Problems

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 \(X\).

Explain what is meant by the parallelogram of forces, and what is meant by the resultant of a system of coplanar forces. \(ABCD\) is a quadrilateral whose opposite sides meet in \(X\) and \(Y\). By considering suitable forces acting along the sides of the quadrilateral, show that the bisectors of the angles \(X\), \(Y\), the bisectors of the angles \(B\), \(D\) and the bisectors of the angles \(A\), \(C\) intersect on a straight line, certain restrictions being made as to which pairs of bisectors are taken.

Two small spherical particles of mass \(m\) are joined by inextensible light strings of length \(a\) to a particle of mass \(M\); the strings lie taut in a straight line on opposite sides of \(M\) on a smooth horizontal table. The particle of mass \(M\) is set in motion by an impulse \(I\) perpendicular to the line of the particles. Show that when the two small spheres collide their relative velocity is \(2I/\sqrt{M(M+2m)}\). The spheres are imperfectly elastic, with coefficient of restitution \(e\). Find the angular velocities of the strings when they are next in line. [You may assume that the strings remain taut throughout the motion.]