In a certain examination the possible marks were integers from 0 to 100; for each such integer there was at least one candidate who obtained that mark. In a second examination, taken by the same candidates, and with the same possible marks, it was found that for each pair of candidates the mark obtained by one was greater than that obtained by the other only if the same had been true in the first examination. Prove that at least one candidate obtained the same mark in both examinations. If a third examination is taken, and the second sentence of the above paragraph remains true when 'third' and 'second' are substituted for 'second' and 'first' respectively, is it necessarily true—
Prove that there cannot exist four (real or complex) numbers, all different, such that the square of each of them is equal to the sum of the cubes of the other three. Prove also that there is one (and only one) set of four distinct numbers such that the square of each is equal to the sum of the fourth powers of the other three. Find the equation whose roots are these numbers.
Let \(p_n\) be the number of ways in which a collection of \(n\) dissimilar objects may be divided into parts, each of which contains at least one object, no regard being paid to order. By convention, the 'division' of the collection into a single part, namely the whole collection, is to be included in this number; also \(p_0\) is to be taken as 1. Prove that \[p_{n+1} = \sum_{m=0}^n \binom{n}{m} p_{n-m},\] where \[\binom{n}{m} = \frac{n!}{m!(n-m)!}\] (0! being taken as 1), and hence, or otherwise, prove that \(p_n/n!\) is the coefficient of \(x^n\) in the expansion in ascending powers of \(x\) of \[e^{x-1}.\]
Given that any solution of the differential equation \[u'' + u = 0\] (where a dash denotes differentiation with respect to \(x\)) must be of the form \(A \sin x + B \cos x\), where \(A\) and \(B\) are constants, and that the function \(f(x)\) satisfies the differential equation \[y'' + \frac{2}{x} y' + y = 0,\] calculate \(\int_a^b xf(x) dx\) in terms of \(f(a)\) and \(f(b)\), indicating any special cases that may arise. Calculate \(\int_a^b x^2f(x) dx\) in terms of \(f'(a)\) and \(f'(b)\).
Nine distinct points, not all collinear, are such that the line joining any two of them passes through a third. Prove that
'The centre of a circle that touches each of two given circles must lie on a certain hyperbola, whose centre is the mid-point of the straight line joining the centres of the given circles.' Examine the above statement; if you find it true, give a proof of it; if not, state and prove a suitably modified version.
Enunciate the principle of virtual work. A light lever \(AOB\) of length \(2a\) can turn freely about its midpoint \(O\), and hangs from the end \(A\). A light rod \(BC\) of length \(h\) is smoothly jointed to \(AB\) and to a weight \(W\) and is restricted by a frictionless constraint to move in the vertical through \(O\). Investigate the positions of equilibrium according as \(a > b\), \(a = b\), or \(a < b\). Show also for the case \(a < b\) that if turning at \(B\) is resisted by a constant frictional couple, there is a position of limiting equilibrium in which the lever makes an angle \(\eta\) with the vertical given by the equation \[4Wa^2 F \sin \theta \cos^2 \theta = (b^2 - a^2)(Wa \sin \theta - F)^2.\]
Define the terms work, energy, and power. A motor-car can travel with speed \(U\) up a slope of 1 in \(a\) and with speed \(V\) down a slope of 1 in \(b\), the power of the engine remaining constant. If the resistance is proportional to the square of the speed, find the maximum speed attainable on a level road. Find also equations for the speed attainable up a slope of 1 in \(c\), and down such a slope. [A slope of 1 in \(a\) is here defined as a slope where the car rises 1 ft. in travelling \(a\) ft. along the road.]
Show that the path of a projectile under gravity is a parabola, and explain the assumptions involved in establishing this. A gun with its barrel inclined at angle \(\alpha\) to the horizontal is wheeled on to a place inclined at angle \(\beta\) to the horizontal, the axles of the gun-carriage being parallel to the lines of greatest slope. Show that the time of flight before a shell fired with velocity \(V\) strikes the plane is \(2V \sin \alpha \sec \beta/g\), and that the direction of the point of impact makes an angle \(\tan^{-1}(\tan \alpha \tan \beta)\) with the horizontal line along the plane through the gun. Find also the ratio of the range to that on a horizontal plane.
Two spheres of masses \(m_1\) and \(m_2\) move with their centres travelling on the same line with velocities \(u_1\) and \(u_2\) and collide. If the coefficient of restitution is \(e\), find the changes of speed produced by the impact. If the initial velocities of the spheres are interchanged and reversed in direction, show that the impulse between the spheres is the same as in the first case. A spherical particle is released at a height \(h\) above a horizontal table with which the coefficient of restitution is \(e\). Show that its average speed, with regard to time, while it is being reduced to rest is \[(\tfrac{1}{2}gh)^{\frac{1}{2}}{(1 + e^2)(1 + e)^{-2}}.\] Explain why this formula does not apply in the case when \(e = 1\).