Prove that any two positive numbers \(a\) and \(b\), of which \(a\) is the greater, can be expressed in the form \[ a = m (x + 1) \div (x - 1), \quad b = m (x - 1) \div (x + 1), \] where \(x\) is a number greater than unity. If \(a_1\) and \(b_1\) are the arithmetic and harmonic means of \(a\) and \(b\), \(a_2\) and \(b_2\) the arithmetic and harmonic means of \(a_1\) and \(b_1\), \(a_3\) and \(b_3\) the corresponding means of \(a_2\) and \(b_2\), and so on, then \[ a_n = \sqrt{(ab)} \frac{(\sqrt{a}+\sqrt{b})^{2^n} + (\sqrt{a} - \sqrt{b})^{2^n}}{(\sqrt{a}+\sqrt{b})^{2^n} - (\sqrt{a} - \sqrt{b})^{2^n}}. \]
A rod, of length \(2a\) and weight \(W\), can slide through a short smooth tube which is inclined at \(60^{\circ}\) to the vertical. A string is attached to the lower end of the rod and passes over a smooth peg at a height \(c\) vertically above the tube, and carries a weight \(P\). Shew, by the method of virtual work, or otherwise, that in equilibrium the middle point of the rod will be at a distance \(c-\frac{1}{2}a\) from the tube, if \(P = \frac{1}{4}\sqrt{7}W\), and \(c < 4a\).
Show that, if \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+px^2+qx+r=0\) then \[ \alpha+\beta+\gamma = -p, \quad \beta\gamma+\gamma\alpha+\alpha\beta = q, \quad \alpha\beta\gamma = -r. \] Show that if \(\alpha\beta+1=0\) then \[ 1+q+pr+r^2 = 0. \]
Discuss generally the question of the existence of maxima or minima of the function \[ y = \frac{x^2+2ax+b}{x^2+2Ax+B}, \] and sketch the various possible forms which the graph of the function may have for different values of the constant coefficients. Illustrate your remarks by reference to the functions \[ \frac{x^2}{x^2+x+1}, \quad \frac{x^2-1}{x^2-4}, \quad \frac{x(x-2)}{(x-1)(x-3)}. \]
An electric train starts with an initial acceleration of 2.5 ft. per sec. per sec., and this acceleration diminishes uniformly with the time until it becomes zero after 30 secs. The train then runs at a uniform speed until the brakes are put on, causing a uniform retardation of 3 ft. per sec. per sec. Find the running time from start to stop when the distance between stations is half a mile.
Prove that, if \(n\) is a prime number,
A man takes a time \(t_1\) to row from a point on one bank of a river to the point directly opposite on the other bank, and a time \(t_2\) to row the same distance down the stream. Shew that the ratio of his velocity in still water to that of the stream is \[ \frac{t_1^2+t_2^2}{t_1^2-t_2^2}. \]
Express \(\tan 2x\) in terms of \(\tan x\), and \(\tan x\) in terms of \(\tan 2x\). Explain the relation between the values of \(x\) which correspond to different determinations of the ambiguous sign in the second expression.
Discuss, giving proofs, graphical methods for finding the magnitude and line of action of the resultant (i) of a coplanar system of forces acting at a point, (ii) of a coplanar system of forces which do not all act at a point, distinguishing the cases of equilibrium and a couple resultant. A light rod \(AB\), 10 feet long, is supported at \(A\) and at another point. A load of 1 lb. is suspended from \(B\), and loads of 5 lbs., 2 lbs. at points 2 ft. and 6 ft. from \(A\). (i) If the second support is at the middle point of \(AB\), find graphically the pressures on the supports. (ii) If the pressure on the support at \(A\) is required to be 4 lbs., find graphically where the second point of support must be.
A shell has velocity 2000 feet per second, and bursts into a great number of fragments of equal masses, the velocity impressed on each fragment being 200 feet per second; find the angle of the cone of dispersion of the fragments.