Let \(C\) be the arc of the parabola \(y = \frac{1}{2}x^2\) between \(x = 0\) and \(x = a\). Calculate the length of \(C\) and the area swept out when \(C\) is rotated about the \(x\)-axis.
(i) Show that if \(|x| < 1\) then \[(1+x)(1+x^2)(1+x^4)(1+x^8)\ldots(1+x^{2^n}) \to \frac{1}{1-x}\] as \(n \to \infty\). (ii) Show that \(\sum_{r=1}^{\infty} \frac{x^{2^r-1}}{1-x^{2^r}}\) converges to \(\frac{x}{1-x}\) if \(|x| < 1\) and to \(\frac{1}{1-x^{-1}}\) if \(|x| > 1\).
Prove that if \(|x| \leq \frac{1}{2}\) then \(x \geq \log (1+x) \geq x-x^2\). By taking logarithms, or otherwise, show that for any positive integer \(k\) \[\left(1-\frac{1}{n^2}\right)\left(1-\frac{2}{n^2}\right)\ldots\left(1-\frac{kn}{n^2}\right) \to e^{-k^2/2}\] as \(n \to \infty\).
Find the straight line which gives the best fit to \(x \cos x\) for \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\); i.e., find constants \(a\), \(b\) such that \[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (x\cos x - ax - b)^2 dx\] is as small as possible.
A point moves in the plane and its position in polar co-ordinates \((r(t), \theta(t))\) is given by \[\frac{d^2r}{dt^2} - r\left(\frac{d\theta}{dt}\right)^2 = -f(r),\] \[r^2\frac{d\theta}{dt} = h,\] where \(h\) is a constant and \(f\) a given function. Show that if \(u = 1/r\), these equations can be written in the form \[\frac{d^2u}{d\theta^2} + u = \frac{1}{h^2u^2}f\left(\frac{1}{u}\right). \tag{*}\] Solve \((*)\) in the cases (i) \(f(r) = 1/r^2\), (ii) \(f(r) = 1/r^3\).
A home-made roulette wheel is divided into 16 sections which are coloured red and black alternately and labelled with the numbers between 1 and 16. The red sections are numbered consecutively with the odd numbers in a clockwise direction, and the black sections are numbered in the opposite direction with the even numbers starting with the number 2 between the numbers 15 and 1. Unfortunately, the wheel is not true, and the probability that the ball lands in the quarter between the numbers 15 and 16 is twice that of each of the adjacent quarters and four times that of the opposite quarter. The probability is uniform within each quarter. Successive rolls are independent. What is the probability that the sum of two consecutive rolls is 28? Given that the sum of two consecutive rolls is 28, what is the probability that the ball landed in a black section both times? Find particular values of \(n\) such that, given that the sum of two successive rolls is \(n\), the probability that the ball has landed in black sections both times is (i) 0, (ii) \(\frac{1}{2}\), (iii) 1.
Balls are drawn successively at random without replacement from a box containing \(R\) red balls and \(B\) blue ones. Find the probability that the number of balls to be drawn in order to obtain \(r\) red ones (\(r \leq R\)) should be \(n\).
Let \(X_1, X_2, \ldots, X_n\) be independent random variables each uniformly distributed on the interval \((0,1)\). Find for \(0 < u < v < 1\) the probability of the event that the smallest of them is between 0 and \(u\) and the largest is between \(u\) and \(v\).
The following is from an advertisement for `X' beer. We've tried our famous `X' Taste Test on twenty beer experts, pouring three glasses, one from the tap, one from the can, and one from the bottle. And then we've asked which is which. Result? No one identified the three correctly. Why? Because all three glasses have the same famous `X' Taste. What confidence can you have in the reasoning in this advertisement?
A large horizontal disc has a toy gun mounted on it in such a way that the barrel of the gun lies in a vertical plane through the centre of the disc and the muzzle of the gun is in the plane of the disc and at a distance \(a\) from its centre. The gun is directed upwards at an angle \(\alpha\) to the horizontal and towards the axis of the disc. The disc is set rotating with angular velocity \(\omega\) about its axis and the gun fires a projectile with velocity \(V\) relative to the gun. Allowing for gravity but ignoring air-resistance, find the value of \(V\) which minimises the distance between the axis of the disc and the point at which the projectile strikes the disc.