If \(\theta\) and \(\phi\) are unequal and less than \(2\pi\), and if \[ (x-a)\cos\theta+y\sin\theta = (x-a)\cos\phi+y\sin\phi = a, \] and \[ \tan\tfrac{1}{2}\theta - \tan\tfrac{1}{2}\phi = 2e, \] prove that \[ y^2 = 2ax - (1-e^2)x^2. \]
Solve for \(x\) and \(y\) the equations \begin{align*} \sin x + \sin y + \sin \alpha &= 0, \\ \cos x + \cos y + \cos \alpha &= 0, \end{align*} and show further that if \(n\) is an integer prime to 3, \[ \cos nx + \cos ny + \cos n\alpha = 0. \]
In any triangle \(ABC\), with the usual notation, prove that \[ r = a \sec\frac{A}{2} \sin\frac{B}{2} \sin\frac{C}{2}. \] If \(AD\) is perpendicular to \(BC\), and if \(\rho_1, \rho_2\) are the radii of the inscribed circles of the triangles \(ABD, ACD\), prove that \[ 2a(\rho_1-\rho_2) = (c-b)(b+c-a). \]
If \(|x|<1\), sum to infinity the series \[ \cos\theta + x\cos 3\theta + x^2\cos 5\theta + \dots + x^n\cos(2n+1)\theta+\dots \] If \[ \log\sin(\theta+i\phi) = \alpha+i\beta, \] prove that \[ 2e^{2\alpha} = \cosh 2\phi - \cos 2\theta. \]
A uniform rectangular board is supported with its plane vertical and with two edges of length \(a\) horizontal, by the pressure of two fingers, one at each of two points \(P\) and \(Q\) in the vertical edges of the board, not at the same horizontal level. If the coefficient of friction of both fingers is \(\mu\), prove that the difference of level of \(P\) and \(Q\) cannot exceed \(\mu a\).
A framework consists of twelve equal rods, six forming the sides of a regular hexagon \(ABCDEF\), and the rest joining the corners of the hexagon to its centre \(O\). All the rods are weightless, inextensible and freely jointed at their ends. The framework is suspended at \(O\) with \(A\) vertically above \(O\), and a weight \(W\) hangs from each corner of the hexagon, in the manner of the Great Wheel of Earl's Court. If the tension in \(OA\) is \(T\), prove that the tension in \(OD\) is \(T+4W\).
A railway truck of mass 10 tons moving at a speed of 4 feet per second collides with a similar stationary truck free to move. The buffer springs, which obey Hooke's law, are such that a force of 5 tons weight between the trucks decreases their distance apart by 9 inches. Find the greatest compression produced in the springs.
\(A\) is the highest point of a fixed smooth sphere whose centre is \(O\). A particle \(P\), starting from rest at \(A\), slides under the action of gravity down the outside of the sphere. Prove that it will leave the sphere when \(3\cos\theta = 2\), where \(\theta\) is the angle \(AOP\). For any smaller value of \(\theta\), find the radial and tangential components of the acceleration of the particle.
A small smooth sphere of mass \(m\) impinges on a small smooth sphere of mass \(m'\) at rest, and \(m'\) starts moving with velocity \(V\). If \(e\) is the coefficient of restitution, prove that, whether the impact is direct or oblique, the kinetic energy dissipated is \[ \frac{m'(m+m')(1-e)V^2}{2m(1+e)}. \]
Masses \(m,m'\) are attached to the ends of a weightless inextensible string \(AOB\) and rest on a smooth horizontal table. The string is in contact with a fixed smooth peg at \(O\), and the portions \(OA(=a)\) and \(OB(=b)\) of the string are straight. The mass \(m\) is now projected horizontally with velocity \(u\) perpendicular to \(OA\). If the string remains in contact with the peg, and all the motion takes place in a horizontal plane, prove that the mass \(m'\) reaches the peg with velocity \[ \frac{u}{a+b}\sqrt{\frac{mb(2a+b)}{m+m'}}. \]