If \(f(a), \phi(a)\) each equal to zero, explain how to find the limit of \(\frac{f(x)}{\phi(x)}\) when \(x \to a\), \(f(x), \phi(x)\) being continuous functions of \(x\). Shew that the limit when \(x \to a\) of \[ \frac{(2a^3x-x^4)^{\frac{1}{2}} - a(a^2x)^{\frac{1}{3}}}{a - (ax^2)^{\frac{1}{3}}} \text{ is } \frac{16}{9}a. \]
A particle is projected inside a smooth straight tube of length \(a\), closed at each end, which lies on a smooth horizontal table, and whose mass is equal to that of the particle. If the coefficient of restitution is \(\frac{1}{2}\), prove that just before the third impact the tube has traversed a distance \(\frac{3}{4}a\), and find the proportion of kinetic energy, which has been lost.
Shew that 80 and 81 are respectively the minimum and maximum values of \(2x^3 - 21x^2+72x\).
A particle of mass \(m\) is tied to the middle point of a light string 26 inches long, whose ends are attached to fixed points in the same horizontal line, 2 ft. apart. If the particle rotates about this line so that it describes a complete circle in a vertical plane, shew that the tension of the string, when the particle is at its lowest point, is necessarily more than \((7.8)mg\).
A plane is drawn dividing a sphere into two parts whose volumes are in the ratio \(3:1\). If \(2\alpha\) is the angle that a diameter of the small circle in which the plane meets the sphere subtends at the centre, shew that \(9\cos\alpha = 4+\cos 3\alpha\), and verify that \(2\alpha\) is approximately equal to \(139^\circ 20'\).
Shew that the whole area enclosed by the curve given by \[ x=a\cos^3\theta, \quad y=b\sin^3\theta \text{ is } \frac{3\pi ab}{8}. \]
Integrate \(\int (1+x^2)e^x dx\); \(\int \sec^3 x dx\); and prove that \[ \int_0^\infty \frac{(3x+4)dx}{(x+2)(x^2+x+1)} = \frac{2}{3}\log_e 2 + \frac{4\pi}{3\sqrt{3}}. \]
A uniform ladder, of length \(l\) and weight \(W\), is to be held with its upper end resting against a smooth vertical wall and with its lower end on a smooth horizontal plane. Shew that the ladder can be held in this position, with a man of weight \(W'\) standing on the top, by a horizontal force \(H\) equal to \((\frac{1}{2}W'+W) \tan\alpha\) applied at the lower end, where \(\alpha\) is the inclination of the ladder to the vertical. If the greatest horizontal force which can be applied is equal to \(P\) (less than \(H\)) shew that the ladder can be kept up by applying a couple in addition; and determine the moment of the couple.
Four lamps, each of weight \(w\), are suspended across a road of width \(5a\), from points B, C, D, E of a string whose ends A, F are fixed at the same level as shewn in the figure. The string CD is horizontal and at a depth \(b\) below AF; shew that the tension in CD is equal to \(3wa/b\).
The distance between the axles of a four-wheeled lorry is equal to \(2a\); the centre of gravity of the lorry and its load is halfway between the axles and at a perpendicular height \(h\) above the road. When the brakes are applied so as to lock the lower pair of wheels, the lorry can just rest on a slope of angle \(\alpha\): prove that the coefficient of sliding friction between the lorry and the road is \(\mu = 2a \tan\alpha / (a+h\tan\alpha)\).