More challenging Questions

Induction 1

1. Factorials are defined inductively by the rule  $$0!=1\quad\text{and}\quad (n+1)!=n!\times(n+1).$$  Then binomial coefficients are defined for $0\leq k\leq n$ by $$\binom{n}{k}=\frac{n!}{k!(n-k)!}.$$ Prove from these definitions that $$\binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1},$$ and deduce the Binomial Theorem: that for any $x$ and $y$, $$(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^ky^{n-k}.$$
2. Prove that  $$\sum_{r=1}^n \frac{1}{r^2}\leq 2-\frac{1}{n}.$$
3. Prove that for $n=1,2,3,\dots$  $$\sqrt{n}\leq \sum_{k=1}^{n}\frac{1}{\sqrt{k}}\leq 2\sqrt{n}-1.$$
4. Let $\displaystyle A=\left(\begin{matrix}        5&-1\\4&1    \end{matrix}\right)$. Show that  $$A^n=3^{n-1}\left(\begin{matrix}        2n+3&-n\\4n&3-2n    \end{matrix}\right)$$ for $n=1,2,3,\dots$. Can you find a matrix $B$ such that $B^2=A$? 
5. Let $k$ be a positive integer. Prove by induction on $n$ that  $$\sum_{r=1}^n r(r+1)(r+2)\dots (r+k-1)=\frac{n(n+1)(n+2)\dots(n+k)}{k+1}.$$ Show now by induction on $k$ that  $$\sum_{r=1}^n r^k = \frac{n^{k+1}}{k+1}+E_k(n)$$ where $E_k(n)$ is a polynomial of degree at most $k$.

Induction 2

1.  Show that $n$ lines in the plane, no two of which are parallel and no three meeting in a point, divide the plane into $$\frac{n^2+n+2}{2}$$ regions.
2. Prove that for every positive integer $n$, that $$3^{3n-2}+2^{3n+1}$$     is divisible by 19.   
3.  
   1. Show that if $u^2-2v^2=1$ then $$(3u+4v)^2-2(2u+3v)^2=1.$$
   2. Beginning with $u_0=3$, $v_0=2$, show that the recursion $$u_{n+1}=3u_n+4v_n \quad \text{and} \quad v_{n+1}=2u_n+3v_n$$ generates infinitely many integer pairs $(u,v)$ which satisfy $u^2-2v^2=1$.
   3. How can this process be used to produce better and better rational approximations to $\sqrt{2}$? How many times need this process be repeated to produce a rational approximation accurate to 6 decimal places?
4. The Fibonacci numbers $F_n$ are defined by the recurrence relation $$F_n=F_{n-1}+F_{n-2}\quad \text{for $n\geq 2$}$$ and $F_0=0$ and $F_1=1$.Prove that for every integer $n\geq 0$, that $$F_n=\frac{\alpha^n-\beta^n}{\sqrt{5}}$$ where $$\alpha=\frac{1+\sqrt{5}}{2},\quad \text{and}\quad \beta=\frac{1-\sqrt{5}}{2}.$$ (Hint: you may find it helpful to show first that the two roots of the equation $x^2=x+1$ are $\alpha$ and $\beta$.)
5. The sequence of numbers $x_0$, $x_1$, $x_2$, $\dots$ begins with $x_0=1$ and $x_1=1$ and is then recursively determined by the equations $$x_{n+2}=4x_{n+1}-3x_n+3^n\quad \text{for $n\geq 0$}.$$
   1. Find the values of $x_2$, $x_3$, $x_4$, and $x_5$.
   2. Can you find a solution of the form $$x_n=A+B\times 3^n +C \times n 3^n$$ which agrees with the values of $x_0,\dots,x_5$ that you have found?
   3. Use induction to prove that this is the correct formula for $x_n$ for all $n\geq 0$.

