Problems

Prove that, if \(0 < x < 1\), \[\pi < \frac{\sin\pi x}{x(1-x)} \leq 4.\]

Suppose, if possible, that \(\pi^2 = a/b\), where \(a\) and \(b\) are positive integers. Let \[f(x) = \frac{x^n(1-x)^n}{n!},\] \[G(x) = b\pi \sum_{r=0}^{n} (-1)^r\pi^{2n-2r}f^{(2r)}(x),\] where \(f^{(2r)}(x)\) denotes the \((2r)\)th derivative of \(f(x)\), and \(n\) is a positive integer. Prove that \[\frac{d}{dx}\{G'(x)\sin\pi x - \pi G(x)\cos\pi x\} = \pi^2 a^n \sin\pi x \cdot f(x),\] and deduce that \[\pi\int_0^1 a^n\sin\pi x \cdot f(x)dx = G(0) + G(1).\] Prove that the right-hand side of this equation is an integer. Show also that, by choice of \(n\) sufficiently large, the left-hand side can be made to lie strictly between 0 and 1. Establish a contradiction to the original supposition that \(\pi^2\) is rational.

\(X\) and \(Y\) are discrete valued random variables, and \[\text{Pr}(X = x, Y = y) = p(x, y), \quad \text{say}.\] The expectation of \(X\) conditional on the value of \(Y\) being \(y\) is defined as \(\mu(y)\), where \[\mu(y) = E(X|Y = y) = \sum_x x \frac{p(x, y)}{b(y)},\] and \[b(y) = \text{Pr}(Y = y),\] so that \[b(y) = \sum_x p(x, y).\] Show that \(E(X) = \sum\mu(y)b(y)\). By taking \(Z = X^2\), find an expression for the variance of \(X\) in terms of \(E(X|Y = y)\) and \(E(X^2|Y = y)\). An ornithologist observes that the number of eggs laid by a sparrow in a nest is distributed approximately as a Poisson random variable with mean \(\lambda\). He suspects that any egg has the same probability \(p\) of hatching, and that they are independent with respect to hatching. Denote by \(X\) the number of fledgelings from a nest, and denote by \(Y\) the number of eggs laid in that nest. Find expressions for \[E(X|Y = y) \quad \text{and} \quad E(X^2|Y = y)\] and hence find the (unconditional) mean and variance of \(X\). A second ornithologist contests that the eggs in a nest are not independent with respect to hatching. He suspects that either, with probability \(\pi\), the whole clutch of eggs hatches, or, with the probability \(1-\pi\), none of the clutch hatches. What are the mean and variance of \(X\) with this model? If you looked at a large sample of sparrows' nests, and found that the mean number of fledgelings per nest was 4, and the sample variance was 12, which ornithologist would you take to be more expert?

A survey is conducted among \(n\) people in order to examine whether there is any association between smoking and lung cancer. The following data are obtained

Here \(n_{11}\) is the number of people who both smoke and have cancer, with \(n_{12}\), \(n_{21}\), \(n_{22}\) defined in the obvious way. It is desired to compress the table \((n_{ij})\) into a single real number \(\delta = \delta(n_{ij})\) which indicates the association between smoking and cancer. Two medical experts are consulted, on the choice of the function \(\delta(n_{ij})\). The first says that \(\delta(n_{ij})\) should be determined uniquely by the pair \((n_{11}/n_{1.}, n_{21}/n_{2.})\), and the second says that \(\delta(n_{ij})\) should be determined uniquely by the pair \((n_{11}/n_{.1}, n_{12}/n_{.2})\). Show that if we choose the function \(\delta(n_{ij})\) to satisfy both the experts, and if \((a_{ij})\), \((b_{ij})\) are two tables for which \[a_{11}a_{22}/a_{12}a_{21} = b_{11}b_{22}/b_{12}b_{21}\] then \(\delta(a_{ij}) = \delta(b_{ij})\). If each of the \(n's\) is fairly large, describe the association between smoking and lung-cancer if

1. \((n_{11}n_{22})/(n_{12}n_{21}) \simeq 1\),
2. \((n_{11}n_{22})/(n_{12}n_{21}) \gg 1\).

Suppose a profit-maximising firm produces a perishable and homogeneous good from which the net revenue per unit sold becomes negative if output exceeds a certain level. If a sales tax of \(t\) pence is imposed on each unit, will the firm produce more or less? Justify your answer.

By vector methods, or otherwise, show that the medians of a triangle are concurrent (at the 'centroid'). Each vertex of a tetrahedron is joined to the centroid of the opposite face. Show that the resulting four lines are concurrent. At each vertex of a tetrahedron a force acts which is towards and perpendicular to the opposite face, and has magnitude proportional to the area of that face. Show that the system of forces is in equilibrium.

Four equal stretched strings \(X_0X_1\), \(X_1X_2\), \(X_2X_3\), \(X_3X_4\), each of natural length \(l\), and modulus of elasticity \(\lambda lm\), lie in a straight line on a smooth horizontal table. The ends \(X_0\), \(X_4\) are fixed, and masses \(m\), \(nm\), \(m\) are attached to the points \(X_1\), \(X_2\), \(X_3\), respectively. The system performs oscillations along the line of the springs. Determine the equations of motion for the masses in terms of their displacements from their equilibrium positions. Show that if all the masses oscillate with the same period \(2\pi/p\), then in order to have a non-trivial solution, either \(p^2 = 2\lambda\), or \(p^2\) satisfies the equation \[(2\lambda - np^2)(2\lambda - p^2) = 2\lambda^2.\]

An axle with perfectly smooth bearings carries a gear-wheel with radius \(a_1\), and the total moment of inertia of the system is \(I_1\). A second similar system is described by parameters \(a_2\) and \(I_2\). The axles are mounted parallel in a rigid piece of machinery, their separation being a little greater than \(a_1 + a_2\), and are rotating with angular velocities \(\omega_1\) and \(\omega_2\). The separation is then reduced, bringing the gear wheels into mesh. Find the loss of energy resulting from the impact. Find also the impulsive couple which must be exerted on the machine as a whole to keep it stationary.

A satellite rotates in a circular orbit around the earth with a period of one day. Find the radius of its orbit. Three such satellites rotate in the earth's equatorial plane. If the satellites lie at the corner of an equilateral triangle, find the largest angle of latitude such that all places on earth at this latitude are visible from at least one of the satellites. [\(g = 9.8\) m/sec\(^2\); radius of the earth \(= 6.4 \times 10^6\) m.]

P and Q are two points on a semi-circle whose diameter is AB; AP and BQ meet in M, AQ and BP meet in N. Prove that MN is perpendicular to AB, and that the circle on MN as diameter cuts the semi-circle orthogonally.