Prove that the set of matrices of the type $\begin{pmatrix} 1 & 0 & 0 \\ x & 1 & 0 \\ y & z & 1 \end{pmatrix}$ with \(x, y, z\) real numbers, forms a group \(G\) under matrix multiplication. [You may assume that matrix multiplication is associative.] Does the subset consisting of those matrices where \(x, y, z\) are restricted to be integers form a subgroup of \(G\)? Is there an element \(a\) in \(G\), with \(a\) not equal to the identity matrix, such that \(ab = ba\) for all \(b\) belonging to \(G\)? Justify your answers.
Prove that \begin{align} (X + Y + Z)(X + \omega Y + \omega^2 Z)(X + \omega^2 Y + \omega Z) = X^3 + Y^3 + Z^3 - 3XYZ, \end{align} where \(\omega = e^{2\pi i/3}\). Hence, or otherwise, find the roots of the equation \begin{align} x^3 - 6x + 6 = 0. \end{align}
A parabola is given by \(x = at^2 + b, y = ct + d\) where \(a\) and \(c\) are not zero. Find the equation of the tangent at the point \(t\). Show that all the points of intersection of pairs of perpendicular tangents lie on the same straight line.
\(P, A, B, C\) are distinct points in three-dimensional Euclidean space, and \(L, M, N\) are the midpoints of \(BC, CA, AB\) respectively. Prove that the lines through \(L, M, N\) parallel to \(PA, PB, PC\) respectively meet in a point.
Find \(a, b\) such that the function \(f(x) = \frac{(ax + b)}{(x - 1)(x - 4)}\) has a stationary value at \(x = 2\) with \(f(2) = -1\). Show that \(f(x)\) has a maximum at \(x = 2\), and sketch the curve.
Let \(I\) be the integral \begin{align} I = \int_{0}^{\pi/2} \ln (\sin x) \, dx. \end{align} Show, by means of changes of variable, that \begin{align} I &= \int_{0}^{\pi/2} \ln (\cos y) \, dy = \int_{\pi/2}^{\pi} \ln (\sin z) \, dz \end{align} By considering \(\int_{0}^{\pi/2} \ln (\sin 2x) \, dx\), or otherwise, prove that \begin{align} I = -\frac{\pi}{2} \ln 2 \end{align} [You may assume that all these integrals converge.]
Seven sunbathers are positioned at equal intervals along a straight shoreline. Each stares fixedly at a nearest neighbour, choosing a neighbour at random if a choice is available. Show that the expected number of unobserved sunbathers is \(\frac{3}{4}\).
A jar contains \(r\) red, \(b\) blue and \(w\) white sweets. A greedy child picks out sweets one by one at random and eats them, until only sweets of a single colour remain. Show, by induction or otherwise, that the probability that only red sweets remain is \(\frac{r}{r + b + w}\).
Solution: Imagine we continue pulling out sweets. Then the last sweet to be removed will be the same colour as all the other sweets which are the same colour. Therefore the question is equivalent to "what is the probability the last sweet is red", but since all possible orders of sweets are equally likely, this is just \(\frac{r}{r+b+w}\)
A chocolate orange consists of a sphere of smooth uniform chocolate of mass \(M\) and radius \(a\), sliced into segments by planes through its axis. It stands on a horizontal table with its axis vertical, and it is held together only by a narrow ribbon around its equator. Show that the tension in the ribbon is at least \(\frac{3}{8\pi}Mg\). [You may assume that the centre of mass of a segment of angle \(\theta\) is at a distance \((3a/2\pi)\sin(\theta/2)\) from the axis.]
A light spring has natural length \(a\) and is such that when compressed a distance \(x\) it produces a force of magnitude \(kx\). It joins two particles of masses \(m_1\) and \(m_2\). The spring is compressed a distance \(b\) and the system released from rest on a smooth table, so that the particles move in a straight line. Find the positions of the particles at time \(t\) later.