Algebra 1

1.  
   1. Find the remainder when $n^2+4$ is divided by 7 for $0\leq n < 7$.
      Deduce that $n^2+4$ is not divisible by 7, for every positive integer $n$. (Hint: write $n=7k+r$ where $0\leq r<7$.)
   2. Now $k$ is an integer such that $n^3+k$ is not divisible by 4 for all integers $n$. What are the possible values of $k$?
2.  
   1. Prove that if $a$, $b$ are positive real numbers then $$\sqrt{ab}\leq \frac{1}{2}(a+b).$$
   2. Now let $a_1$, $a_2$, $\dots$, $a_n$ be positive real numbers. Let $S=a_1+a_2+\dots+a_n$ and $P=a_1a_2\cdots a_n$. Suppose that $a_i$ and $a_j$ are distinct. Show that replacing $a_i$ and $a_j$ with $(a_i+a_j)/2$ and $(a_i+a_j)/2$ increases $P$ without changing $S$.
      Deduce that $$(a_1a_2\cdots a_n)^{1/n}\leq \frac{a_1+a_2+\cdots+a_n}{n}.$$
3.  
   1. Let $n$ be a positive integer. Show that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\cdots + xy^{n-2}+y^{n-1}).$$
   2. Let $a$ also be a positive integer. Show that if $a^n-1$ is prime then $a=2$ and $n$ is prime.
      Is it true that if $n$ is prime then $2^n-1$ is also prime?
4. Let $a$, $b$, $r$, $s$ be rational numbers with $s\neq 0$. Suppose that the number $r+s\sqrt{2}$ is a root of the quadratic equation $$x^2+ax+b=0.$$ Show that $r-s\sqrt{2}$ is also a root.
5.  
   1. The cubic equation $ax^3+bx^2+cx+d=0$ has roots $\alpha$, $\beta$, $\gamma$ and so factorises as $$a(x-\alpha)(x-\beta)(x-\gamma).$$ Determine $$\alpha+\beta+\gamma, \quad \alpha\beta+\beta\gamma+\gamma\alpha, \quad \alpha\beta\gamma,$$ in terms of $a$, $b$, $c$, $d$. What does $\alpha^2+\beta^2+\gamma^2$ equal?
   2. Show that $\cos 3\theta=4\cos^3 \theta - 3 \cos \theta$.
   3. By considering the roots of the equation $4x^3-3x-\cos 3\theta=0$ deduce that $$\cos \theta \cos(\theta+2\pi/3)\cos(\theta+4\pi/3)=\frac{\cos(3\theta)}{4}.$$ What do $$\cos\theta+\cos(\theta+2\pi/3)+\cos(\theta+4\pi/3)\quad \text{and}\quad \cos^2\theta+\cos^2(\theta+2\pi/3)+\cos^2(\theta+4\pi/3)$$ equal?

Algebra 2

1. Under what conditions on the real numbers $a$, $b$, $c$, $d$, $e$, $f$ do the simultaneous equations $$ax+by=e\quad \text{and}\quad cx+dy=f$$ have (a) a unique solution, (b) no solution, (c) infinitely many solutions in $x$ and $y$.
   Select values of $a$, $b$, $c$, $d$, $e$, $f$ for each of these cases, and sketch on separate axes the lines $ax+by=e$ and $cx+dy=f$.
2. For what values of $a$ do the simultaneous equations $$\begin{align*}x+2y+a^2z&=0,\\x+ay+z&=0,\\x+ay+a^2z&=0,\end{align*}$$ have a solution other than $x=y=z=0$? For each such $a$ find the general solution to the above equations.
3. Do $2\times 2$ matrices exist satisfying the following properties? Either find such matrices or show that no such exist.
   1. $A$ such that $A^5=I$ and $A^i\neq I$ for $1\leq i\leq 4$,
   2. $A$ such that $A^n\neq I$ for all positive integers $n$,
   3. $A$ and $B$ such that $AB\neq BA$,
   4. $A$ and $B$ such that $AB$ is invertible and $BA$ is singular (i.e. not invertible)
   5. $A$ such that $A^5=I$ and $A^{11}=0$.
4. Let $$A=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\quad \text{and let}\quad A^T=\left(\begin{matrix}a&c\\b&d\end{matrix}\right)$$ be a $2\times2$ matrix and its transpose. Suppose that $\det A=1$ and $$A^TA=\left(\begin{matrix}1&0\\0&1\end{matrix}\right).$$ Show that $a^2+c^2=1$, and hence that $a$ and $c$ can be written as $$a=\cos \theta \quad\text{and}\quad c=\sin \theta$$ for some $\theta$ in the range $0\leq \theta < 2\pi$. Deduce that $A$ has the form $$A=\left(\begin{matrix}\cos\theta&-\sin\theta\\ \sin\theta & \cos\theta\end{matrix}\right).$$
5.  
   1. Prove that $$\det(AB)=\det(A)\det(B)$$ for any $2\times2$ matrices $A$ and $B$.
   2. Let $A$ denote the $2\times 2$ matrix $$\left(\begin{matrix}a&b\\c&d\end{matrix}\right).$$ Show that $$\begin{equation}A^2-(\operatorname{trace}A)A+(\det A)I=0 \qquad \qquad \text{(1)}\end{equation}$$ where
      • $\operatorname{trace}A=a+d$ is the trace of $A$, that is the sum of the diagonal elements,
      • $\det A=ad-bc$ is the determinant of $A$,
      • $I$ is the $2\times 2$ identity matrix.
   3. Suppose now that $A^n=0$ for some $n\geq 2$. Prove that $\det A=0$. Deduce using equation (1) that $A^2=0$.

