Suppose that \(u_n\) satisfies the recurrence relation \[u_{n+2} = \alpha u_{n+1} + \beta u_n,\] and \(v_n\) satisfies the relation \[v_{n+2} = (\alpha^2 + 2\beta)v_{n+1} - \beta^2 v_n \quad (n \geq 0)\] with \(v_0 = u_0\), \(v_1 = u_2\). Show that \(v_n = u_{2n}\) for all positive integers \(n\). Hence or otherwise show that, if \(v_0 = v_1 = 1\) and \(v_{n+2} = 3v_{n+1} - v_n\), then \(v_n = 2^n \quad (n\geq 2)\).
The equation \(x^3 + ax^2 + bx + c\) (\(c \neq 0\)) has three distinct roots which are in geometric progression and whose reciprocals may be rearranged to form an arithmetic progression. Find \(b\) and \(c\) in terms of \(a\).
Mr and Mrs Pinkeye have three babies: Albert, Bertha and Charles, who sleep in separate rooms. Albert wakes up during the night twice as often as Bertha and Bertha wakes up twice as often as Charles. Albert cries on 20\% of the occasions when he wakes up, Bertha cries on 50\% of the occasions when she wakes up and Charles on 80\%. For the purposes of this question it may be assumed that they wake their parents up but not each other. The parents wake as soon as a child starts crying. To which child should Mr Pinkeye go first and what is the probability that, if he does so, he has gone to the right child?
The manufacturers claim that 4 people out of 5 cannot tell `Milkoflave' from cows' milk. The Milk Marketing Board claims on the contrary that 4 out of 5 people can tell the difference. A consumers' magazine asks you to decide between the two claims. Explain how you would try to do so. Bear in mind that the magazine will wish to know in advance how many tests you propose to make and that tests cost money. [You are not asked to find a best procedure but to suggest a reasonable procedure.]
Let \(X_1, X_2, ...\) be independent random variables uniformly distributed on \([1, 2]\). Show that \[\Pr(a < (X_1X_2 ... X_n)^{1/n} < b) \to 1 \text{ as } n \to \infty\] if and only if \(a < 4/e < b\). You may use any results from probability theory that you know.
Show that the integral \[I_n = \int_{-\infty}^{+\infty} x^{2n}e^{-x^2}dx\] (where \(n\) is a positive integer) obeys the recurrence relation \[I_{n+1} = (n + \tfrac{1}{2})I_n\] By expanding \(\cos ax\) as a power series in \(x\), or otherwise, show that \[\int_{-\infty}^{+\infty} e^{-x^2}\cos ax \, dx = \pi^{1/2}e^{-a^2/4}.\] [You may assume that you may integrate the infinite series term by term.]
Find the limit of \[\left(\frac{\beta x^{\beta-1}}{x^\beta - a^\beta} - \frac{1}{x-a}\right)\] as \(x \to a\).
The equation \[\sin x = \lambda x\] (where \(\lambda > 0\), \(x > 0\)) has a finite number \(N\) of non-zero solutions \(x_i\), \(i = 1, \ldots, N\), where \(N\) depends on \(\lambda\), provided \(\lambda < 1\).
A uniform circular cylinder of mass \(m\) and radius \(a\) moves under the action of a horizontal force \(P\) on a rough horizontal table. The force \(P\) is directed through the centre of mass of the cylinder. The table moves with acceleration \(f\) in the same direction as the force \(P\). Both \(P\) and \(f\) are perpendicular to the axis of the cylinder. The coefficient of friction between the cylinder and the table is \(\mu\). Initially the cylinder and the table are at rest, and there is no slipping in the subsequent motion. When the cylinder has moved a distance \(x\) perpendicular to its axis, it has rotated through an angle \(\theta\) about its axis. Show that \[\ddot{x} + a\ddot{\theta} = f\] and find \(\ddot{\theta}\). Show that \[\mu \geq \tfrac{1}{3}|mf - P|/mg\] where \(g\) is the acceleration due to gravity.
A small bead can slide on the spoke of a wheel of radius \(b\) that is constrained to rotate about its axle with angular velocity \(\omega\). Initially the bead is at rest relative to the wheel, at a distance \(d > 0\) from the centre. Show that if gravity can be ignored, the bead will always slide to the rim of the wheel, whatever the coefficients \(\mu_1\) and \(\mu_2\) of static and sliding friction. When its distance from the centre of the axle is \(a\), the bead has an outward velocity of magnitude \([\sqrt{(1 + \mu_1^2)} + \mu_2]a\omega\). Show that the bead hits the rim of the wheel at a time \[t = \frac{\ln(b/a)}{[\sqrt{(1 + \mu_1^2)} - \mu_2]\ \omega}\] later. [You may assume the spoke to lie in the radial direction.]