Problems

Concorde flies the distance \(d\) from London to New York in an average time \(t_1\) and makes the return journey in an average time \(t_2\) where \(t_2 < t_1\). Assuming that the earth is flat and that Concorde flies at a uniform speed \(V\) in still air, find the speed of the prevailing wind and the angle it makes with the straight line joining New York to London.

Particles of mud are thrown off the tyres of the wheels of a cart travelling at constant speed \(V\). Neglecting air resistance, show that a particle which leaves the ascending part of a tyre at a point above the hub will be thrown clear of the wheel provided its height above the hub at the instant when it leaves the tyre is greater than \(a^2/V^2\), where \(a\) is the radius of the tyre.

Show that the second order differential equation \[\frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = f(x),\] with \(b\) and \(c\) constants, can be written as the pair of equations \[\frac{dp}{dx} - m_1 p = f(x),\] \[\frac{dy}{dx} - m_2 y = p,\] where \(m_1\), \(m_2\) are constants to be determined. Hence, or otherwise, find the general solution of \[\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + y = e^{-x}\]

Define \(f_n(x) = n^2 x (1-x)e^{-nx}\) for \(0 \leq x \leq 1\), \(n = 0, 1, 2 \ldots\). Show that, for each \(x\) such that \(0 < x \leq 1\), \[\lim_{n \to \infty} f_n(x) = 0.\] Show also that \[\lim_{n \to \infty} \int_0^1 f_n(x) dx \neq \int_0^1 \lim_{n \to \infty} f_n(x) dx.\] [You may assume that \(\lim_{n \to \infty} nb^{-n} = 0\) for \(b > 1\).]

Show, by induction or otherwise, that, if \(n\) consecutive integers have arithmetic mean \(m\), then the sum of their cubes is \[mn\{m^2 + \frac{1}{4}(n^2-1)\}\] Find an expression in terms of \(m\) and \(n\) for the sum of their squares. Let \(s_1\) be the sum of \(n\) consecutive integers, \(s_2\) the sum of their squares and \(s_3\) the sum of their cubes. Prove that \[9s_2^2 \geq 8 s_1 s_3\]

Let \(G\) be the set of all \(2 \times 2\) real matrices of the form \[\begin{pmatrix} 1 & 0 \\ a & h \end{pmatrix}\] with \(h \neq 0\). Let \(A\) and \(H\) respectively be the set of all real matrices of the form \(\begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix}\) and the set of all real matrices of the form \(\begin{pmatrix} 1 & 0 \\ 0 & h \end{pmatrix}\) with \(h \neq 0\). Show that \(G\) is a group with respect to matrix multiplication, and that \(A\) and \(H\) are subgroups of \(G\). [You may assume that matrix multiplication is associative.]

1. [(i)] Show that every element of \(G\) may be written as a product of an element of \(A\) and an element of \(H\).
2. [(ii)] Prove that if \(X \in A\) and \(Y \in G\) then \(Y^{-1}XY \in A\). Prove also that if \(X_1, X_2 \in A\) and neither \(X_1\) nor \(X_2\) is the identity matrix, then there exists a matrix \(Z \in H\) such that \(Z^{-1}X_1Z = X_2\).

Using (i) and (ii), or otherwise, show that the only subgroups of \(G\) containing \(H\) are \(G\) and \(H\).

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 \(\frac{\pi}{4} < \int_0^{\pi/4} (\cos \theta)^{-1/2} d\theta < \{\frac{\pi}{4} \ln(1+\sqrt{2})\}^{1/2}\)