An exhibition hall in contemporary style consists of a concrete structure forming the surface of a paraboloid of revolution; the surface is obtained by rotating the parabola \(x = at^2\), \(y = 2at\) about the \(x\)-axis, which is vertically downwards. The greatest height of the hall is \(h\). Find the volume of the hall, and show that \(S\), the area of the roof surface, is given by \[S = \frac{8\pi a^2}{3}\left\{\left(1+\frac{h}{a}\right)^\frac{3}{2} - 1\right\}.\] Show that if the hall is designed with a fixed height \(h\), then the increase in \(S\) corresponding to an increase of the parameter \(a\) from \(h\) to \(h(1 + \delta)\), where \(\delta\) is small, is approximately \[\frac{4\pi h^2}{3}(5\sqrt{2}-4)\delta.\]
Show that \[2\sin\frac{1}{2}x \sum_{n=1}^{N} \cos nx = \sin(N + \frac{1}{2})x - \sin\frac{1}{2}x.\] Let \[S_N(x) = \sum_{n=1}^{N} \frac{\sin nx}{n}.\] Find the values of \(x\) at the turning points of \(S_N(x)\) in the interval \(0 < x < \pi\), and show that \(S_N(x) \geq S_{N-1}(x)\) at each of them. Using induction, or otherwise, show that \(S_N(x) > 0\) for all \(x\) such that \(0 < x < \pi\).
A village contains two shops, \(X\) and \(Y\), which compete with one another to supply its needs. A local schoolmaster, wishing to amuse himself by trying to forecast their prospects, supposes that each shop has allotted to it a number, called its 'prosperity rating', which varies with time; he denotes the prosperity ratings of \(X\), \(Y\) at time \(t\) by \(x\), \(y\) respectively. He then invents a theory which implies that the rates of change of \(x\), \(y\) are \[\lambda(N - x) + \mu(x - y), \lambda(N - y) + \mu(y - x)\] respectively, where \(\lambda\), \(\mu\), \(N\) are positive constants. Initially \(x\) has the higher prosperity rating. Show that, according to his theory, the prosperity rating of \(Y\) can never overtake that of \(X\), but that it almost does so in due course if a certain condition on \(\lambda\) and \(\mu\) is satisfied. Suppose now that (1) is replaced by \[\lambda(N - x) + \mu(x - y)^{\frac{1}{2}}, \lambda(N - y) + \mu(y - x)^{\frac{1}{2}}.\] By considering the sign of \(dz/dt\), where \(z = x - y\), discuss whether you would expect \(z\) to become small when \(t\) is large.
Two lines in a plane meet in \(P\). Prove that successive reflexion in the two lines is equivalent to a rotation about \(P\). \(P_1\) and \(P_2\) are two distinct points of a plane. By considering the effect on the points \(P_1\) and \(P_2\), or otherwise, prove that a rotation of the plane about \(P_1\) followed by an equal and opposite rotation about the original position of \(P_2\) is equivalent to a translation. The four points \(A\), \(B\), \(C\), \(D\) lie on a circle of radius \(r\). Prove that successive reflexion in \(AB\), \(BC\), \(CD\), \(DA\) is equivalent to a translation through a distance \(AC \cdot BD/r\).
A model of hyperbolic (non-Euclidean) geometry is given as follows. The points (called \(h\)-points) of the geometry are the points \((x, y)\) of the Euclidean plane satisfying \(y > 0\). The lines (called \(h\)-lines) of the geometry are the parts of circles or straight lines which lie in the half-plane \(y > 0\) and which are orthogonal to the \(x\)-axis. The measurement of angles in the hyperbolic geometry is the same as in the Euclidean geometry. Prove that the following statements are true in the hyperbolic geometry. (a) There is a unique \(h\)-line through any two distinct \(h\)-points. (b) Any two distinct \(h\)-lines meet in at most one \(h\)-point. (c) If \(P\) is an \(h\)-point not on an \(h\)-line \(L\), then there are infinitely many \(h\)-lines through \(P\) which do not meet \(L\). (d) If \(P\) is an \(h\)-point not on an \(h\)-line \(L\), then there is a unique \(h\)-line \(L'\) through \(P\) that cuts \(L\) orthogonally.
