A finite number of circles, not intersecting or touching each other, are drawn on the surface of a sphere, thus dividing the surface into a number of regions. Prove that it is always possible to colour the surface with two colours in such a way that each region is of a single colour, and adjacent regions are of different colours. Given such a set of circles and such a colouring of the resulting regions, show that it is always possible to draw a further circle in such a way that a single recolouring of one of the new regions will restore the colour property; and that, provided there are already at least two circles present, then a further circle may be drawn in such a way that a single recolouring will not suffice.
\(R\) is a ring with identity. A relation \(\sim\) is defined on \(R\) by \(x \sim y\) if and only if there is an element \(z\) having an inverse in \(R\) such that \(x = zy\). Prove that \(\sim\) is an equivalence relation. Let \(R\) be the ring of all \(3 \times 3\)-matrices of the form $$Z = \begin{pmatrix} a & b & c \\ c & a & b \\ b & c & a \end{pmatrix},$$ where \(a, b, c\) are integers. Prove that $$\det Z = \frac{1}{3}(a+b+c)[(b-c)^2 + (c-a)^2 + (a-b)^2],$$ and hence find the invertible elements of \(R\). Determine the number of elements in the various equivalence classes of \(R\) under \(\sim\).
Planners have to route a motorway from a point 20 miles due east of the centre of a town to a point 20 miles due west. Construction costs amount to £1m per mile, and the cost of compulsory acquisition is given in £m per mile by a function \(f(r)\) of the distance \(r\) from the centre. It is decided to build the motorway as two straight east-west sections, together with a semicircular ring road concentric with the town. Calculate the total cost of the motorway as a function of the radius of the ring road, and obtain an equation from which the values of the radius for which the cost is stationary may be found. Describe the cheapest planned route (i) if \(f(r) = k \cdot |20-r|\), and (ii) if \(f(r) = k \cdot |10-r|\), where \(k\) is a constant.
A vector space is said to be finite-dimensional if there exists a finite number of vectors \(x_1, x_2, \ldots, x_n\) such that each vector in the space can be written as a linear combination $$c_1 x_1 + c_2 x_2 + \ldots + c_n x_n$$ with \(c_1, c_2, \ldots, c_n\) scalars. The vector space \(V_1\) consists of all sequences $$x = (\xi_1, \xi_2, \ldots)$$ of real numbers which have only a finite number of terms \(\xi_i\) non-zero. Addition and scalar multiplication are defined by $$(\xi_1, \xi_2, \ldots) + (\eta_1, \eta_2, \ldots) = (\xi_1 + \eta_1, \xi_2 + \eta_2, \ldots),$$ $$c(\xi_1, \xi_2, \ldots) = (c\xi_1, c\xi_2, \ldots).$$ \(V_2\) consists of all real sequences, the definitions of addition and scalar multiplication being the same. Prove that neither \(V_1\) nor \(V_2\) is finite-dimensional.
A particle of mass \(m\) moves in a plane under the action of a force of magnitude \(f(r)\) directed towards a fixed point \(O\) in the plane, where \(r\) is the distance of the particle from \(O\). Deduce from the equations of motion that the angular momentum about \(O\) and the total energy \(\frac{1}{2}mv^2 + \int f(r)dr\) remain constant. If \(f(r) = kr\), and initially \(r = r_0, v = v_0\) and the direction of motion is at right angles to the radius vector, find the value of \(r\) when the direction of motion is next at right angles to the radius vector.
A straight rigid uniform hair lies on a smooth table. At each end of the hair sits a flea. Show that, if the mass of the hair is not too great relative to that of the fleas, then in simultaneous jumps with the same velocity and angle of take-off they will be able to change ends without colliding in mid-air.
When an e.m.f. \(E(t)\) is applied to an inductor of constant inductance \(L\) and resistance \(R\), the current \(I\) is governed by the equation $$L \frac{dI}{dt} + RI = E.$$ Given that \(I = 0\) at time \(t_0\), find \(I(t)\) in the following cases: (i) \(E(t) = \begin{cases} 0 & \text{for } t < t_0 \\ E_0 \sin \omega t & \text{for } t > t_0 \end{cases}\) (\(E_0\) constant); (ii) \(E(t) = \begin{cases} 0 & \text{for } t < t_0 \\ E_0 \sin \omega t & \text{for } t_0 < t < t_1 \\ 0 & \text{for } t > t_1 \end{cases}\)
The measurement of a certain physical quantity \(Q\) involves the use of the unit of length. Let \(q\) denote the measure of \(Q\) when the unit of length is taken to be \(\mathbf{u}\), and \(\tilde{q}\) when it is taken to be \(\mathbf{\tilde{u}}\). Assume that, if \(\mathbf{u} = \lambda \mathbf{\tilde{u}}\), then $$\tilde{q} = f(\lambda, q).$$ Now suppose that the sum \(Q_1 + Q_2\) has a meaning independent of the choice of the unit of length. Then we must have $$q_1 + q_2 = q_3 \Rightarrow \tilde{q}_1 + \tilde{q}_2 = \tilde{q}_3.$$ Therefore $$f(\lambda, q_1) + f(\lambda, q_2) = f(\lambda, q_1 + q_2)$$ for all \(q_1, q_2\) and all positive \(\lambda\). Prove that \(f(\lambda, q)\) must be of the form \(\phi(\lambda)q\). By considering two successive changes of unit show that $$\phi(\lambda \lambda') = \phi(\lambda)\phi(\lambda')$$ and deduce the form of the function \(\phi(\lambda)\). (Assume that all the functions considered are differentiable.)
Three unequal rods \(A_0 A_1\), \(A_1 A_2\) and \(A_2 A_3\) are smoothly jointed at \(A_1\) and \(A_2\). The ends \(A_0\) and \(A_3\) can slide along a smooth horizontal rail. Find the position of stable equilibrium. Investigate the equilibrium of a similar system consisting of a chain of \(n\) rods \(A_0 A_1, \ldots, A_{n-1} A_n\), and show that there is precisely one stable configuration.
Assume that, if impulsive forces are applied to a rigid body at rest, the centre of mass \(G\) acquires a velocity \(F/M\) and the body acquires an angular velocity \(L/I\), where \(F\) is the resultant of the impulsive forces, \(L\) is their moment about \(G\), \(M\) is the mass, and \(I\) the moment of inertia about \(G\). Two uniform rods \(AB\), \(BC\) of masses \(M\), \(N\) and lengths \(2a\), \(2b\) respectively, smoothly jointed at \(B\), are placed in a straight line on a smooth horizontal surface. A horizontal impulsive force \(F\), at right angles to the line of the rods, is applied at \(A\). Find the initial speed \(V\) of \(C\). Show that if \(F\) is applied at \(C\) instead the initial speed of \(A\) is \(V\).