A perfectly elastic particle is dropped from a point on a fixed vertical circular hoop, shew that after two rebounds it will rise vertically if \[ 2\sin 4\theta = \tan\theta, \] where \(\theta\) is the angular distance of the point from the highest point of the hoop.
A particle of mass \(m\) is placed on a smooth wedge of mass \(M\) with one face vertical and the other inclined to the horizontal at an angle \(\alpha\) resting on a smooth horizontal table. If the wedge moves parallel to its lines of greatest slope and encounters a fixed obstacle which brings it to rest at the instant when its velocity is \(V\), shew that the magnitude of the impulse between the wedge and the obstacle is \(V(M+m\sin^2\alpha)\), and find the impulse between the wedge and the particle. Shew also that the velocity of the particle relative to the wedge is reduced instantaneously in the ratio \(M+m\sin^2\alpha:M+m\).
Determine \(\sin\frac{\pi}{10}\) and \(\sin\frac{\pi}{5}\), and prove that \[ 8\sin\frac{\pi}{10}\sin\frac{3\pi}{10} = \sqrt{(30-6\sqrt{5})} - \sqrt{5}-1. \]
Two flag-staffs of heights \(a\) and \(b\) stand on level ground at points \(A\) and \(B\). At a point \(P\) on the ground directly between \(A\) and \(B\) they subtend equal angles \(\alpha\) and at another point \(Q\) on the ground distant \(c\) from \(P\) they subtend equal angles \(\beta\). \(PQ\) makes with \(PA\) an angle \(\gamma\). Prove that \[ \frac{1}{a}-\frac{1}{b} = \frac{2}{c}\cot\alpha\cos\gamma, \] also that \[ \cot^2\beta - \cot^2\alpha = c^2/ab. \]
Prove that, when \(n\) is an even integer, \[ \cos n\theta = 1 - \frac{n^2}{2!}\sin^2\theta + \frac{n^2(n^2-2^2)}{4!}\sin^4\theta - \frac{n^2(n^2-2^2)(n^2-4^2)}{6!}\sin^6\theta - \dots. \] Prove that \[ \operatorname{cosec}^2\frac{\pi}{20} + \operatorname{cosec}^2\frac{3\pi}{20} + \operatorname{cosec}^2\frac{7\pi}{20} + \operatorname{cosec}^2\frac{9\pi}{20} = 48. \]
Expand \(\cos x\) in ascending powers of \(x\), and prove that \[ \cos x \cosh x = 1 - \frac{2^2x^4}{4!} + \frac{2^4x^8}{8!} - \dots. \]
Find the maximum and minimum values of \(y\), where \(y^2=x^2(x-1)^3\).
If \(y\) is a function of \(x\) and \(x\) is a function of \(t\), express \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) in terms of differential coefficients with respect to \(t\). Change the independent variable from \(x\) to \(t (=\log x)\) in the equation \[ x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0. \]
Find the equation of the normal at any point of the curve \(x=f(t), y=F(t)\). Shew that the centre of curvature at the point \(t\) on the curve \(x=at^3, y=at^2\) is \[ x=\frac{1}{2}at(1+3t^2), \quad y=-\frac{1}{6}at^2(2+9t^2). \]
Shew how to find the envelope of the curves \(f(x,y,\alpha)=0\), where \(\alpha\) is an arbitrary parameter. Find the envelope of the circles which pass through the vertex of the parabola \(y^2=4ax\) and have their centres on the curve.