Problems

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.

1. [(i)] Evaluate \(\displaystyle \int_0^1 \sin^{-1}x\, dx\).
2. [(ii)] For \(y = 4\tan^{-1}(e^x)\), evaluate the integral \begin{equation*} \int_{-\infty}^{\infty}\left[\frac{1}{2}\left(\frac{dy}{dx}\right)^2 + (1 - \cos y)\right]dx. \end{equation*} [Take \(-\pi/2 < \tan^{-1}x < \pi/2\).]

Find all the stationary values of the function \(y(x)\) defined by \begin{equation*} \frac{ay + b}{cy + d} = \sin^2x + 2\cos x + 1 \end{equation*} where \(ad \neq bc\), \(a \neq 3c\) and \(a \neq -c\). Assume that \(a/c > 3\) or \(a/c < -1\) and show that \(y(x)\) is then a bounded function for all \(x\).

Sketch the curve given by the equations \begin{align*} x &= a(\theta + \sin\theta)\\ y &= a(1 - \sin\theta), \quad a > 0. \end{align*} Find the area under the curve between two successive points where \(y = 0\).

Juggins enjoys playing the following game: he throws a die repeatedly. The game stops when he throws a 1; alternatively he can stop it after any throw. His score is the value of his last throw. How should Juggins play to maximise his expected score?

A die is thrown until an even number appears. What is the expected value of the sum of all the scores?

The ``logistic'' difference equation is \begin{equation*} x_{n+1} = ax_n(1 - x_n), \end{equation*} where \(1 < a < 4\). Show that if either \(x_1 < 0\) or \(x_1 > 1\), then \(x_n \to -\infty\) as \(n \to \infty\), but if \(0 < x_1 < 1\), then \(0 < x_n < 1\) for all \(n\). Show further that if \(x_n\) tends to a finite limit \(x\) as \(n \to \infty\), then \(x = 0\) or \(x = 1 - 1/a\). By writing \(x_n = x + \epsilon_n\), and considering \(\epsilon_{n+1}/\epsilon_n\), or otherwise, show that sequences \(x_n\) with \(x_1\) sufficiently close to \(1 - 1/a\) get steadily closer to \(1 - 1/a\) provided \(a < 3\).

Find the largest volume which can be attained by a circular cone inscribed in a sphere of radius \(R\).

A bifilar pendulum consists of two point masses at the ends of a light horizontal rigid rod of length \(2L\). This rod is suspended symmetrically by two thin vertical threads of length \(l\), separation \(2d < 2L\). Show that the frequency of small oscillations in which the system rotates about a vertical axis through the centre of the rod is smaller than that when the whole system performs small oscillations perpendicular to its equilibrium plane. [Vertical displacements may be neglected.]

A fine chain of mass \(\rho\) per unit length has length \(l\) and is suspended from one end so that it hangs vertically at rest with the lower end just touching a horizontal plane. The chain is released so that it falls freely and collapses inelastically onto the plane. Find as a function of time the force exerted on the plane.