If \(0 \geq a_i \geq -1\) for all \(i\) show that \(\displaystyle \prod_{r=1}^{n}(1 + a_r) \geq 1 + \sum_{r=1}^{n} a_r\). Show that the inequality is also true if \(a_i \geq 0\) for all \(i\). Is it true if \(a_i > -1\) for all \(i\)?
A curve is given in polar coordinates by \(r(\theta)\) for \(0 \leq \theta \leq \pi\), and it is rotated about the axis \(\theta = 0\) to form a solid of revolution. Derive a formula for the surface area of the solid. Calculate the area of the surface so formed in the case \(r(\theta) = ae^{k\theta}\).
A set of functions \(y_n (n = 0, 1, 2, ...)\) is defined for \(|x| \leq 1\) by \[y_n(x) = \cos(n \cos^{-1} x).\]
Show that the graph of \(y = ax^3 + bx^2 + cx + d\) (\(a \neq 0\)) can be transformed into precisely one of the forms \begin{align*} (a)&~ y = x^3 + x \\ (b)&~ y = x^3 \\ (c)&~ y = x^3 - x \end{align*} by means of a finite sequence of plane transformations of the types \((x, y)\mapsto(\lambda x, \mu y)\) and \((x, y)\mapsto(x + \alpha, y + \beta)\).
Suppose that \(u_n\) satisfies the recurrence relation \[u_{n+2} = \alpha u_{n+1} + \beta u_n,\] and \(v_n\) satisfies the relation \[v_{n+2} = (\alpha^2 + 2\beta)v_{n+1} - \beta^2 v_n \quad (n \geq 0)\] with \(v_0 = u_0\), \(v_1 = u_2\). Show that \(v_n = u_{2n}\) for all positive integers \(n\). Hence or otherwise show that, if \(v_0 = v_1 = 1\) and \(v_{n+2} = 3v_{n+1} - v_n\), then \(v_n = 2^n \quad (n\geq 2)\).
The equation \(x^3 + ax^2 + bx + c\) (\(c \neq 0\)) has three distinct roots which are in geometric progression and whose reciprocals may be rearranged to form an arithmetic progression. Find \(b\) and \(c\) in terms of \(a\).
Mr and Mrs Pinkeye have three babies: Albert, Bertha and Charles, who sleep in separate rooms. Albert wakes up during the night twice as often as Bertha and Bertha wakes up twice as often as Charles. Albert cries on 20\% of the occasions when he wakes up, Bertha cries on 50\% of the occasions when she wakes up and Charles on 80\%. For the purposes of this question it may be assumed that they wake their parents up but not each other. The parents wake as soon as a child starts crying. To which child should Mr Pinkeye go first and what is the probability that, if he does so, he has gone to the right child?
The manufacturers claim that 4 people out of 5 cannot tell `Milkoflave' from cows' milk. The Milk Marketing Board claims on the contrary that 4 out of 5 people can tell the difference. A consumers' magazine asks you to decide between the two claims. Explain how you would try to do so. Bear in mind that the magazine will wish to know in advance how many tests you propose to make and that tests cost money. [You are not asked to find a best procedure but to suggest a reasonable procedure.]
Let \(X_1, X_2, ...\) be independent random variables uniformly distributed on \([1, 2]\). Show that \[\Pr(a < (X_1X_2 ... X_n)^{1/n} < b) \to 1 \text{ as } n \to \infty\] if and only if \(a < 4/e < b\). You may use any results from probability theory that you know.
Show that the integral \[I_n = \int_{-\infty}^{+\infty} x^{2n}e^{-x^2}dx\] (where \(n\) is a positive integer) obeys the recurrence relation \[I_{n+1} = (n + \tfrac{1}{2})I_n\] By expanding \(\cos ax\) as a power series in \(x\), or otherwise, show that \[\int_{-\infty}^{+\infty} e^{-x^2}\cos ax \, dx = \pi^{1/2}e^{-a^2/4}.\] [You may assume that you may integrate the infinite series term by term.]