\(\Sigma, \Sigma', \Sigma''\) are three conics each touching two given straight lines. The other pair of tangents common to \(\Sigma', \Sigma''\) meet in \(A\); the other pair of tangents common to \(\Sigma'', \Sigma\) meet in \(A'\); the other pair of tangents common to \(\Sigma, \Sigma'\) meet in \(A''\). Prove that \(A, A', A''\) are collinear.
Show that two parabolas can be drawn to touch the sides of a triangle \(ABC\) and to pass through an assigned point \(D\). If these two parabolas intersect at right angles at \(D\), show that \(A, B, C, D\) are concyclic.
Find expressions for the roots of the equation \[ z^6+z^5+z^4+z^3+z^2+z+1=0, \] and mark these roots on an Argand diagram. If \(\alpha=2\pi/7\), prove that \(\cos 3\alpha + \cos 2\alpha + \cos \alpha = -\frac{1}{2}\). Find the roots of the cubic equation \(\xi^3+\xi^2-2\xi-1=0\).
Find the equation whose roots are the squares of the roots of the cubic equation \(a_0x^3+a_1x^2+a_2x+a_3=0\). If the values of the coefficients \(a_0, a_1, a_2, a_3\) are given numerically, and the roots \(\alpha, \beta, \gamma\) of the equation are real, and such that \(|\alpha| > |\beta| > |\gamma|\), show that the continued repetition of this process will yield an approximate value of \(|\alpha|\). Suggest a method of finding the other roots.
Evaluate \(\int_0^\pi \sin^m x dx\) in the cases where \(m\) is an odd or an even positive integer. Show that \(M/n > \pi > M/(n+\frac{1}{4})\), where \(n\) is a positive integer and \[ M = \left\{ \frac{2^{2n}(n!)^2}{(2n)!} \right\}^2. \]
If \(\phi(x,t)\) is a function of the variables \(x, t\), and is expressed as a function \(\psi(u,v)\) by means of the relations \(u=ct+x, v=ct-x\), where \(c\) is a constant, show that \begin{align*} \text{(i)} \quad & \frac{\partial\phi}{\partial x}\frac{\partial\phi}{\partial t} = c^2\left\{ \left(\frac{\partial\psi}{\partial u}\right)^2 - \left(\frac{\partial\psi}{\partial v}\right)^2 \right\}, \\ \text{(ii)} \quad & \frac{\partial^2\phi}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2} = 4\frac{\partial^2\psi}{\partial u\partial v}. \end{align*} Show that the general solution of \(c^2\frac{\partial^2\phi}{\partial x^2} = \frac{\partial^2\phi}{\partial t^2}\) is \(f(ct+x)+g(ct-x)\), where \(f(z)\) and \(g(z)\) are arbitrary functions of \(z\).
A pile consists of \(M\) red cards and \(N\) black cards, all distinguishable from one another. Write down the number of ways in which a hand of \(n\) cards can be made up, of which \(r\) are red and \(n-r\) black. Show that, if \(n\) is given, and \(M\) and \(N\) are large compared with \(n\), the numbers of ways corresponding to \(0, 1, 2, \dots n\) red cards are approximately proportional to the terms of the binomial expansion of \((q+p)^n\), where \(p=M/(M+N)\), \(q=N/(M+N)\). If this ``binomial distribution'' holds exactly, show that the average number of red cards for all the different possible ways will be \(np\).
A shell explodes at a vertical height \(h\) above a plane which is inclined at an angle \(\beta\) to the horizontal, and the initial speed \(V\) of the fragments is the same in all directions. Show that the distance between the highest and the lowest points of the plane that can be reached by the fragments is \[ \frac{2V^2 \sec\beta}{g} \left( \sec^2\beta + \frac{2gh}{V^2} \right)^\frac{1}{2}. \] [If any formula relating to the motion of a projectile is quoted, it should be proved.]
A uniform rod \(AB\) of mass \(m\) and length \(2a\) is suspended by light inextensible strings \(AC\) and \(BD\), each of length \(l\), from fixed points \(C\) and \(D\) which are at the same height and \(2a\) apart. Initially the rod is horizontal and the strings vertical. The rod is then twisted through an angle \(\theta\) and held in equilibrium in this position by a couple consisting of two horizontal forces applied to the rod. Calculate (i) the moment of the couple, (ii) the vertical displacement of the centre of the rod. The rod is allowed to oscillate freely about its equilibrium position, the motion consisting of a small angular rotation about a vertical axis together with a vertical displacement of the centre of the rod. Determine the frequency of the oscillations.
A uniform rod \(AB\), of mass \(m\) and length \(2l\), rests on a smooth horizontal table, to which it is freely pivoted at \(A\). A particle, of mass \(m\), moving horizontally with velocity \(V\), strikes the rod normally at \(B\) and adheres to it. Find the kinetic energy \(T_1\) of the system after impact. Find also the impulsive reaction at \(A\). If the rod \(AB\) had been completely free to move on the table, find what the kinetic energy \(T_2\) would have been, and show that \(T_2/T_1 = 16/15\).