A uniform sphere of radius \(a\) and mass \(m\) with centre \(B\) has a particle of mass \(m\) embedded in it at a point \(A\) just below its surface. It is placed upon a perfectly rough fixed sphere of radius \(a\) and centre \(O\) in such a way that \(O\), \(A\) and \(B\) are in the same vertical plane. Let \(\alpha\) be the angle between \(OB\) and the upward vertical, and let \(\beta\) be the angle between \(BA\) and the downward vertical. Use a geometrical argument to derive a condition for equilibrium in terms of \(\alpha\) and \(\beta\). Show that there is a position of equilibrium for any fixed \(\alpha\) in the range \(0 \leq \alpha \leq \pi/6\). For rolling displacements of the movable sphere upon the fixed sphere in which \(A\) remains in the same vertical plane as \(O\) and \(B\), show that \(d\beta/d\alpha = 2\) and evaluate \(dV/d\alpha\), where \(V\) is the potential energy of the system; hence show that the positions of equilibrium of the system are unstable.
A rocket is programmed to burn its propellant fuel and eject it at a variable rate but at a constant velocity \(u\) relative to the rocket. Its initial mass is \(M_0\) and its mass at time \(T\) after all its fuel has been burned is \(M_0(1-e)\), where \(e\) is a constant, \(e < 1\). The rocket is launched from rest in a vertical direction under the influence of a constant gravitational acceleration \(-g\). Show that the velocity \(w\) of the rocket at time \(T\) is given by \begin{align*} w = -gT - u\log_e(1-e) \end{align*} independently of all details of the fuel burning program other than the fact that the burning takes time \(T\). In the special case where the mass of the rocket at time \(t\) is \(M_0(1-pt)\) for \(0 \leq t \leq T\), \(p\) being constant, show that the rocket rises to a height \(H\) given by \begin{align*} H = -\frac{1}{2}gT^2 - \frac{u}{p}[pT\log_e(1-pT) - pT - \log_e(1-pT)] \end{align*} at time \(T\). Show that \(H\) is certainly positive if \(up > g\).
Show that \(n\) coplanar lines in 'general position' (i.e. no two lines parallel, no three lines concurrent) divide the plane into \(\frac{1}{2}(n^2+n+2)\) regions. Show, also, that the regions may be coloured, each either red or blue, in such a way that no two regions whose boundaries have a line segment in common have the same colour. Find the number of regions into which \(n\) planes, in general position, divide three-dimensional space.
Show that the operation of matrix multiplication on the set \(M_2\) of real \(2 \times 2\) matrices is associative but not commutative. Let \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\), \(O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\) and let \(A\), \(B\) be members of \(M_2\). Prove that
Let \(N = \{1, 2, 3, \ldots\}\) and let \(F\) be the set of all real-valued functions \(f\) on \(N\) such that \(f(1) = 1\). For members \(f\), \(g\) of \(F\), define the function \(f * g\) on \(N\) by \[(f * g)(n) = \sum_{d|n} f(d) g(n/d)\] for all \(n\) of \(N\), where the summation is over all divisors \(d\) of \(n\) (including 1 and \(n\)). Prove that \((F, *)\) is an abelian (i.e. commutative) group, with identity element \(e\) given by \[e(n) = \begin{cases} 1 & (n = 1),\\ 0 & (n > 1). \end{cases}\] [N.B. You may assume, without proof, that each element of \(F\) has an inverse.] Let \(s\), \(\mu\) be the elements of \(F\) given by \[s(n) = 1 \quad \text{(all \(n\) of \(N\))},\] \[\mu(n) = \begin{cases} 1 & (n = 1),\\ (-1)^k & \text{(if \(n = p_1 \ldots p_k\), a product of \(k\) distinct primes)},\\ 0 & \text{(otherwise)}. \end{cases}\] Prove that \(s*\mu = e\) and deduce that, if \(g\) belongs to \(F\) and if \(f(n) = \sum_{d|n} g(d)\) (all \(n\) of \(N\)), then \(g(n) = \sum_{d|n} \mu(d)f(n/d)\) (all \(n\) of \(N\)), where the summation is taken over all divisors \(d\) of \(n\) in both cases.
