Prove, by use of the identity \[ \frac{1}{1+x+x^2} = \frac{1-x}{1-x^3}, \] or otherwise, that \[ 1 - (n-1) + \frac{(n-2)(n-3)}{1.2} - \frac{(n-3)(n-4)(n-5)}{1.2.3} + \dots \] (the series being continued so long as all the factors in the numerator are positive) is equal to \(-1, 0, \text{ or } 1\): and give rules for distinguishing between the different cases.
A light string \(ABCDE\), of length 100 inches, is divided into four equal parts at \(B, C, D\). The ends \(A\) and \(E\) are fixed in the same horizontal at a distance 88 inches apart, and weights \(W, W'\) and \(W\) are hung from \(B, C\) and \(D\). If in equilibrium the depth of \(C\) below \(AE\) is 22 inches, shew that \(W:W' = 11:14\), and that the tensions in \(AB\) and \(BC\) are in the ratio 6:5.
Find the sum of the cubes of the first \(n\) natural numbers. Find the sum to \(2n+1\) terms of the series \(1^3-2^3+3^3-4^3+\dots\).
The equation of a rational algebraic curve of the \(n\)th degree being written in the form \[ x^n f_0\left(\frac{y}{x}\right) + x^{n-1} f_1\left(\frac{y}{x}\right) + x^{n-2} f_2\left(\frac{y}{x}\right) + \dots = 0, \] obtain the asymptotes of the curve; and find the conditions that it may have (i) an inflexional asymptote, (ii) two parallel asymptotes, (iii) a parabolic asymptote. Trace the curve \(x(y^2-4x^2)-x-2y+3=0\), and shew that \((\frac{1}{2},2)\) is a double point.
A well-known safety device for lifts consists of an extension of the lift shaft below ground level; the floor of the lift is made to fit this well closely, so that a pneumatic buffer is provided. A lift weighing 3000 lbs. falls from a height of 30 ft. above ground level into such a safety pit 10 ft. deep, the base of the lift being 8 ft. by 5 ft. Find approximately how far the lift will descend before it is stopped, neglecting air leakage and assuming that the pressure of the air varies inversely as its volume. The resulting equation may be solved graphically: take atmospheric pressure as 15 lbs. per sq. in.
Prove that \[ \begin{vmatrix} 1 & 1 & 1 \\ \sec A & \sec B & \sec C \\ \cosec A & \cosec B & \cosec C \end{vmatrix} = 16 \frac{\sin\frac{B-C}{2} \sin\frac{C-A}{2} \sin\frac{A-B}{2}}{\sin 2A \sin 2B \sin 2C} \{\sin(B+C) + \sin(C+A) + \sin(A+B)\}. \]
A tripod, formed of three equal rods each of weight \(2W\) smoothly hinged together at one end, stands on a smooth horizontal plane, and a weight \(W\) is suspended from the hinge. Equilibrium is maintained by light strings joining the feet \(A, B, C\) of the rods. Shew that the pressure on the plane at \(A\) is \(W(3-2\cot B \cot C)\), and that the tensions of the strings are given by \(T_1 \cos A = T_2 \cos B = T_3 \cos C = 2W \tan\theta \cot A \cot B \cot C\), where \(\theta\) is the inclination of each rod to the vertical.
Find correct to two decimal places the real root of the equation \[ x^3+x^2+x-100=0. \]
Write an essay on the determination of the state of stress in a plane frame built up by light rigid bars, which are smoothly jointed. Consider (i) the relation between the number of joints and the number of bars in a frame, which is just stiff,
A railway motor-car, weighing 30 tons, is driven by a petrol engine direct coupled to a dynamo, which supplies motors geared to the driving axles. If the total tractive resistance be equivalent to 20 lbs. per ton, and if the maximum speed on the level be 36 miles an hour, find (1) the efficiency of the motors and gearing, (2) the efficiency of the engine and dynamo: given that the maximum current is 360 amperes at 165 volts, and that for this output the engine consumes 55 lbs. of petrol an hour, the fuel having a calorific value of 11,000 thermal units per pound. One thermal unit is equivalent to 1400 ft. lbs.