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 \(A\):
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*}
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
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
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
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 \(c > (5^{1/2} - 1)/2\).
(a) Evaluate \[\int_0^{\infty} \frac{1}{(1+t^2)^2} dt.\] (b) Show that \[\int_a^b \left\{\left(1-\frac{a}{x}\right)\left(\frac{b}{x}-1\right)\right\}^{1/2} dx = \pi\left\{\frac{a+b}{2} - (ab)^{1/2}\right\}\] where \(0 < a < b\). [The substitution \(t^2 = (x-a)/(b-x)\) is suggested.]
Find the general solution, for \(x > 0\), of the differential equation \[x^2y'' - 4xy' + 6y = 0\] by searching for solutions of the form \(y = x^{\lambda}\). Find, similarly, a particular solution of the equation \[x^2y'' - 4xy' + 6y = Cx^k\] provided \(k \neq 2\), \(k \neq 3\). Hence find the general solution of (1), and the solution that satisfies \begin{align} y(1) = 1, \quad y'(1) = 0. \end{align} Write \(k = 3+\varepsilon\), and obtain a tentative solution to (1) and (2) in the exceptional case \(k = 3\) by carefully taking the limit of your last result as \(\varepsilon \to 0\). Verify that it does indeed satisfy both the differential equation (1) and conditions (2).