Calculus 1

1. Sketch the graph of the curve $$y=\frac{x^2+1}{(x-1)(x-2)}$$ carefully labelling any turning points and asymptotes.
2. The parabola $x=y^2+ay+b$ crosses the parabola $y=x^2$ at $(1,1)$ making right angles.
   Calculate the values of $a$ and $b$.
   On the same axes, sketch the two parabolas.
3. The curve $C$ in the $xy$-plane has equation $$x^2+xy+y^2=1.$$ By solving $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=0$, show that the maximum and minimum values taken by $y$ are $\displaystyle \pm \frac{2}{\sqrt{3}}$.
   By changing to polar co-ordinates ($x=r\cos\theta$, $y=r\sin\theta$), sketch the curve $C$.
   What is the greatest distance of a point on $C$ from the origin?
4. Sketch the curve $y=x^3+ax+b$ for a selection of values of $a$ and $b$.
   Suppose now that $a$ is negative. Find the co-ordinates of the turning points of the graph and deduce that $y=0$ has exactly two roots when $$b=\pm\frac{2a}{3}\sqrt{\frac{-a}{3}}$$ For what values of $b$ does the equation $y=0$ have three distinct real roots?
5. On separate $xu$- and $yu$-axes sketch the curves $u=8(x^3-x)$ and $u=e^y/y$ labelling all turning points.
   [Harder] Hence sketch the curve $e^y=8y(x^3-x)$.

Calculus 2

1. Use calculus, or trigonometric identities, to prove the following inequalities for $\theta$ in the range $0<\theta<\frac{\pi}{2}$;
   1. $\sin\theta<\theta$,
   2. $\theta<\tan\theta$,
   3. $\cos 2\theta < \cos^2\theta$.
      Hence, without directly calculating the following integrals, rank them in order of size. $$\text{(a)}\quad \int_0^1 x^3\cos x \,\mathrm{d}x,\qquad\text{(b)}\quad \int_0^1 x^3\cos^2x \,\mathrm{d}x,\qquad\text{(c)}\quad \int_0^1 x^2\sin x \cos x \,\mathrm{d}x,\qquad\text{(d)}\quad \int_0^1 x^3 \cos 2x \,\mathrm{d}x.$$
2. Show that the equation $$\sin x=\frac{1}{2}x$$ has three roots. Using Newton-Raphson, or a similar numerical method, find the positive root to 6 d.p.
   The equation $\sin x =\lambda x$ has three real roots when $\lambda=\alpha$ or when $\beta<\lambda<1$ for two  real numbers $\alpha<0<\beta$. Plot, on the same axes, the curves $$y=\sin x,\qquad y=\alpha x,\qquad y=\beta x.$$
3. Let $S$ denote the circle in the $xy$-plane with centre $(0,0)$ and radius 1. A regular $m$-sided polygon $I_m$ is inscribed in $S$ and a regular $n$-sided polygon $C_n$ is circumscribed about $S$.
   1. By considering the perimeter of $I_m$ and the area bounded by $C_n$, prove that $$m\sin\left(\frac{\pi}{m}\right)<\pi < n \tan \left(\frac{\pi}{n}\right)$$ for all natural numbers $m,n\geq 3$.
   2. Archimedes showed (using this method) that $3\frac{10}{71}<\pi <3 \frac{1}{7}$. What are the smallest values of $m$ and $n$ needed to verify Archimedes' inequality?
4. Find the coefficients of $1$, $x$, $x^2$, $x^3$, $x^4$ in the power series expansion (Taylor's series expansion) for $f(x)=\sec x$.
   Use this approximation to make an estimate for $\sec\frac{1}{10}$. With the aid of a calculator, find to how many decimal places the approximation is accurate.
5. Show that $\int \ln x \,\mathrm{d}x=x\ln x - x +\text{constant}$.
   Sketch the graph of the equation $y=\ln x$. By consideration of areas on your graph, show that  $$n\ln n -n+1<\sum_1^n \ln r < (n+1)\ln (n+1)-n$$
   Let $G_n=\sqrt[n]{n!}$ denote the geometric mean of $1$, $2$, $\dots$, $n$. Show that $G_n/n$ approaches $1/e$ as $n$ becomes large.

