If a polygon of an even number of sides be inscribed in a circle, shew that the products of the perpendiculars drawn from any point on the circle on the alternate sides are equal.
Obtain Newton's formulae connecting the coefficients of the equation \[ x^n + p_1x^{n-1} + p_2x^{n-2} + \dots + p_n = 0 \] and the sums \(s_1, s_2, \dots\) of the powers of the roots of the equation. Express \(p_r\) in terms of \(s_1, s_2, \dots, s_r\), and \(s_r\) in terms of \(p_1, p_2, \dots, p_r\). Define the order \(r\), and the weight \(w\), of a rational symmetric function of the roots; and shew that it can be expressed as a rational function of degree \(r\) in the coefficients, the sum of the suffixes in each term being the same and equal to \(w\). Find the value of \(\Sigma (a_1^2 a_2^2 a_3)\), where \(a_1, a_2, \dots, a_n\) are the roots of the above equation.
A uniform wooden pole, of specific gravity 0.64, is floating on water and one end is lifted out of the water. What fraction of the rod's length remains immersed?
Shew (by induction or otherwise) that if \(n\) and \(k\) are positive integers, then \[ f_{n,k} = x^n - k (x+y)^n + \frac{k(k-1)}{1.2}(x+2y)^n - \frac{k(k-1)(k-2)}{1.2.3}(x+3y)^n + \dots \] contains no term in \(x\) of degree higher than \(n-k\); and deduce that \[ f_{n,n} = (-1)^n n! y^n. \]
A straight uniform rod of weight \(w\) and length \(l\) is laid on a rough horizontal table, the coefficient of friction being \(\mu\). A horizontal force is applied at one end at right angles to the rod, and the force is gradually increased until the rod begins to move. Shew that the rod begins to turn about a point distant \(l/\sqrt{2}\) from that end, and that the magnitude of the force then is \(\mu w (\sqrt{2}-1)\).
Any two points \(P, Q\) are taken on two non-intersecting straight lines, shew that the locus of the middle point of \(PQ\) is a plane.
Write a short essay on complex numbers, starting from the beginning and erecting a series of definitions and theorems sufficient to justify the manipulation of complex numbers in accordance with all the laws of elementary algebra.
A simple engine governor consists of a parallelogram of jointed rods each \(9''\) in length: it rotates about a vertical diameter and carries a pair of balls at the side joints each 5 lbs. in weight, whilst the lowest joint carries a collar of 10 lbs. weight sliding on the vertical axis. Find the limits of speed between which the centres of the two balls will rotate at a radius of \(4\frac{1}{2}''\), if there is a frictional force of 2 lbs. weight tending to prevent sliding at the collar.
Shew that the number of divisors (unity and the number itself included) of the number \[ N = p_1^{a_1} p_2^{a_2} \dots p_n^{a_n}, \] where \(p_1, p_2, \dots p_n\) are different primes, is \[ (a_1+1)(a_2+1)\dots(a_n+1) \equiv d(N). \] If \(f(N)\) is the number of different primes, and \(F(N)\) the total number of primes, including repetitions, in the above expression of \(N\), then \[ 2^{f(N)} \leq d(N) < 2^{F(N)}. \] In what circumstances can the signs of equality occur?
A circular cylinder of weight \(W\) rests on a rough inclined plane, being partly supported by a fine thread, which is wrapped round it and, passing over the top of the cylinder, is attached to a fixed point in the plane (the other end is tied to the cylinder and the whole string is in a vertical plane perpendicular to the axis of the cylinder). The string makes an angle \(\gamma\) with the plane. Shew that, if the plane be tilted, the equilibrium is limiting when the angle of inclination, \(\alpha\), of the plane is given by the equation \(\tan\alpha = \mu (1+\cos\gamma)/(1-\mu\sin\gamma)\), and that the tension of the string is then \(\frac{1}{2} W \sin\alpha \sec^2\frac{1}{2}\gamma\).