Problems

A light rod of length \(a\) rests horizontally with its ends on equally rough fixed planes inclined at angles \(\alpha\) and \(\beta\) to the horizontal. The vertical plane through the rod is perpendicular to the line of intersection of the rough planes. The coefficient of friction between the ends of the rod and the planes on which they lie is \(\tan \lambda\), and \(\alpha < \lambda < \pi/2 - \alpha\), \(\beta < \lambda < \pi/2 - \beta\). Show that the length of that section of the rod on which a weight can be placed without disturbing the equilibrium is \[\frac{a \sin 2\lambda \cos(\alpha - \beta)}{\sin(\alpha + \beta)}.\]

An harmonious population with ample space and food is liable to grow at a rate proportional to its size. However, disunity induces mortal combat, so that in practice the ratio \(x_n\) of the number in any generation to a fixed number \(k\) satisfies \[x_{n+1} = \alpha x_n(1 - x_n)\] where \(\alpha\) is a positive constant. It is known that under certain circumstances the solution to this equation is of the form \begin{align*} x_n = x_{n+2} = x_{n+4} = \ldots = p,\\ x_{n+1} = x_{n+3} = x_{n+5} = \ldots = q. \end{align*} Show that aside from the trivial solution \(p = q = 0\), the relation \[\alpha^2(1-p)(1-q) = 1\] can then be satisfied, together with either \(p = q\) or \[\alpha(p+q-1) = 1.\] Hence, or otherwise, establish the ranges of \(\alpha\) for which (a) the population can oscillate with a period of 2 generations and (b) a non-zero steady state exists.

In a manufacturing process it is required to determine the shape of a truncated circular cone, of given height \(h\) and base radius \(a\), whose surface area (excluding the flat top and bottom) is least. The shape can be changed only by varying the radius \(c\) of the top, and the value of \(c\) may be taken as zero if necessary. Find the optimal value of \(c\) in the two cases (i) \(h^2 = 3a^2/8\); (ii) \(h^2 = 15a^2/32\), and sketch the relationship between \(c\) and surface area in each case.

The elements of the \(n \times n\) matrix \(A = (a_{ij})\) are all equal to either 1 or \(-1\). Prove or disprove the following assertions concerning the determinant \(\delta\) of \(A\):

1. \(\delta = 0\) only if there are two rows of \(A\) which are multiples of one another.
2. \(\delta\) is divisible by \(2^{n-1}\).
3. \(\delta\) can only take the values \(0, \pm 2^{n-1}\).

Let \(X\) be a non-empty set with an associative binary operation \(*\). Suppose that \begin{align*} (a) &~\text{there is an } e \in X \text{ such that } e*x = x \text{ for all } x \in X,\\ (b) &~\text{for each } x \in X \text{ there is a } y \in X \text{ such that } x*y = e. \end{align*}

1. Show that \(\{x*e : x \in X\}\) is a group under \(*\).
2. By considering two-element sets \(X\) or otherwise, show that \(\{x : x \in X\}\) need not be a group under \(*\).

An even integer \(2n\) is said to be \(k\)-powerful if the set \(\{1, 2, \ldots, 2n\}\) can be partitioned into two disjoint sets \(\{a_1, a_2, \ldots, a_n\}\), \(\{b_1, b_2, \ldots, b_n\}\) such that \[\sum_{r=1}^{n} a_r^j = \sum_{r=1}^{n} b_r^j \quad \text{(for all \(j = 1, 2, \ldots, k\)).}\] Show that

1. \(2n\) is 1-powerful if and only if \(n\) is even;
2. if \(2n\) is \(k\)-powerful, then \(4n\) is \((k+1)\)-powerful.

A tetrahedron has vertices at the origin, and at points \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\). The inscribed sphere lies inside the tetrahedron and touches all four faces. Show that this sphere has radius \[\frac{|[\mathbf{a}, \mathbf{b}, \mathbf{c}]|}{|\mathbf{a} \times \mathbf{b}| + |\mathbf{b} \times \mathbf{c}| + |\mathbf{c} \times \mathbf{a}| + |(\mathbf{a} \times \mathbf{b}) + (\mathbf{b} \times \mathbf{c}) + (\mathbf{c} \times \mathbf{a})|}\] where \([\mathbf{a}, \mathbf{b}, \mathbf{c}]\) denotes the scalar triple product of \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\).

On the first Thursday of May Professors Addem, Bakem and Catchem visit the Botanic Garden to admire the cow-parsley bed. Professor A invariably arrives at 2 pm and leaves at 3 pm. Professors B and C arrive independently at uniformly distributed random times between 1 pm and 3 pm, spend 12 minutes in rapt contemplation and then depart. Find

1. The probability that A and B meet.
2. The probability that B and C meet.
3. The probability that A, B and C meet together.
4. The probability that none of them meet.

Micro chips are produced in large batches. The engineer in charge believes that \[\Pr \text{(\(n\) defective chips in a batch)} = \frac{(4-n)^2}{30} \quad [0 \leq n \leq 2]\] \[\Pr \text{(3 or more defective chips)} = \frac{1}{30}.\] The results of testing 1000 batches are recorded as follows

Do these figures bear out the engineer's belief? Do you have a better idea, and, if so, can you explain why you think it better?

Farmer Jones' meadow may be regarded as the square \(0 \leq x \leq 1, 0 \leq y \leq 1\). At time \(t = 0\), Jones enters at \((1,0)\) and walks at constant velocity \((0, c)\). At the same moment his dog, Spot, enters at \((0,0)\) and runs at unit speed, directed always towards the instantaneous position of Jones. Show that Spot's path satisfies \[(1-x)\frac{dp}{dx} = c(1 + p^2)^{\frac12}\] where \(p = \frac{dy}{dx}\). Hence show that Spot does not overtake Jones inside the meadow if \(c > (5^{1/2} - 1)/2\).