Calculus 3

1. Evaluate $$\int\frac{\ln x}{x}\,\mathrm{d}x,\qquad \int x \sec^2 x \,\mathrm{d}x, \qquad \int_3^\infty \frac{\mathrm{d}x}{(x-1)(x-2)},\qquad \int_0^1 \tan^{-1}x\,\mathrm{d}x,\qquad \int_0^1 \frac{\mathrm{d}x}{e^x+1}.$$
2. Evaluate, using trigonometric and/or hyperbolic substitutions, $$\int \frac{\mathrm{d}x}{x^2+1},\qquad \int_1^2 \frac{\mathrm{d}x}{\sqrt{x^2-1}}, \qquad \int \frac{\mathrm{d}x}{\sqrt{4-x^2}},\qquad \int_2^\infty\frac{\mathrm{d}x}{(x^2-1)^{3/2}}.$$
3. By completing the square in the denominator, and using the substitution $$x=\frac{\sqrt{2}}{3}\tan \theta -\frac{1}{3}$$ evaluate $$\int\frac{\mathrm{d}x}{3x^2+2x+1}.$$ By similarly completing the square in the following denominators, and making appropriate trigonometric and/or hyperbolic substitutions, evaluate the following integrals $$\int\frac{\mathrm{d}x}{\sqrt{x^2+2x+5}},\qquad \int_0^\infty \frac{\mathrm{d}x}{4x^2+4x+5}.$$
4. Let $t=\tan \frac{1}{2}\theta$. Show that $$\sin\theta =\frac{2t}{1+t^2},\qquad \cos\theta=\frac{1-t^2}{1+t^2},\qquad \tan \theta = \frac{2t}{1-t^2}$$ and that $$\mathrm{d}\theta=\frac{2\mathrm{d}t}{1+t^2}.$$ Use the substitution $t=\tan \frac{1}{2}\theta$ to evaluate $$\int_0^{\pi/2}\frac{\mathrm{d}\theta}{(1+\sin\theta)^2}.$$
5. Let $$I_n=\int_0^{\pi/2} x^n\sin x \,\mathrm{d}x.$$
   Evaluate $I_0$ and $I_1$.Show, using integration by parts, that $$I_n=n\left(\frac{\pi}{2}\right)^{n-1}-n(n-1)I_{n-2}.$$ Hence, evaluate $I_5$ and $I_6$.

Calculus 4

1. Find the general solutions of the following separable differential equations. $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x^2}{y}, \qquad\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\cos^2x}{\cos^2 2y},\qquad\frac{\mathrm{d}y}{\mathrm{d}x}=e^{x+2y}.$$
2. Find the solution of the following initial value problems. On separate axes sketch the solution to each problem.
   $$\begin{align*}\frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{1-2x}{y},\qquad y(1)=-2,\\\frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{x(x^2+1)}{4y^3},\qquad y(0)=\frac{-1}{\sqrt{2}},\\\frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{1+y^2}{1+x^2},\qquad y(0)=1.\end{align*}$$
3. The equation for Simple Harmonic Motion, with constant frequency $\omega$, is $$\frac{\mathrm{d}^2x}{\mathrm{d}t^2}=-\omega^2 x.$$ Show that $$\frac{\mathrm{d}^2x}{\mathrm{d}t^2}=v\frac{\mathrm{d}v}{\mathrm{d}x}$$ where $\displaystyle v=\frac{\mathrm{d}x}{\mathrm{d}t}$ denotes velocity. Find and solve a separable differential equation in $v$ and $x$ given that $x=a$ when $v=0$.
   Hence show that $$x(t)=a\sin\left(\omega t+\varepsilon\right)$$  for some constant $\varepsilon$.
4. Find the most general solution of the following homogenous constant coefficient differential equations:
   $$\begin{align*}\frac{\mathrm{d}^2y}{\mathrm{d}x^2}-y&=0,\\\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+4y&=0,\quad\text{where $y(0)=y'(0)=1$.}\\\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+3\frac{\mathrm{d}y}{\mathrm{d}x}+2y&=0,\\\frac{\mathrm{d}^2y}{\mathrm{d}x^2}-4\frac{\mathrm{d}y}{\mathrm{d}x}+4y&=0,\quad\text{where $y(0)=y'(0)=1$.}\end{align*}$$
5. Write the differential equation $$(2x+y)+(x+2y)\frac{\mathrm{d}y}{\mathrm{d}x}=0$$ in the form $$\frac{\mathrm{d}}{\mathrm{d}x}\left(F(x,y)\right)=0$$ where $F(x,y)$ is a polynomial in $x$ and $y$. Hence find the general solution of the equation.
   Use this method to find the general solution of $$\left(y\cos x +2xe^y\right)+\left(\sin x +x^2e^y-1\right)\frac{\mathrm{d}y}{\mathrm{d}x}=0.$$

