Give definitions of, and proofs of the simplest properties of, the hyperbolic functions \(\cosh x, \sinh x, \tanh x\). Draw the graphs of the functions and of the inverse functions; and express the inverse functions in terms of logarithms. Explain the parallelism between formulae involving the hyperbolic functions and the corresponding formulae involving the trigonometrical functions \(\cos x, \sin x, \tan x\).
Two ladders of equal length but unequal weights, hinged together, form a step-ladder, the weights of the two parts being \(W_1\) and \(W_2\) respectively; and equilibrium is maintained by friction between the ladders and the ground, the coefficient of friction being \(\mu\). Find the inclination of the ladders at which slipping is on the point of taking place; and calculate the action at the hinge when this is the case. Assume the centre of gravity of each ladder to be at its middle point.
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.