State Simpson's rule for the numerical evaluation of \(\int_0^a f(x) \, dx\), and show that it is exact when \(f(x)\) is a cubic polynomial. By applications of this rule using three ordinates to $$\int_0^{1/3} \frac{dx}{\sqrt{1-x^2}} \quad \text{and} \quad \int_0^1 \frac{dx}{\sqrt{1-x^2}}$$ find expressions approximating to \(\frac{1}{4}\pi\) and \(\frac{1}{2}\pi\). Which result would you expect to yield the closer approximation, and why?
The real pairs \((x,y)\) and \((u,v)\) are related by $$x + iy = k(u + iv)^2 \qquad (k \text{ real}).$$ Identify the curves in the \((x,y)\) plane which correspond to \(u = \text{constant}\) and \(v = \text{constant}\), and show that they intersect at right angles.
A closed curve is given in polar coordinates by the equation $$r = a(1 - \cos \theta).$$ Show that the tangent at the point \(\theta\) is inclined at an angle \(\psi = \frac{1}{2}\theta\) to the axis \(\theta = 0\). Find the radius of curvature at the point \(\theta\).
The polynomial \(T_n(x)\), where \(n\) is a non-negative integer, satisfies $$(1-x^2) \frac{d^2 T_n}{dx^2} - x \frac{dT_n}{dx} + n^2 T_n = 0;$$ $$T_0(1) = 1; \quad T_n(x) = (-1)^n T_n(-x).$$ By substituting \(x = \cos \theta\) and solving the transformed equation, obtain \(T_n(x)\) in simple form as a function of \(\theta\) and hence show that $$T_0(x) = 1, \quad T_1(x) = x,$$ and that $$T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x) \quad \text{for } n = 1, 2, 3, \ldots$$
The position vector, \(\mathbf{r}(t)\), of a moving point \(P\) relative to a fixed origin satisfies the equation $$\ddot{\mathbf{r}} = \mathbf{k} \times \mathbf{r},$$ where \(\mathbf{k}\) is a constant vector and \(\dot{\mathbf{r}} = d\mathbf{r}/dt\). Show that the locus of \(P\) is a circle. Describe the motion of \(P\) when \(\mathbf{r}\) satisfies the equation $$\ddot{\mathbf{r}} = \mathbf{k} \times \dot{\mathbf{r}}.$$
Let \(u\) be a function of \(x\) and \(y\). If \(x\) and \(y\) are related by \(u(x,y) = \text{constant}\), prove that $$\frac{dy}{dx} = -\frac{\partial u / \partial x}{\partial u / \partial y}.$$ Deduce that the partial differential equation $$2yu \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$$ has solutions given by $$u = f(x - y^2 u),$$ where \(f\) is an arbitrary function. Find the solution such that \(u = x\) when \(y = 0\).
Prove that, if \(g(x) > 0\), then $$\int_a^b f(x)g(x) \, dx \leq \max_{a \leq x \leq b} \{f(x)\} \int_a^b g(x) \, dx$$ and hence that $$\left| \int_a^b f(x)g(x) \, dx \right| \leq \max_{a \leq x \leq b} \{|f(x)|\} \int_a^b g(x) \, dx.$$ Give an example of functions \(f(x)\) and \(g(x)\) for which $$\left| \int_a^b f(x)g(x) \, dx \right| > \max_{a \leq x \leq b} \{|f(x)|\} \int_a^b g(x) \, dx.$$ The function \(h(x)\) vanishes at \(x = 0\), and possesses a first derivative. Show that $$\int_0^a h(x) \, dx = \int_0^a (a-x)h'(x) \, dx,$$ and deduce that $$\left| \int_0^a h(x) \, dx \right| \leq \frac{1}{2}a^2 M,$$ where $$M = \max_{0 \leq x \leq a} \{|h'(x)|\}.$$
A computer data tape is prepared with the numbers $$n, x_1, y_1, x_2, y_2, \ldots, x_n, y_n,$$ where \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\) are pairs of observations of two variables \(X\) and \(Y\). Write a program in any standard language, or draw a flow diagram for such a program, which will read in the data, and print out the mean and variance of \(X\) and of \(Y\) and the (product-moment) correlation coefficient of \(X\) and \(Y\).
Three players \(A\), \(B\) and \(C\) each throw three fair dice in turn until one of them wins by making a score of 15 or more on the three dice. Show that the players' respective probabilities of winning are in the ratio $$(54)^2 : 54 \times 49 : (49)^2.$$
Explain what is meant by the term 'standard error of the mean'. Matches are put into a box five at a time until the weight of the box and matches combined reaches \(M\) grams, when the box is said to be full. The weight of an individual match is normally distributed with mean \(m\) grams and standard deviation \(\sigma\) grams. The weight of an empty match-box is normally distributed with mean \(5m\) grams and standard deviation \(2\sigma\) grams. Find the value of \(M\) such that there is only one chance in a hundred that a full match-box contains fewer than 50 matches.