Calculus 5

1. Find all solutions of the following separable differential equations: $$\begin{align*}\frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{y-xy}{xy-x}. \\\frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{\sin^{-1}x}{y^2\sqrt{1-x^2}},\quad y(0)=0. \\\frac{\mathrm{d}^2y}{\mathrm{d}x^2}&=(1+3x^2)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2\quad\text{where $y(1)=0$ and $y'(1)=\frac{-1}{2}$.}\end{align*}$$
2. Use the method of integrating factors to solve the following equations with initial conditions
   $$\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x}+xy &= x,\quad \text{where $y(0)=0$}.\\         2x^3 \frac{\mathrm{d}y}{\mathrm{d}x} -3x^2y &=1,\quad \text{where $y(1)=0$}.\\         \frac{\mathrm{d}y}{\mathrm{d}x} - y\tan x &=1,\quad \text{where $y(0)=1$}.     \end{align*}$$
3. Find the most general solution of the following inhomogeneous constant coefficient differential equations:
   $$\begin{align*}         \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+3\frac{\mathrm{d}y}{\mathrm{d}x}+2y &= x,\\         \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+3\frac{\mathrm{d}y}{\mathrm{d}x}+2y &= \sin x,\\         \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+3\frac{\mathrm{d}y}{\mathrm{d}x}+2y &= e^x,\\         \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+3\frac{\mathrm{d}y}{\mathrm{d}x}+2y &= e^{-x}.     \end{align*}$$
4.  
   1. By making the substitution $y(x)=xv(x)$ in the following homogeneous polar equations, convert them into separable differential equation involving $v$ and $x$, which you should then solve.
      $$\begin{align*}             \frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{x^2+y^2}{xy}.\\             x\frac{\mathrm{d}y}{\mathrm{d}x}&=y+\sqrt{x^2+y^2}. \end{align*}$$
   2. Make substitutions of the form $x=X+a$, $y=Y+b$, to turn the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x+y-3}{x-y-1}$$ into a homogeneous polar differential equation in $X$ and $Y$. Hence find the general solution of the above equation.
5. A particle $P$ moves in the $xy$-plane. Its co-ordinates $x(t)$ and $y(t)$ satisfy the equations $$\frac{\mathrm{d}y}{\mathrm{d}t}=x+y\quad\text{and}\quad \frac{\mathrm{d}x}{\mathrm{d}t}=x-y,$$ and at time $t=0$ the particle is at $(1,0)$. Find, and solve, a homogeneous polar equation relating $x$ and $y$.
   By changing to polar co-ordinates ($r^2=x^2+y^2$, $\tan\theta=y/x$), sketch the particle's journey for $t\geq 0$.

Complex Numbers

1. By writing $\omega= a +ib$ (where $a$ and $b$ are real), solve the equation $$\omega^2=-5-12i.$$ Hence find the two roots of the quadratic equation $$z^2-(4+i)z+(5+5i)=0.$$
2. By substituting $z=x+iy$ or $z=re^{i\theta}$ into the following equations and inequalities, sketch the following regions of the complex plane on separate Argand diagrams:
   1. $|z-3-4i|<5$
   2. $\displaystyle  \arg(z)=\frac{\pi}{3}$
   3. $\displaystyle 0\leq \operatorname{Re}\left(\frac{iz+3}{2}\right)<2$
   4. $e^z=1$
   5. $\operatorname{Im}(z^2)<0$
3. Find the image of the point $z=2+it$ under each of the following transformations.
   1. $z\mapsto iz$
   2. $z\mapsto z^2$
   3. $z\mapsto e^z$
   4. $\displaystyle z\mapsto \frac{1}{z}$
      By letting $t$ vary over all real values find the image of the line $\operatorname{Re}z=2$ under the same transformations.
4.  
   1. Given that $e^{i\theta}=\cos \theta + i\sin \theta$, prove that $$\cos(\alpha+\beta)=\cos \alpha\cos\beta - \sin\alpha \sin \beta.$$
   2. Use De Moivre's Theorem to show that $$\cos 5\theta = 16 \cos^5 \theta -20 \cos^3 \theta +5 \cos \theta.$$
5.  
   1. Let $z=\cos\theta+i\sin\theta$ and let $n$ be an integer. Show that $$2\cos\theta = z+\frac{1}{z}\quad \text{and that} \quad 2i\sin\theta=z-\frac{1}{z}.$$ Find expressions for $\cos n\theta$ and $\sin n\theta$ in terms of $z$.
   2. Show that $$\cos^5 \theta = \frac{1}{16}\left(\cos 5 \theta + 5 \cos 3 \theta +10\cos \theta \right)$$ and hence find $\int_0^{\pi/2}\cos^5 \theta \,\mathrm{d}\theta$.

