\(ABCDE\) is a regular pentagon inscribed in a circle, and \(A'\) is the other extremity of the diameter through \(A\). Prove that \[DC^2 - A'C^2 = \frac{1}{4}A'A^2.\]
A circle touches the ellipse \(x^2/a^2 + y^2/b^2 = 1\) at its intersections with the line \(x = c\). Find its centre and radius. Interpret your results when \(c\) is formally put equal to (i) \(a\), (ii) a value strictly between \(a\) and \(a/e\), (iii) \(a/e\), where \(e\) is the eccentricity of the ellipse.
The Cartesian coordinates of a particle \(P\) at time \(t\) are \((x(t), y(t))\), where \[x = u(1+t), \quad (u > 0),\] \[\frac{dy}{dx} = \frac{y}{x} + \frac{x}{(x^2+y^2)^{\frac{1}{2}}}.\] Initially the particle is on the \(x\) axis; if \(O\) is the origin \((0, 0)\), prove that the slope of \(OP\) increases with time, and show that \(4y = 3x\) after a time \[t = \sqrt{2}\exp(15/32) - 1.\]
By applying the Taylor expansion to the function \(f(x) \equiv (x^2-1)^n\), or otherwise, prove that for all \(x\), and \(h \neq 0\), \[\left[\frac{(x^2-1) + 2hx + h^2}{h}\right]^n = \sum_{r=0}^{2n} \frac{h^{r-n}}{r!}\left(\frac{d}{dx}\right)^r[(x^2-1)^n].\] Write \((x^2-1)/h\) for \(h\) on each side of the above equation, and show that for \(1 \leq m \leq n\), \[\frac{1}{(n-m)!}\left(\frac{d}{dx}\right)^{n-m}[(x^2-1)^n] = \frac{1}{(n+m)!}(x^2-1)^m\left(\frac{d}{dx}\right)^{n+m}[(x^2-1)^n].\] Deduce that \[y(x) = \left(\frac{d}{dx}\right)^n[(x^2-1)^n]\] satisfies the differential equation \[\frac{d}{dx}\left[(x^2-1)\frac{dy}{dx}\right] - n(n+1)y = 0.\]
For each integer \(n \geq 1\), write \(t_n\) for the number of ways of placing \(n\) people into groups (so that \(t_1 = 1\), \(t_2 = 2\), \(t_3 = 5\), etc.). Defining \(t_0 = 1\), show that \[t_{n+1} = \sum_{r=0}^{n} \binom{n}{r}t_{n-r},\] for \(n \geq 0\), and hence show that \(t_n/n!\) is the coefficient of \(x^n\) in the Maclaurin expansion of \(\exp(\exp x - 1)\), for each \(n \geq 1\).
In an election there are three candidates, \(A, B\) and \(C\), and \(N\) voters. Each voter acts independently of the others, and is equally likely to vote for any one of the candidates. Each voter votes exactly once. Suppose the voters are numbered \(1, 2, \ldots, N\) and define the random variables \(A_1, \ldots, A_N, B_1, \ldots, B_N\) by \begin{align*} A_j &= \begin{cases} 1 & \text{if the \(j\)th voter votes for \(A\),} \\ 0 & \text{otherwise}; \end{cases} \\ B_j &= \begin{cases} 1 & \text{if the \(j\)th voter votes for \(B\),} \\ 0 & \text{otherwise}. \end{cases} \end{align*} Let \(X, Y\) be the total number of votes cast for \(A, B\) respectively. Find the distribution of \(X\), and the covariance of \(X\) and \(Y\). Why would you expect this covariance to be negative?
(i) The real numbers \(a_1, \ldots, a_n\) satisfy the constraint \begin{equation*} \sum_{i=1}^{n} a_i = C, \end{equation*} where \(C\) is a given constant. Show that \(\sum_{i=1}^{n} a_i^2\) is minimised subject to (*) by \(a_i = C/n\) for \(i = 1, \ldots, n\). (ii) In an experiment to determine the mean body-weight \(\mu\) of a species of moth, \(n\) moths of this species are weighed, and their weights \(x_1, \ldots, x_n\) recorded. It may be assumed that \(x_1,\ldots, x_n\) are uncorrelated and have common mean \(\mu\) and common variance \(\sigma^2\), where \(\sigma^2\) is known. We wish to find the best linear unbiased estimator of \(\mu\), that is the function \(\sum_{i=1}^{n} a_i x_i\) which has expectation \(\mu\) and smallest variance. Assuming (i), find the appropriate values of the set \(\{a_i\}\), and find the variance of the best linear unbiased estimator.
If the moment of inertia of a body of mass \(m\) about an axis which passes through the centre of mass is \(mk^2\), show that the moment of inertia about a parallel axis a distance \(l\) from the first is \(m(k^2+l^2)\). A thin uniform rod of mass \(m\) is attached to a smooth hinge at one end. The rod falls from rest in the horizontal position. If the maximum strain which the hinge can take in any direction is \(mg\), show that the hinge will snap when the rod makes an angle \(\sin^{-1} \left(\frac{2\sqrt{3}}{3}\right)\) with the vertical. Describe, without explicit calculation, the motion of the rod after the hinge has broken.
A two-stage rocket carries a payload of mass \(m\). Each stage has mass \(M\) including fuel of mass \(\lambda M\), where \(0 < \lambda < 1\). When the fuel is ignited, it burns at a constant rate \(k\), and exhaust gases are ejected at constant speed \(w\) relative to the rocket. Justify carefully the following equation of motion, which ignores gravity, during the burning of the first stage: \begin{equation*} (2M + m - kt)\frac{dv}{dt} = wk, \end{equation*} where \(v\) is the speed of the rocket. If the first stage drops off when it is burnt out, and the second stage then ignites, find the velocity of the rocket when both stages are fully burnt. Find also the corresponding velocity for a single stage rocket, with the same properties \(k, w\), which has mass \(2M\) (including fuel of mass \(2\lambda M\)) and carries a payload of mass \(m\).
A uniform solid cylinder is projected up a rough plane with speed \(v\) in such a way that it has initially no rotation. The plane is inclined at an angle \(\alpha\) to the horizontal, and the coefficient of friction is \(\frac{1}{3}\tan\alpha\). Show that the frictional force acts down the plane for all times \(t\) less than \(t_1 = 2V\textrm{cosec}\alpha/3g\). Show also that at this time \(t_1\) a pure rolling motion cannot commence, and that at all later times the frictional force acts up the plane.