A pile of mass 4 tons is to be driven into the muddy bottom of a canal, the resistance varying directly as the depth already penetrated. It is found that the pile sinks a distance of 3 inches under its own weight. Shew that when the pile-hammer of mass \(\frac{1}{2}\) ton falls on the pile through a distance of 12 ft, the pile and the hammer sink a further distance of 1.47 inches.
Prove that the path of a projectile in a vacuum would be a parabola. A small elastic spherical ball is dropped on to a fixed hemispherical dome, whose base rests on a horizontal plane, striking it at a point where the radius from the centre of the dome to the point of contact makes an angle 30\(^\circ\) with the vertical. Shew that the ball will strike the plane at a distance \(R\) from the dome if it is dropped from a height \(\frac{R}{4}(9-3\sqrt{3})\) above the horizontal plane, where \(R\) is the radius of the dome and the coefficient of elasticity is \(\frac{1}{\sqrt{3}}\).
A particle of mass \(m\), lying on a smooth horizontal table, is attached to two elastic strings whose natural lengths are \(l\) and \(l'\) and moduli \(\lambda\) and \(\lambda'\) respectively. The other ends of the strings are fixed to two points on the table at a distance apart greater than \(l+l'\). Shew that if the particle vibrates in the line of the strings, its period will be \[ 2\pi \sqrt{\frac{m}{\frac{\lambda}{l}+\frac{\lambda'}{l'}}}. \]
Find the conditions that \(ax^2+2hxy+by^2+2gx+2fy+c\) should
Shew how to express \(ax^2+2bx+c\) and \(a'x^2+2b'x+c'\) simultaneously in the forms \(p(x-\alpha)^2+q(x-\beta)^2\) and \(p'(x-\alpha)^2+q'(x-\beta)^2\). Apply your method to \[ 7x^2-22x+28 \quad \text{and} \quad 27x^2-62x+68. \]
Prove that \[ \log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots. \] Sum the series \[ \sum_{n=0}^{n=\infty} \frac{(3n+2)}{n^2+3n+2}x^n. \]
Prove the rule for finding successive convergents of the continued fraction \[ \frac{1}{a_1+} \frac{1}{a_2+} \frac{1}{a_3+} \dots. \] In the recurring continued fraction \[ \frac{1}{1+} \frac{1}{2+} \frac{1}{3+} \frac{1}{1+} \frac{1}{2+} \frac{1}{3+} \dots, \] prove that \(3p_{3n}=2q_{3n}+q_{3n-3}\).
Prove Wilson's theorem that \((n-1)!+1\) is divisible by \(n\) when \(n\) is a prime. Prove that \(\frac{72!}{(36!)^2}-1\) is divisible by 73.
State Maclaurin's theorem on the expansion of a function \(f(x)\) in ascending powers of \(x\). If \(y=e^{\sin^{-1}x}\) and \(y_n\) denote the \(n\)th differential coefficient of \(y\) with respect to \(x\), prove that, when \(x=0\), \(y_{n+2}=(n^2+1)y_n\), and hence expand \(y\) in ascending powers of \(x\).
Explain how the maxima and minima values of a function \(f(x)\) may be obtained. A right circular cone with axis vertical is drawn to circumscribe a sphere of radius \(a\) resting on a horizontal plane, the base of the cone also resting on the plane. Prove that the minimum volume of the cone is twice the volume of the sphere. Shew also that the sum of the areas of the curved surface of the cone and of its plane base is a minimum at the same time.