Geometry

1. Describe the regions of space given by the following vector equations. In each, $\mathbf{r}$ denotes the vector $x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$; $\cdot$ and $\wedge$ denote the scalar (dot) and vector (cross) product:
   1. $\mathbf{r}\wedge\left(\mathbf{i}+\mathbf{j}\right)=\left(\mathbf{i}-\mathbf{j}\right)$,
   2. $\mathbf{r}\cdot \mathbf{i}=1$,
   3. $|\mathbf{r}-\mathbf{i}|=|\mathbf{r}-\mathbf{j}|$,
   4. $|\mathbf{r}-\mathbf{i}=1$,
   5. $\mathbf{r}\cdot \mathbf{i}=\mathbf{r}\cdot \mathbf{j}=\mathbf{r} \cdot \mathbf{k}$,
   6. $\mathbf{r}\wedge\mathbf{i}=\mathbf{i}$.
2. Find the shortest distance between the lines $$\frac{x-1}{2}=\frac{y-3}{3}=\frac{z}{2}\qquad\text{and}\qquad x=2,\quad \frac{y-1}{2}=z$$ (Hint: parametrise the lines and write down the vector between two arbitrary points on the lines; then determine when this vector is normal to both lines.)
3. Let $L_\theta$ denote the line through $(a,b)$ making an angle $\theta$ with the $x$-axis. Show that $L_\theta$ is a tangent of the parabola $y=x^2$ if and only if $$\tan^2\theta - 4a \tan \theta +4b=0.$$ (Hint: parametrise $L_\theta$ as $x=a+\lambda \cos\theta$ and $y=b+\lambda \sin \theta$ and determine when $L_\theta$ meets the parabola precisely once.)
   Show that the tangents from $(a,b)$ to the parabola subtend an angle of $\pi/4$ if and only if $$1+24b+16b^2=16a^2.$$ (Hint: use the formula $\tan(\theta_1-\theta_2)=(\tan\theta_1-\tan\theta_2)/(1+\tan\theta_1\tan\theta_2)$.)
   Sketch the curve $1+24y+16y^2=16x^2$ and the original parabola on the same axes.
4. What transformations of the $xy$-plane do the following matrices represent?
   1. $\left(\begin{matrix}x\\y\end{matrix}\right)\mapsto \left(\begin{matrix}1&0\\0&-1 \end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)$
   2. $\left(\begin{matrix}             x\\y         \end{matrix}\right)\mapsto \left(\begin{matrix}             2&0\\0&1         \end{matrix}\right)\left(\begin{matrix}             x\\y         \end{matrix}\right)$
   3. $\left(\begin{matrix}             x\\y         \end{matrix}\right)\mapsto \left(\begin{matrix}             1/2&1/2\\1/2&1/2         \end{matrix}\right)\left(\begin{matrix}             x\\y         \end{matrix}\right)$
   4. $\left(\begin{matrix}             x\\y         \end{matrix}\right)\mapsto \left(\begin{matrix}             \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta         \end{matrix}\right)\left(\begin{matrix}             x\\y         \end{matrix}\right)$
      Which, if any, of these transformations are invertible?
5. The cycloid is the curve given parametrically by the equations $$x(t)=t-\sin t, \quad \text{and}\quad y(t)=1-\cos t \quad \text{for $0\leq t\leq 2\pi$}.$$
   1. Sketch the cycloid.
   2. Find the arc-length of the cycloid.
   3. Find the area bounded by the cycloid and the $x$-axis.
   4. Find the area of the surface of revolution generated by rotating the cycloid around the $x$-axis.
   5. Find the volume enclosed by the surface of revolution generated by rotating the cycloid around the $x$-axis.