Problems

Show that, for \(r \geq 10\), \[(r-\frac{1}{2})(r+\frac{1}{2}) < r^2 < (r-\frac{39}{80})(r+\frac{41}{80}).\] Deduce that \[\frac{80}{761} \leq \sum_{r=10}^{\infty} \frac{1}{r^2} \leq \frac{2}{19}.\]

Lady Bracknell is holding a dinner party. She has arranged the six diners around a circular table, with Algernon next to Cecily. It is the custom at Bracknell Hall for those dining to change places several times during the meal, in order to vary the conversation. Let \(G\) be the set of those rearrangements of the six diners after which Algernon and Cecily are still sitting next to each other. (Two rearrangements are to be considered the same if one can be obtained from the other merely by rotating the diners around the table.) Show that \(G\) forms a group, under the operation of performing one rearrangement after another. How many elements does \(G\) have? Now suppose that, in the initial arrangement, Cecily is seated on Algernon's right. Let \(H\) be the set of those elements of \(G\) after which Cecily is still on Algernon's right. Show that whenever \(g\) is an element of \(G\) and \(h\) an element of \(H\), \(ghg^{-1}\) is an element of \(H\).

Evaluate the determinant \[A = \begin{vmatrix} 1 & z & z^2 & 0 \\ 0 & 1 & z & z^2 \\ z^2 & 0 & 1 & z \\ z & z^2 & 0 & 1 \end{vmatrix}.\] Plot in the Argand diagram the points satisfying \(A = 0\).

State Pythagoras's Theorem. Two circles \(\alpha\), \(\beta\) with centres \(A\) and \(B\) and radii \(a\) and \(b\), lie in different planes \(\pi\) and \(\varpi\) respectively which meet in a line \(l\). Show that the two circles will lie on the same sphere if and only if \(AB\) is perpendicular to \(l\) and \[AP^2-BP^2 = a^2-b^2\] for every point \(P\) on \(l\).

Prove that the straight line \[ty = x+at^2\] touches the parabola \(y^2 = 4ax\) (\(a \neq 0\)), and find the coordinates of the point of contact. The tangents from a point to the parabola meet the directrix in points \(L\) and \(M\). Show that, if \(LM\) is of a fixed length \(l\), the point must lie on \[(x+a)^2(y^2-4ax) = l^2x^2.\]

Positive numbers \(p\) and \(q\) satisfy \[\frac{1}{p}+\frac{1}{q} = 1,\] and \(y\) is defined by \(y = x^{p-1}\), for \(x > 0\). Express \(x\) in terms of \(y\) and \(q\). By considering \(\int_0^s ydx\) and \(\int_0^t xdy\) as areas, or otherwise, show that if \(s > 0\) and \(t > 0\) then \[st \leq \frac{s^p}{p}+\frac{t^q}{q}.\] When does equality hold?

A circular arc subtends an angle \(2\alpha(< \pi)\) at the centre of a circle of radius \(R\). A surface is generated by rotating the arc about the line through its end points. Prove that the area of this surface is \(4\pi R^2(\sin\alpha-\alpha\cos\alpha)\).

A function \(f(x)\) is defined, for \(x > 0\), by \[f(x) = \int_{-1}^1 \frac{dt}{\sqrt{(1-2xt+x^2)}}.\] Prove that, if \(0 \leq x \leq 1\), then \(f(x) = 2\). What is the value of \(f(x)\) if \(x > 1\)? Has \(f(x)\) a derivative at \(x = 1\)?

Consider a group of students who have taken two examination papers. Suppose that 80\% of these students pass on Paper I. Suppose further that any student who passes on Paper I has a 70\% chance of passing on Paper II while those who failed Paper I have only a 20\% chance. What is the probability that a student who passes on Paper II did not pass on Paper I?

Let the random variable \(X\) have the exponential distribution with parameter \(\lambda > 0\), that is \[P\{X \leq x\} = \begin{cases} 1-e^{-\lambda x}, & \text{if}~ x \geq 0,\\ 0, & \text{if}~ x < 0. \end{cases}\] Let \(Y\) be a random variable having the exponential distribution with parameter \(\mu\), and suppose that \(X\) and \(Y\) are independent. Find the distribution of min\((X, Y)\) and the probability that \(Y\) exceeds \(X\).