Problems

Three sets \(A\), \(B\), \(C\) are chosen at random in such a way that: (i) For any one of the sets \(A\), \(B\), \(C\) the event that it has non-empty intersection with either of the other two sets is independent of the event that it has non-empty intersection with the third set, and this event has constant probability. (ii) For any two of \(A\), \(B\), \(C\) the probability that they have non-empty intersection is \(p_1\). (iii) The probability that each pair of \(A\), \(B\), \(C\) has non-empty intersection is \(p_2\). Show that the probability that no two of \(A\), \(B\), \(C\) have non-empty intersection is \[p_0 = 1 - 3p_1(1-p_1) - p_2.\]

Analyse the following cases by any methods you think suitable, explaining briefly in each case the principles underlying your method.

1. In order to test whether a certain treatment increases the strength of a material, an experiment was carried out on 12 specimens of the material, of which 6 received the treatment and 6 did not, yielding the following results. \begin{array}{lrrrrrr} \text{With treatment} & 96 & 94 & 102 & 98 & 98 & 100 \\ \text{Without treatment} & 102 & 104 & 108& 110 & 106 & 106 \end{array} (Higher figures correspond to higher strength.)
2. The following data were obtained on the occurrences of 80 objects among 5 classes. \begin{array}{crrrrr} \text{Class} & A & B & C & D & E \\ \text{Expected numbers} & 10 & 10 & 20 & 20 & 20 \\ \text{Observed numbers} & 9 & 12 & 19 & 19 & 21 \end{array}

\(P\) is a passenger on a roundabout at a fair. When the roundabout is rotating uniformly, a given point \(A\) on the roundabout moves in a circle of radius \(3a\) about the central axis of the roundabout with constant angular velocity \(\omega\). The passenger \(P\) is moving relative to the roundabout in a circle of radius \(a\) about \(A\) with constant angular velocity \(2\omega\). How many times is \(P\) stationary during one revolution of \(A\)? Find the distance travelled by \(P\) between two points when he is at rest.

A square-wheeled bicycle is ridden at constant horizontal speed \(V\). The sides of one wheel are always parallel to the sides of the other, and the wheels do not slip on the ground. If the wheels remain in contact with the ground show that \[V^2 < ag/16,\] where \(a\) is the circumference of each wheel.

Identical ball-bearings \(A\), \(B\), \(C\), of diameter \(a\), are collinear. \(B\) and \(C\) are initially at rest with their centres a distance \(b\) apart, and \(A\), moving co-linearly towards them, strikes \(B\). The coefficient of restitution is \(e\). Show that \(A\) will strike \(B\) again, after \(A\) has travelled a further distance \[\frac{1+e}{1-e}(b-a).\]

Suppose that the coefficient of friction between two surfaces is directly proportional to the velocity difference between the surfaces (the 'slipping velocity'). Show that a body can slide down an inclined plane with a constant velocity which depends on the inclination of the plane. A cylinder of radius \(a\) and radius of gyration \(k\), with its axis horizontal, rolls (with slipping of the above character) down an inclined plane. If the cylinder starts from rest, determine the subsequent motion. Show that the frictional force tends to the force which would be necessary to keep the cylinder rolling without slipping, but that the slipping velocity at the point of contact does not tend to zero.

A particle of unit mass moves, in the absence of gravity, in the plane of a disc of unit radius and moment of inertia \(k^2\). The particle is attached by a light inextensible string of length \(l_0\) to a point on the rim of the disc. The particle's motion is such that the string is always taut, and wraps itself round the disc as the particle moves in a clockwise sense. (i) If the disc is fixed, show that the kinetic energy \(T\) of the system is \(\frac{1}{2}l^2\dot{\theta}^2\), and its angular momentum \(h\) is \(l^2\dot{\theta}\), where \(l\) is the length of the unwrapped portion of the string. Which, if either, of \(T\) and \(h\) is constant? (ii) If the disc is free to rotate about its axis, with angular velocity \(\dot{\phi}\) which is positive if the body rotates anticlockwise, show that \[h = l^2\dot{\theta}+(1+k^2+l^2)\dot{\phi},\] and find \(T\). Which, if either, of \(T\) and \(h\) is now constant? Show that \[h^2+l^2\dot{\theta}^2(1+k^2) = 2T(1+k^2+l^2).\]

Show that \(|{\bf a} \wedge {\bf b}|^2 = a^2b^2 - ({\bf a} \cdot {\bf b})^2\). If \({\bf a} \wedge {\bf b} \neq 0\), and if \[\alpha{\bf a} + \beta{\bf b} + \gamma{\bf a} \wedge {\bf b} = \alpha'{\bf a} + \beta'{\bf b} + \gamma'{\bf a} \wedge {\bf b},\] show that \(\alpha = \alpha'\), \(\beta = \beta'\), \(\gamma = \gamma'\). For some \(\lambda\), and for some non-zero \({\bf x}\), \[({\bf a}\cdot{\bf x}){\bf a}+({\bf b}\cdot{\bf x}){\bf b} = \lambda{\bf x}.\] By looking for solutions of the form \({\bf x} = \alpha{\bf a} + \beta{\bf b} + \gamma{\bf a} \wedge {\bf b}\), or otherwise, show that either \(\lambda = 0\) or \[\lambda^2-(a^2+b^2)\lambda + |{\bf a} \wedge {\bf b}|^2 = 0.\]

A point moves on the (fixed) set of points in the plane having integer coordinates \((m, n)\) with \(m \geq n\). The point starts at the origin \((0, 0)\) and at each step can move either one unit in the \(x\)-direction or one unit in the \(y\)-direction (provided this is possible); thus from \((m, n)\) it goes either to \((m + 1, n)\) or to \((m, n + 1)\) if \(m > n\), while it necessarily goes to \((m+1,n)\) if \(m = n\). Let \(h_p\) denote the number of distinct paths from \((0, 0)\) to \((p,p)\) for \(p \geq 0\) (so that \(h_0 = 1\)), and let \(k_p\) denote the number of distinct paths from \((0, 0)\) to \((p,p)\) for \(p > 0\) that do not include any of the points \((m, m)\) for \(0 \leq m \leq p - 1\). Show that \(h_p = \sum_{q=1}^{p} k_q h_{p-q}\) and that \(k_p = h_{p-1}\) for \(p \geq 1\). Writing \(H(x) = \sum_{p=0}^{\infty} h_p x^p\) and \(K(x) = \sum_{p=1}^{\infty} k_p x^p\), obtain an expression for \(H(x)\) as a function of \(x\), and hence show that \(h_p = \frac{1}{p+1}\binom{2p}{p}\).

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The points \(A,...,J\) are called nodes; two nodes are adjacent if they are joined by a straight line in the figure. Prove that each node belongs to just two collections of four mutually non-adjacent nodes. Show that there are five such collections, any two of which have just one node in common. Deduce that the permutations of the nodes which preserve adjacency form a group isomorphic to \(S_5\), the group of permutations of five objects.