Problems

Prove that \[ \frac{(1-\sin\theta)(1+\sin 15\theta)}{(1+\sin 3\theta)(1-\sin 5\theta)} = (16\sin^4\theta - 8\sin^3\theta - 16\sin^2\theta+8\sin\theta+1)^2, \] and find the values of \(\theta\) for which the expression on the right vanishes. (Note: The numbers in the trigonometric functions on the LHS are very difficult to read from the scan and may be inaccurate.)

The centres of the circumcircle and the inscribed circle of a triangle are \(O\) and \(I\), the radii are \(R\) and \(r\). Prove that \[ OI^2 = R^2 - 2Rr. \] Triangles are inscribed in a circle, centre \(O\), and circumscribed to a circle, centre \(I\). Show that the centres of their escribed circles lie on a circle of radius \(2R\), whose centre \(I'\) is such that \(O\) bisects \(II'\).

Find the asymptotes of the curve \[ x^2(x+y)=x+4y, \] and trace the curve.

If \(w\) is a function of \(x\) and \(y\), and if \[ x=u^3-3uv^2, \quad y=3u^2v-v^3, \] prove that \begin{align*} 3(u^2+v^2)\left(u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}\right) &= u\frac{\partial w}{\partial u} - v\frac{\partial w}{\partial v}, \\ 9(u^2+v^2)^2\left(\frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial y^2}\right) &= \frac{\partial^2 w}{\partial u^2} + \frac{\partial^2 w}{\partial v^2}. \end{align*}

If \[ u_n = \int_0^{\pi/2} \sin n\theta \cos^{n+1}\theta \operatorname{cosec}\theta \, d\theta, \] find the relation between \(u_n\) and \(u_{n-1}\), and hence evaluate \(u_n\).

If \[ \phi(c-x) = \phi(x), \] show that \[ \int_0^c x^3\phi(x)dx = \frac{3c}{2}\int_0^c x^2\phi(x)dx - \frac{c^3}{4}\int_0^c \phi(x)dx. \]

The function \(\log x\) is defined for real positive values of \(x\) by the equation \[ \log x = \int_1^x \frac{dt}{t}. \] Prove that

1. [(i)] \(\log x^{-1} = -\log x\);
2. [(ii)] if \(k>0\), \[ \frac{\log x}{x^k} \to 0 \text{ as } x\to\infty. \]

Define Simple Harmonic Motion and obtain an expression for the periodic time. Consider the case of a bead sliding on a smooth wire in the form of a cycloid which is fixed with its axis vertical and vertex downwards, and shew that the equation of the hodograph of the motion can be written in the form \[ (x^2+y^2)^2 = (C+4ga)x^2+Cy^2, \] where \(C\) is a constant and \(a\) is the radius of the generating circle of the cycloid.

Obtain the expressions \(v\frac{dv}{ds}\) and \(\frac{v^2}{\rho}\) for the tangential and normal components of the acceleration of a particle which is describing a plane curve. A smooth groove in the form of the catenary \(s=c\tan\psi\) is fixed in a vertical plane with the line \(\psi=0\) horizontal and vertex downwards. A particle of mass \(m\) starts from rest in the position \(\psi=\frac{\pi}{4}\) and moves freely in the groove. Prove that the particle does not leave the groove in the subsequent motion and find the greatest and least values of the reaction between the particle and the groove.

A uniform circular hoop of radius \(r\) in a horizontal plane is spinning about its centre with uniform angular velocity \(4\omega\), and is supported by a smooth horizontal table, when a point \(B\) of the hoop is momentarily brought to rest. Find the change in the angular velocity of the hoop and shew that the distance between the position of \(B\) when brought to rest and its position \(\frac{\theta}{\omega}\) units of time later is \[ 2r\{ \sin^2\theta + \theta^2 - \theta\sin 2\theta \}^{\frac{1}{2}}. \] \subsubsection*{Geometry and Trigonometry} In analytical geometry questions it may be assumed that the axes are rectangular.