A weight \(W\) is attached to a fixed point by four light strings. At the mid-point of each string, which is of length \(2l\), is fixed a light particle. Any particle repels any other with a force equal to \(\lambda\) times the reciprocal of the square of their distance apart. Show that, if the particles are in equilibrium at the vertices of a square of side \(x\), \[ W^2 x^6 - (9+4\sqrt{2})\lambda x^2 (2l^2 - x^2) = 0. \] Prove that this equation has one and only one positive root.
Obtain formulae for the number of permutations \({}^n P_r\) and the number of combinations \({}^n C_r\) of \(n\) unlike things \(r\) at a time. There are \(n\) cards numbered from 1 to \(n\) in black and from \(n\) to 1 in red so that the two numbers on each card add up to \(n+1\). Prove that the number of groups of \(r\) cards which can be formed so that the \(2r\) numbers on them are all different is \[ \frac{n(n-2)(n-4)\dots(n-2r+2)}{r!} \] if \(n\) is even, and find the corresponding formula when \(n\) is odd. Hence, or otherwise, find the probability that, when four of the numbers \(1, 2, 3, 4, 5, 6, 7, 8\) are taken at random, no two of them add up to 9.
If \begin{align*} \cos\theta &= \cos\alpha\cos\phi, \\ \sin(\theta+\phi) &= \lambda\sin\alpha\cos\phi, \\ \text{and prove that} \\ \sin(\theta-\phi) &= \lambda^{-1}\sin\alpha\cos\phi, \end{align*} and hence, or otherwise, express \(\sin\theta\) and \(\sin\phi\) in terms of \(\lambda\) and \(\alpha\). (It is assumed that \(\lambda \sin\alpha, \cos\alpha, \cos\phi\) are all different from 0.)
Prove that the asymptotes of a rectangular hyperbola bisect the angles between any pair of conjugate diameters. A conic \(\Sigma\) has a focus at the centre of a rectangular hyperbola. A pair of conjugate diameters of the hyperbola meet \(\Sigma\) in \(P, Q, R, S\). Prove that the poles with respect to \(\Sigma\) of \(PR, QR, PS, QS\) lie on the asymptotes of the hyperbola.
A variable point \(P\) lies in a fixed plane containing a fixed point \(A\). A particle at \(P\) is under the action of a force of magnitude \(\lambda/AP\) (where \(\lambda\) is a constant) directed along \(\overrightarrow{AP}\). Prove that, if the particle is displaced along a straight line from \(P_1\) to \(P_2\), the work done by the force in the displacement is \(\lambda \log\frac{AP_2}{AP_1}\). If \(B\) is another fixed point of the plane, and an additional force of magnitude \(\lambda/BP\) directed along \(\overrightarrow{BP}\) acts on the particle, prove that the work done by the resultant force in the displacement from \(P_1\) to \(P_2\) is \(\lambda \log\frac{AP_2.BP_1}{AP_1.BP_2}\).
Prove de Moivre's theorem \((\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta\) for any integer \(n\) (positive, negative, or zero). Explain how this result can be used to deduce trigonometrical identities from algebraical identities, and illustrate by proving the identity \[ \frac{2.4.\dots.2n}{1.3.\dots.(2n-1)}\cos^{2n}\theta = 1 + 2\sum_{r=1}^n \frac{n(n-1)\dots(n-r+1)}{(n+1)(n+2)\dots(n+r)}\cos 2r\theta, \] where \(n\) is any positive integer.
Prove that, if \(m\) and \(n\) are fixed positive integers, then \[ \frac{m}{x^m-1} - \frac{n}{x^n-1} \] tends to a limit when \(x\) tends to 1, and find the limit. By putting \(y=x^\lambda\), or otherwise, prove that, if \(\lambda\) is a fixed positive rational number, then \[ \frac{y^\lambda-1}{y-1} \] tends to the limit \(\frac{1}{2}(1-\lambda)\) when \(y\) tends to 1. [The positive value of \(y^\lambda\) for \(y>0\) is to be taken.]
The conic \(S\), the line \(l\) and the point \(A\) are fixed. A variable line \(\lambda\) through \(A\) meets \(l\) in \(P\), and \(Q\) is the point on \(\lambda\) conjugate to \(P\) with regard to \(S\). Prove that the locus of \(Q\) is a conic passing through \(A\), the points of intersection of \(l\) and \(S\), and the pole of \(l\) with regard to \(S\).
Define the angular velocity of a lamina moving in its own plane. Two circular cylinders \(A, B\) have radii \(a, b\) respectively \((a>b)\) and a common axis. A circular cylinder \(C\) of diameter \(a-b\) touches \(A\) and \(B\), each along a generator. \(A\) rolls on a fixed plane with angular velocity \(\omega\), and \(B\) rotates with angular velocity \(\omega'\) (measured in the same sense). If the surfaces in contact do not slip, show that (i) \(C\) has angular velocity \((a\omega - b\omega')/(a-b)\), and (ii) the plane containing the axes has angular velocity \((a\omega + b\omega')/(a+b)\).
If \[ I_m = \int_0^{\pi/2} \cos^m x \,dx, \] evaluate \(I_{2n}\) and \(I_{2n+1}\) for all non-negative integers \(n\). Prove that \(I_{2n+2} < I_{2n+1} < I_{2n}\), and deduce that \[ \frac{2.4.6.\dots.2n}{1.3.5.\dots.(2n-1)}\frac{1}{\sqrt{n}} \] tends to the limit \(\sqrt{\pi}\) as \(n\) tends to infinity.