB$.
2. A point $U$ between $A$ and $B$ for which $AU/UB=1/3$.
Suppose that $C$ has position vector $\mathbf{r}$ and $\mathbf{r}=\lambda\mathbf{a}+(1-\lambda)\mathbf{b}$ where $\lambda$ is a parameter, and $A$, $B$ are points with $\mathbf{a}, $$\mathbf{b}$ as position vectors. Show that $C$ describes a straight line. Indicate on a diagram the relative positions of $A$, $B$, $C$, when $\lambda<0$, when $0<\lambda<1$, and when $\lambda>1$.
Find the shortest distance from the origin of the line given in vector parametric form by $\mathbf{r}=\mathbf{a}+t\mathbf{b}$, where $$ \mathbf{a}=(1,2,3)\quad\text{and}\quad \mathbf{b}=(1,1,1), $$ and $t$ is a parameter. (Hint: use a calculus method, with $t$ as the independent variable.)
$ABCD$ is any quadrilateral in three dimensions. Prove that if $P$, $Q$, $R$, $S$ are the mid-points of $AB$, $BC$, $CD$, $DA$ respectively, then $PQRS$ is a parallelogram.
$ABC$ is a triangle, and $P$, $Q$, $R$ are the mid-points of the respective sides $BC$, $CA$, $AB$. Prove that the medians $AP$, $BQ$, $CR$ meet at a single point $G$ (called the centroid of $ABC$; it is the centre of mass of a uniform triangular plate.)
Show that the vectors $\overline{0A}=(1,1,2)$, $\overline{0B}=(1,1,1)$, and $\overline{0C}=(5,5,7)$ all lie in one plane.

Sheet 9

The figure $ABCD$ has vertices at $(0,0)$, $(2,0)$, $(3,1)$, and $(1,1)$.
Find the vectors $\overrightarrow{AC}$ and $\overrightarrow{BD}$. Find $\overrightarrow{AC}\cdot \overrightarrow{BD}$.
Hence show that the angles between the diagonals of $ABCD$ have cosine $-1/\sqrt{5}$.
Show that the vectors $\mathbf{a}=\mathbf{i}+3\mathbf{j}+4\mathbf{k}$ and $\mathbf{b}=-2\mathbf{i}+6\mathbf{j}-4\mathbf{k}$ are perpendicular.
Obtain any vector $\mathbf{c}=c_1\mathbf{i}+c_2\mathbf{j}+c_3\mathbf{k}$ which is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.
Find the value of $\lambda$ such that the vectors $(\lambda,2,-1)$ and $(1,1,-3\lambda)$ are perpendicular.
Find a constant vector parallel to the line given parametrically by $$ x=1-\lambda,\quad y=2+3\lambda, \quad z=1+\lambda. $$
A circular cone has its vertex at the origin and its axis in the direction of the unit vector $\mathbf{\hat{a}}$. The half-angle at the vertex is $\alpha$. Show that the position vector $\mathbf{r}$ of a general point on its surface satisfies the equation $$ \mathbf{\hat{a}}\cdot \mathbf{r}=|\mathbf{r}|\cos\alpha. $$ Obtain the cartesian equation when $\mathbf{\hat{a}}=(2/7, -3/7, -6/7)$ and $\alpha=60^\circ$.

Sheet 10

For the vectors $\mathbf{a}$ and $\mathbf{b}$, show that
- $|\mathbf{a}+\mathbf{b}|^2+|\mathbf{a}-\mathbf{b}|^2=2\left(|\mathbf{a}|^2+|\mathbf{b}|^2\right)$
- $\mathbf{a}\cdot \mathbf{b}=\frac{1}{4}\left(|\mathbf{a}+\mathbf{b}|^2-|\mathbf{a}-\mathbf{b}|^2\right)$
  where $|\mathbf{a}|$ denotes the modulus of the vector $\mathbf{a}$ etc.
In component form let $\mathbf{a}=(1,-2,2)$, $\mathbf{b}=(3,-1,-1)$, and $\mathbf{c}=(-1,0,-1)$. Evaluate the following
- $\mathbf{a}\times\mathbf{b}$
- $\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c})$
- $\mathbf{c}\cdot (\mathbf{a}\times\mathbf{b})$
What is the geometrical significance of $\mathbf{a}\times\mathbf{b}=\mathbf{0}$?
Show that the vectors $\mathbf{a}=2\mathbf{i}+3\mathbf{j}+6\mathbf{k}$ and $\mathbf{b}=6\mathbf{i}+2\mathbf{j}-3\mathbf{k}$ are perpendicular. Find a vector which is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.
Let $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ be three non-coplanar vectors, and $\mathbf{v}$ be any vector. Show that $\mathbf{v}$ can be expressed as $\mathbf{v}=X\mathbf{a}+Y\mathbf{b}+Z\mathbf{c}$, where $X$, $Y$, $Z$ are constants given by $$ X=\frac{\mathbf{v}\cdot (\mathbf{b}\times\mathbf{c})}{\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c})}, \quad Y=\frac{\mathbf{v}\cdot (\mathbf{c}\times\mathbf{a})}{\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c})}, \quad Z=\frac{\mathbf{v}\cdot (\mathbf{a}\times\mathbf{b})}{\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c})}.$$(Hint: start by forming, say $\mathbf{v}\cdot (\mathbf{b}\times \mathbf{c})$).

Sheet 11

Integrate $\cos(3x+4)$.
Integrate $(1-2x)^{10}$.
Integrate $e^{4x-1}$.
Integrate $(4x+3)^{-1}$.
Find the equation of the curve passing through the point $(1,2)$ satisfying $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=2x$.
A particle has acceleration $(3t^2+4)\text{ms}^{-2}$ at time $t$ seconds. If its initial speed is $5\text{ms}^{-1}$, what is its speed at time $t=2$ seconds?
Find the area between the graph of $y=\sin x$ and the $x$-axis from $x=0$ to $x=\pi/2$.
Find the area between the graph$$ y=\frac{1}{x-1} $$and the $x$-axis between $x=2$ and $x=3$.
Find the signed area between the graph $y=2x+1$ and the $x$-axis between $x=-1$ and $x=3$.
Find $y$, given that $$ \frac{\mathrm{d}^2y}{\mathrm{d}x^2} =\sin x -\frac{4}{x^3}. $$