If \(y = e^{-x}\sin(x\sqrt{3})\), prove that \begin{align} \frac{d^n y}{dx^n} = (-2)^n e^{-x} \sin(x\sqrt{3} - \frac{1}{3}n\pi) \end{align} Hence, or otherwise, show that \begin{align} y = \frac{\sqrt{3}}{2}\sum_{n=0}^{\infty}\left(\frac{(2x)^{3n+1}}{(3n+1)!} - \frac{(2x)^{3n+2}}{(3n+2)!}\right) \end{align}
Using the fact that \begin{align} \lim_{n\to\infty}\left(\frac{b-a}{n}\sum_{m=1}^{n}f(a+m[b-a]/n)\right) = \int_{a}^{b}f(x)dx \end{align} show that \begin{align} \lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+n}\right) = \ln 2 \end{align} Evaluate \begin{align} \lim_{n\to\infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+...+\frac{n}{n^2+n^2}\right) \end{align}
Suppose that \(f(n)\) is a polynomial with rational coefficients of degree \(k > 0\) in \(n\) where \(n\) is an integer. Show that the function \begin{align} g(n) = f(n) - f(n-1) \end{align} is a polynomial of degree \(k-1\). Show also that if \(f(n)\) has the form \(g(n)\cdot g(n+1)\cdot g(n+2)\) (for some polynomial \(g(n)\)), then \(g(n)\) is divisible by \(g(n) \cdot g(n+1)\). Evaluate the sum \begin{align} f(n) = \sum_{r=1}^{n} \frac{1}{r^3(r+1)^3(r^2+r+1)} \end{align} and hence show that \(f(n)\) is a perfect cube for all values of \(n\).
Using the substitution \(x = e^t\) or otherwise solve \begin{align} x^2\frac{d^2y}{dx^2} - 4x\frac{dy}{dx} + 6y = 6\ln x - 5 \text{ for } x > 0 \end{align} given \(y(1) = 0\) and \(y(e) = 2\).
The mountain villages \(A\), \(B\), \(C\), \(D\) lie at the vertices of a tetrahedron, and each pair of villages is joined by a road. After a snowfall the probability that any road is blocked is \(p\), and is independent of the conditions on any other road. Find the probability that it is possible to travel from any village to any other village by some route after snowfall. In the case \(p = \frac{1}{2}\) show that this probability is \(\frac{19}{32}\).
\(A\) and \(B\) play the following game. \(A\) throws two unbiased four-sided dice (each has the numbers 1 to 4 on its sides), and notes the total \(Y\). \(B\) tries to guess this number, and guesses \(X\). If \(B\) guesses correctly he wins \(X^2\) pounds, and if he is wrong he loses \(\frac{1}{2}X\) pounds. (a) Show that \(B\)s average gain if he always guesses 8 is \(\frac{1}{4}\). (b) He decides that he will always guess the same value of \(X\). Which value of \(X\) would you advise him to choose, and what is his average gain in this case?
The lifetime in days, \(X\), of a safety component in a chemical plant is given by the negative exponential distribution \begin{align} P(X \leq t) = 1 - e^{-\lambda t} \text{ for } t \geq 0 \end{align} Find the mean lifetime of the component. The component is checked at 8 o'clock every morning, and if faulty is replaced immediately. Let \(Y\) be the length of time, in days, between the component failing and being replaced. Show that the probability that the component fails on the \(n\)th day and is replaced within \(24y\) hours, where \(0 \leq y \leq 1\), is \((e^{\lambda y} - 1)e^{-\lambda n}\) for \(n = 1, 2, ...\). Hence prove that \begin{align} P(Y \leq y) = \frac{e^{\lambda y} - 1}{e^{\lambda} - 1} \end{align} and calculate the mean of \(Y\).
Let \(X_1, ..., X_m\) be independent normally distributed random variables, with mean \(\mu\) and variance \(\sigma^2\). Let \(X > 0\), and let \(Y\) be the number of observations falling in the range \((a-X, \mu+X)\). Give an expression for \(P(Y = r)\) for \(r = 0, 1, ..., m\). If \(\alpha = \frac{1}{2}\) and \(m = 10\), what is \(P(Y \leq 2)\)? (You may leave your answer in a form suitable for calculation.)
A uniform rod \(AB\) of length \(l\) lies on a rough horizontal table. A string is attached to the rod at \(B\) and is pulled in a horizontal direction perpendicular to the rod. Show that, as the tension in the string is gradually increased, the rod begins to turn about a point whose distance from \(A\) is \(l(1 - 1/\sqrt{3})\), and find the value of the tension when that occurs, in terms of the weight of the rod and the coefficient of friction between the rod and the table.
An inclined plane makes an angle \(\alpha\) with the horizontal. A small, perfectly elastic sphere is projected up the plane at an angle of elevation \(\beta\) relative to the plane. Its second bounce occurs at the point of projection. Show that \(2 \tan \alpha \tan \beta = 1\).