In San Theodoros execution is by firing squad at dusk. Executions take place at any time between 6 and 7 pm with equal probability, and as darkness falls the aim of the soldiers worsens at a steady rate, so that at 6 pm their aim is perfectly true, at 6.30 they miss their target with probability \(\frac{1}{2}\) and by 7 pm they always miss. General Tapioca, ruler of San Theodoros, a liberal, has ordered on humanitarian grounds that on exactly half the executions the firing squad shall use blank rounds. Tintin, a reporter, is sentenced to die by firing squad but survives. What is the probability that he faced live rounds?
Solution: \begin{align*} && \mathbb{P}(\text{live rounds} | \text{survives}) &= \frac{\mathbb{P}(\text{live rounds and survives})}{\mathbb{P}(\text{survives})} \\ &&&= \frac{\mathbb{P}(\text{choose live rounds})\mathbb{P}(\text{missed}|\text{live rounds})}{\mathbb{P}(\text{choose live rounds})\mathbb{P}(\text{missed}|\text{live rounds})+\mathbb{P}(\text{choose blank rounds})\mathbb{P}(\text{missed}|\text{blank rounds})} \\ &&&= \frac{\frac12 \cdot \frac12 }{\frac12 \cdot \frac12 + \frac12 \cdot 1} \\ &&&= \frac{1}{3} \end{align*} The probability that he is missed is \(\frac12\) since it is equally likely they shoot with \(p\) or \(1-p\) for each \(p\) over the hour.
A shell is fired from a gun with a muzzle velocity \(V\) and an elevation of \(45^{\circ}\) to the horizontal. At the top of its flight the shell splits into two equal fragments which separate with a relative velocity of magnitude \(\sqrt{2}V\) and elevation \(\alpha\) in the plane of the trajectory. Show that the range of one fragment is \begin{equation*} \frac{V^2}{2g}[1 + (1 + \cos\alpha)(\sin\alpha + \sqrt{\sin^2\alpha + 1})], \end{equation*} and find the range of the other.
A particle of mass \(m\) moves along a straight line in a resistive medium. It experiences a retarding force of magnitude \(\lambda v^3 + kv\), where \(v\) is its velocity and \(\lambda\) and \(k\) are positive constants. Given that the initial velocity of the particle is \(w\), find \(v\) as a function of time. Find \(v\) as a function of \(s\), the distance travelled, and show that \(s\) never exceeds \begin{equation*} \frac{m}{\sqrt{\lambda k}}\tan^{-1}\left(w\sqrt{\frac{\lambda}{k}}\right) \end{equation*}
A uniform rod of mass \(m\) and length \(4a\) can rotate freely in a smooth horizontal plane about its midpoint. Initially the rod is at rest. A particle of mass \(m\) travelling in the plane with velocity \(u\) at right angles to the rod collides perfectly elastically with the rod at a distance \(a\) from the centre. Find the velocity of the particle and angular velocity of the rod after collision. Do the particle and the rod undergo a subsequent collision?
If \(y = \cos(m \sin^{-1} x)\), show that \begin{equation*} (1 - x^2)\left(\frac{dy}{dx}\right)^2 - m^2(1 - y^2) = 0 \end{equation*} \begin{equation*} (1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + m^2y = 0. \end{equation*} Using Leibniz' theorem, or otherwise, show that, for integer \(n \geq 0\), \begin{equation*} (1 - x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n + 1)x\frac{d^{n+1}y}{dx^{n+1}} + (m^2 - n^2)\frac{d^ny}{dx^n} = 0. \end{equation*} By considering the Taylor series of \(\cos(m \sin^{-1} x)\) about \(x = 0\), show that \begin{equation*} \cos mx - \cos(m \sin^{-1} x) = \frac{m^2x^4}{3!} + \text{higher order terms.} \end{equation*}
By evaluating the integral, sketch \begin{equation*} f(x) = \int_0^{\pi} \frac{\sin\theta\, d\theta}{(1 - 2x\cos\theta + x^2)^{1/2}} \end{equation*}
The sequence \(u_0, u_1, u_2, \ldots\) is defined by \(u_0 = 1\), \(u_1 = 1\), and \(u_{n+1} = u_n + u_{n-1}\) for \(n \geq 1\). Prove that \begin{equation*} u_{n+2}^2 + u_{n-1}^2 = 2(u_{n+1}^2 + u_n^2). \end{equation*} Using this result and induction, or otherwise, show that \begin{equation*} u_{2n} = u_n^2 + u_{n-1}^2 \quad \text{and} \end{equation*} \begin{equation*} u_{2n+1} = u_{n+1}^2 - u_{n-1}^2 \end{equation*}
If \(\alpha, \beta, \gamma\) are the roots of the equation \begin{equation*} x^3 - s_1x^2 + s_2x - s_3 = 0, \end{equation*} show that either \(\alpha\beta\gamma = 0\), or \(\frac{s_2}{s_3} = \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). Deduce that the equation \(x^3 - ax^2 + 7bx - 2b = 0\) cannot have three strictly positive integer roots. Find three pairs of numbers \(a, b\) such that for each pair, the equation \(x^3 - ax^2 + bx - b = 0\) has three strictly positive integer roots.
Let \(a\) be a non-zero real number and define a binary operation on the set of real numbers by \begin{equation*} x * y = x + y + axy. \end{equation*} Show that the operation \(*\) is associative. Prove that \(x * y = -1/a\) if and only if \(x = -1/a\) or \(y = -1/a\) Let \(G\) be the set of all real numbers except \(-1/a\). Show that \((G, *)\) is a group.