For any real number \(x\), \([x]\) denotes the greatest integer not exceeding \(x\). Evaluate, for positive integers \(n\), \(r\), the number of multiples of \(r\) in the set \(\{1, 2, \ldots, n\}\). Positive integers \(a_1, \ldots, a_k\) satisfy \(a_i \leq N\) (\(i = 1, \ldots, k\)) and L.C.M.\((a_i, a_j) > N\) whenever \(i \neq j\). (L.C.M.\((a_i, a_j)\) is the least common multiple of \(a_i\) and \(a_j\).) By showing that \[\sum_{i=1}^{k} [N/a_i] \leq N,\] prove that \[\sum_{i=1}^{k} \frac{1}{a_i} \leq 2.\] Deduce that, for any real \(x > 2\), \[\sum_{\sqrt{x} < p \leq x} \frac{1}{p} \leq 2,\] where the summation is over prime numbers \(p\) in the given range, and deduce that if \(r\) is a positive integer and \(N = 2^{2^r}\), then \[\sum_{1 < p \leq N} \frac{1}{p} \leq 2r.\]
The churches of St Aldate, St Buryan and St Cett stand on the flat East Anglian plane, and their tall steeples (all of different heights) can be seen for miles around. As part of the university rag, three parties of students each choose a different patron saint and set out to push a bed along the path from which the two steeples belonging to the churches of the other two saints have the same apparent height. Show that the students will go round in circles and that the centres of the three circles will be collinear. Is it possible for the heights to be such that the three circles meet at a point? Is it possible for the heights to be such that none of the circles intersect? If two of the steeples were of the same height would you advise the rag committee to run this stunt? In each case give your reasons.
Let \(P\) and \(Q\) be points on the same side of a line \(l\). Let \(Q'\) be the reflection of \(Q\) in \(l\). Show that if \(O\) is a point on the line \(l\) then \(PO + OQ = PO + OQ'\). Deduce that \(PO + OQ\) is a minimum when \(O\) is so chosen that \(OP\) makes the same angle with \(l\) as does \(OQ\). Suppose now that \(ABC\) is an equilateral triangle and \(P\), \(Q\), \(R\) and \(T\) are points on \(BC\), \(CA\), \(AB\) and \(AB\) respectively. (Note that this means \(P\) lies between \(B\) and \(C\) inclusive; similar restrictions apply to \(Q\), \(R\) and \(T\).)
Each day a factory produces \(x_1\) tons of product \(A\), \(x_2\) tons of product \(B\), \(x_3\) tons of product \(C\) and \(x_4\) tons of waste \(D\). The nature of the process is such that \[x_1 + x_2 + x_3 = \lambda_1,\] \[x_4 - \log_e x_2 = \lambda_2,\] and \(x_1, x_2, x_3, x_4 \geq 0\). For what range of values of the constants \(\lambda_1\) and \(\lambda_2\) is it possible to find \(x_1, x_2, x_3\) and \(x_4\) satisfying the conditions above? The daily profit of the factory is \(2 \tan^{-1} x_1 + x_2\) (in thousands of pounds). Show how to choose \(x_1, x_2, x_3\) and \(x_4\) to maximise this profit.
Suppose \(x\) is a continuous function with continuous derivative satisfying \[\dot{x}(t) + x(t) = 0 \quad \text{for } |x(t)| \leq 1,\] \[\dot{x}(t) + 4x(t) = 0 \quad \text{for } 1 < |x(t)|,\] \[x(0) = 0, \quad \dot{x}(0) = v.\] Giving an account of your reasoning but without necessarily resorting to detailed calculation, show that \(x\) is periodic for all choices of \(v\). Give a rough sketch of how the period varies with \(v\), indicating the main features of your sketch and explaining why they occur (again exact numerical detail is not required). How would your various conclusions be altered (if at all) for the general initial conditions \(x(0) = u\), \(\dot{x}(0) = v\)?