A solid is constructed by cutting the corners off a cube in such a way that its set of faces consists of six identical squares and eight identical equilateral triangles. At each vertex two triangular faces and two square faces meet. Find the cosine of the angle \(\theta\) between any two triangular faces which meet at a vertex.
A point moves in a plane in such a way that its least distances from two fixed non-intersecting circles in that plane are equal. Describe the locus of the point in each of the various cases which may arise and justify your answers.
Show that, if \(f(x)\) is an increasing positive function for \(0 \leq x \leq 1\), then \[\frac{1}{n} \sum_{r=0}^{n-1} f\left(\frac{r}{n}\right) \leq \int_0^1 f(x) dx \leq \frac{1}{n} \sum_{r=1}^n f\left(\frac{r}{n}\right).\] Deduce that, for \(k \geq 0\), \[\left|n^{-k-1} \sum_{r=0}^n r^k - \frac{1}{k+1}\right| \leq \frac{1}{n}.\] Use similar arguments to show that \[\left|\sum_{r=n}^{2n} \frac{1}{r} - \log 2\right|\] can be made as small as we like by taking \(n\) large enough.
British Rail have found that their income from a route is given by \(I(v) = hv\), where \(v\) is the average speed (in appropriate units) of trains over the route, and \(h\) is constant. The capital cost \(C\) of improvements to track and signalling to obtain a speed \(v\) is estimated as follows:
The expenditures \(x(t)\) and \(y(t)\) on armaments at time \(t\) of two countries are governed by the equations \[\frac{dx}{dt} = -ax+by+k_1,\] \[\frac{dy}{dt} = -ay+bx+k_2,\] where \(a > 0\), \(b > 0\), \(k_1 > 0\), \(k_2 > 0\); also \(x(0) = y(0) = 0\). Show that the total expenditure \(x(t) + y(t)\) on armaments will increase without bound as \(t \to \infty\) if \(b \geq a\), but will tend to a limit if \(b < a\). Find \(x(t)\) and \(y(t)\) when \(b \neq a\).
Let \(f(x) = 2\cos x^2 - \frac{1}{x^2}\sin x^2\) for \(x \geq 1\). Find \(\int_1^t f(x)dx\) for \(t \geq 1\), and show that it tends to a limit as \(t \to \infty\) but that \(f(x)\) does not tend to zero as \(x \to \infty\). Give an example of a function \(h(x) \geq 0\), defined for \(x \geq 1\) and such that \(\int_1^t h(x)dx \to \infty\) as \(t \to \infty\) but \(h(x) \to 0\) as \(x \to \infty\). Give an example of a function \(l(x) \geq 0\), defined for \(x \geq 1\), and such that \(dl/dx > 0\) for all \(x \geq 1\) but \(l(x) \leq 1\) for all \(x \geq 1\). [It may be helpful to sketch such an \(l(x)\); but an explicit formula should be given.]
Show by using the binomial expansion or otherwise that \((1 + x)^n \geq nx\) whenever \(x \geq 0\) and \(n\) is a positive integer. Deduce that if \(y > 1\) then, given any number \(K\), we can find an \(N\) such that \(y^n \geq K\) for all integers \(n \geq N\). Show similarly that if \(y > 1\) then, given any \(K\), we can find an \(N\) such that \(\frac{y^n}{n} \geq K\) for all integers \(n \geq N\).
A paraboloidal bucket is formed by rotating the curve \(ay = x^2\) (\(0 \leq y \leq a\)) about the \(y\)-axis which is vertical. Water runs out of the bucket, initially full, through a small hole at \(y = 0\). The volume of water issuing per unit time is proportional to \(h^\alpha\), where \(h\) is the depth of the water remaining in the bucket at time \(t\), and \(\alpha\) is a constant (\(0 < \alpha < 2\)). At time \(t_1\) the bucket is half-empty (in terms of volume); it becomes totally empty at time \(t_2\). Find \(t_1/t_2\), showing that it depends on \(\alpha\) only.
By repeated integration by parts, or otherwise, show that \[\frac{1}{n!} \int_0^1 (1-t)^n e^t dt = e - \sum_{r=0}^n \frac{1}{r!}.\] Prove that \(0 \leq \int_0^1 (1-t)^n e^t dt \leq 1\) (\(n \geq 1\)). Deduce that \[\left|e - \sum_{r=0}^n \frac{1}{r!}\right| \leq \frac{1}{n!}.\] [By convention, \(0! = 1\).]
Sketch the `\(2m\)-rose' defined in polar coordinates by \(r = |\sin m\theta|\), for \(m = 1, 2, 3\). Show that for all integers \(m > 0\) the total area of the petals is independent of \(m\), and evaluate this area.