Problems

Let \(a\) and \(b\) be real numbers with \(a > 0\). Successive terms in the sequence \(\{f_n\}\) of real numbers are related by \[f_{n+1} = af_n + b\]

1. [(i)] If \(r\) is any real root of the polynomial \(x^3 - ax - b\), prove that \(f_n - r\) has the same sign for all values of \(n\).
2. [(ii)] Now suppose that \(x^3 - ax - b\) has three real roots \(r_1, r_2, r_3\) with \(r_1 < r_2 < r_3\). Prove that \(\{f_n\}\) is an increasing sequence if \(f_1 < r_1\) or \(r_2 < f_1 < r_3\) but is decreasing or constant otherwise.

Explain how complex numbers can be represented on an Argand diagram and demonstrate how to obtain from the positions of \(z_1\) and \(z_2\) in the diagram the positions of \(z_1 + z_2\) and \(z_1 z_2\). Interpret geometrically the inequality \[|z_1 + z_2| \leq |z_1| + |z_2|\] Prove that, if \(|a_i| \leq 2\) for \(i = 1, 2, \ldots n\), then the equation \[a_1 z + a_2 z^2 + \ldots + a_n z^n = 1\] has no solution with \(|z| \leq \frac{1}{4}\)

In a certain card game, a hand consists of \(n\) cards. Each card is either a Pip, a Queen or a Rubbish, and these occur independently of each other with probabilities \(p\), \(q\), \(r\) respectively. Calculate the expected number of Pips. The value of a hand is the product of the number of Pips with the number of Queens. Show that the expected value of a hand is \(n(n-1)pq\). [Hint: \((a + b + c)^m\) is the sum of all terms of the form \(\frac{m!}{r!s!t!}a^r b^s c^t\), where \(r, s, t\) are non-negative integers with \(r + s + t = m\).]

Using the inequality \(\int_a^b [f(x) + \lambda g(x)]^2 dx \geq 0\) for all \(\lambda\), where \(b > a\), show that \[\left(\int_a^b f(x)g(x) dx\right)^2 \leq \left(\int_a^b [f(x)]^2 dx\right)\left(\int_a^b [g(x)]^2 dx\right).\] Show that \(\frac{\pi}{4} < \int_0^{\pi/4} (\cos \theta)^{-1/2} d\theta < \{\frac{\pi}{4} \ln(1+\sqrt{2})\}^{1/2}\)

The function \(B(x, y)\) is defined by the equation, \[B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt,\] for positive \(x\) and \(y\). Show that

1. [(i)] \(B(x, y) = B(y, x)\),
2. [(ii)] \(B(x, y) = B(x+1, y) + B(x, y+1)\),
3. [(iii)] \(xB(x, y+1) = yB(x+1, y)\).

Hence calculate \(B(x, y)\) when both \(x\) and \(y\) are positive integers.

Sketch the curve whose equation in polar coordinates is \[r = 1 - \frac{5}{6} \sin \theta.\] Find the range of real values of \(b\) for which the simultaneous equations \[(x^2 + y^2 + \frac{5}{6}y)^2 = x^2 + y^2\] \[y = b\] have a real solution.

Show that \[x^2 y'' + 2x(x+2)y' + 2(x+1)^2 y = e^{-x}\] can be transformed to a second order linear differential equation with constant coefficients by using the substitution \(y = zx^n\) for a suitable value of \(n\). Find the general solution of the original equation.

The number of messages to be sent by carrier pigeon during a week is a random variable whose distribution is Poisson with parameter \(\mu\). Each bird released with a message has probability \(p\) of evading predatory hawks and arriving at its destination. At the end of each week, all birds which arrived are returned safely to the starting point. Assuming no shortage of pigeons, find the distribution of the net loss of birds per week, and hence write down its mean and variance. Suppose \(\mu = 50\). What is the smallest number of pigeons that needs to be available in order that one may be 99\% confident that all messages to be sent during a week can be carried?

A uniform solid sphere of radius \(r\) and mass \(m\) is drawn slowly and without slipping up a flight of steps by a horizontal force which is always applied to the highest point of the sphere and is always perpendicular to the vertical planes which form the faces of the steps. If each step is of height \(\frac{1}{2}r\) (and of breadth greater than \(r\)), prove that the coefficient of friction between the sphere and the edge of the steps must exceed \(\tan (\pi/6)\), and find the maximum horizontal force throughout the movement.

The annual frisbee-throwing competition between Oxford and Cambridge mathematicians takes place on an infinitely large horizontal plain in perfect weather. According to the rules, each frisbee must be a flat thin disc, so that the air resistance to motion parallel to the plane of the disc is negligible, but resistance to motion perpendicular to the disc is exceedingly large. It must be thrown from ground level with a given speed \(V\) and at a given inclination \(\alpha (>0)\) to the horizontal (i.e the axis of the disc makes an angle \(\alpha\) to the vertical). The angle \(\beta\) between its initial velocity and a horizontal line in the plane of the disc may be chosen freely. As they throw, the competitors give the frisbee a spin about its axis, and the resulting gyroscopic effect is such that the direction of the axis is constant during flight. The longest throw wins. Show that the competitor who chooses \(\beta\) closest to \(\pi/4\) wins.