Prove that \[F(x) \equiv x^{n+1} - (n+1)x + n \geq 0\] for all positive numbers \(x\) and positive integers \(n\). For what values of \(x\) does equality occur? Suppose \(a_1\), \(a_2\), ..., \(a_{n+1}\) are positive numbers whose geometric mean is \(G_{n+1}\). Suppose also that \(G_n\) is the geometric mean of the first \(n\). By considering \(F(x)\) with \(x^{n+1} = a_{n+1}/G_n^n\), show that \[a_{n+1} \geq (n+1)G_{n+1} - nG_n.\] When does equality occur? Deduce that the arithmetic mean of a finite number of positive numbers is greater than the geometric mean, unless they are all equal. [The geometric mean of positive numbers \(b_1\), \(b_2\), ..., \(b_m\) is \((b_1 b_2...b_m)^{1/m}\) and the arithmetic mean is \((b_1 + b_2 + ... + b_m)/m\).]
Give a multiplication table for the group of symmetries of a square, expressing each entry in the form \(a^m b^n\) where \(a\) is a rotation through a right angle and \(b\) is a reflexion in one of the diagonals. [MISSING DIAGRAM] Let \(C_1\), \(C_2\), \(C_3\) be squares whose vertices are numbered as in the diagram. By considering the symmetry group of each of the squares as a group of permutations of the symbols \(\{1, 2, 3, 4\}\), show that \(\Sigma_4\), the group of all permutations of four symbols, has (at least) three subgroups of order 8.
\(S\) is the set of points in the plane represented by pairs of integers \((n, m)\). The axis is the set of points \((n, 0)\); a point \((n, m)\) is above the axis if \(m > 0\), and below the axis if \(m < 0\). A path in \(S\) is a succession of ascents \((n, m) \to (n + 1, m + 1)\) and descents \((n, m) \to (n + 1, m - 1)\). \(N_{n,m}\) is the number of possible paths from \((0, 0)\) to \((n, m)\). For fixed \(n > 0\), for what values of \(m\) is \(N_{n,m}\) non-zero? If \(n = a + d\), \(m = a - d\), calculate \(N_{n,m}\). Let \(A\) and \(B\) be points above the axis, and \(A'\) be the reflexion of \(A\) in the axis. Prove that the number of paths from \(A\) to \(B\) that hit the axis is equal to the total number of paths from \(A'\) to \(B\). [Consider the first point at which such a path hits the axis.] The winning candidate in a ballot polls \(a\) votes, the loser \(d\) votes. Show that the probability that throughout the counting the winner is always ahead is \((a-d)/(a+d)\).
For a given function \(f(x)\) define \[F_n(x, f) = \frac{1}{n!}\int_0^x (x-t)^n f(t)dt\] where \(n \geq 0\) and \(0! = 1\). Show that \[(n+1)F_{n+1}(x, f) = xF_n(x, f) - F_n(x, g)\] where \(g(x) = xf'(x)\) and \(n \geq 0\). Show by induction that for all \(f\) \[\frac{d}{dx}F_n(x, f) = F_{n-1}(x, f),\] and hence evaluate \[\frac{d^k}{dx^k}F_n(x, f)\] at \(x = 0\) for \(k = 1, 2, ..., n+1\).
The ends of a uniform rod of length \(2b\) are constrained to lie on a smooth wire in the form of a parabola, which is fixed with its axis vertical and vertex downwards. The length of the latus rectum of the parabola is \(4a\). Prove that, when the rod is inclined at an angle \(\theta\) to the horizontal, the height of its centre of gravity above the vertex of the parabola is \[\frac{1}{4a}(b^2\cos^2\theta + 4a^2\tan^2\theta).\] Hence, or otherwise, find the positions of equilibrium and discuss their stability.