(i) Prove that if \(A_1\) and \(A_2\) are any two events $$P(A_1 \cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2).$$ (ii) State the corresponding result for four events. (iii) Four letters are placed at random in four envelopes. Assuming that one and only one letter is right for each envelope, use the result in (ii) to find the probability that all four letters are placed in the wrong envelope. [\((A_1 \cap A_2)\) means that both \(A_1\) and \(A_2\) occur, and \((A_1 \cup A_2)\) means that at least one of \(A_1\) and \(A_2\) occur.]
Each of four players is dealt 13 cards from a pack of 52 which contains 4 aces. Player \(A\) looks at his hand and winks at his partner, Player \(B\), which is a pre-arranged signal that his hand has at least one ace. Player \(B\) winks back to show that he has at least one ace as well. Player \(C\) looks at his hand and sees that he has just one ace. From Player \(C\)'s point of view what is the probability that his partner, Player \(D\), also has at least one ace if (i) he saw the winks and understood their meaning; (ii) he knows nothing about his opponents' signals?
A process for obtaining a new sequence \(v_0, v_1, \ldots\) from a given sequence \(u_0, u_1, \ldots\) is defined as follows: Write down the sequence \(u_0, u_1, \ldots\) and below it write the sequence of first differences \(u_1 - u_0, u_2 - u_1, \ldots\); below that write the sequence of second differences, and so on. The sequence \(v_0, v_1, \ldots\) is then read off down the left-hand vertical column. So, for example, starting with \(1, 1, 2, 6, 24, \ldots\) we get: \begin{align*} 1 \quad 1 \quad 2 \quad 6 \quad 24 \quad \ldots \\ 0 \quad 1 \quad 4 \quad 18 \quad \ldots \\ 1 \quad 3 \quad 14 \quad \ldots \\ 2 \quad 11 \quad \ldots \\ 9 \quad \ldots \end{align*} and the new sequence is \(1, 0, 1, 2, 9, \ldots\) If \(u_n\) is defined by the recurrence relation $$u_{n+1} = (n+1)u_n, \quad u_0 = 1,$$ prove that \(v_n\) is defined by the recurrence relation $$v_{n+1} = (n+1)v_n + (-1)^{n+1}, \quad v_0 = 1,$$ and that \(v_n/u_n \to e^{-1}\) as \(n \to \infty\).
Find the greatest value of \(2^{\frac{1}{2}}(p+q)^{\frac{1}{2}}(1-s)^{\frac{1}{2}}+(s-p)^{\frac{1}{2}}(s-q)^{\frac{1}{2}}\) in the three-dimensional region \(p, q, s \geq 0, p+q \leq s \leq \frac{1}{2}\).
Evaluate $$\int_0^\infty \int_0^\infty e^{-x} \frac{\sin cx}{x} dx dc,$$ and hence evaluate $$\int_0^\infty \frac{\sin x}{x} dx.$$
In a plane three circles of equal radii are drawn through a point. Prove that the circle through their other three intersections has the same radius.
Find a necessary and sufficient condition for the pair of straight lines $$px^2 + qxy + ry^2 = 0$$ to be perpendicular. A variable chord \(PQ\) of a conic \(S\) subtends a right angle at a fixed point \(O\) in the plane of \(S\). Prove that the locus of the foot of the perpendicular from \(O\) to \(PQ\) is in general a circle. Under what circumstances is the locus a straight line?
A finite number of circles, not intersecting or touching each other, are drawn on the surface of a sphere, thus dividing the surface into a number of regions. Prove that it is always possible to colour the surface with two colours in such a way that each region is of a single colour, and adjacent regions are of different colours. Given such a set of circles and such a colouring of the resulting regions, show that it is always possible to draw a further circle in such a way that a single recolouring of one of the new regions will restore the colour property; and that, provided there are already at least two circles present, then a further circle may be drawn in such a way that a single recolouring will not suffice.
\(R\) is a ring with identity. A relation \(\sim\) is defined on \(R\) by \(x \sim y\) if and only if there is an element \(z\) having an inverse in \(R\) such that \(x = zy\). Prove that \(\sim\) is an equivalence relation. Let \(R\) be the ring of all \(3 \times 3\)-matrices of the form $$Z = \begin{pmatrix} a & b & c \\ c & a & b \\ b & c & a \end{pmatrix},$$ where \(a, b, c\) are integers. Prove that $$\det Z = \frac{1}{3}(a+b+c)[(b-c)^2 + (c-a)^2 + (a-b)^2],$$ and hence find the invertible elements of \(R\). Determine the number of elements in the various equivalence classes of \(R\) under \(\sim\).
Planners have to route a motorway from a point 20 miles due east of the centre of a town to a point 20 miles due west. Construction costs amount to £1m per mile, and the cost of compulsory acquisition is given in £m per mile by a function \(f(r)\) of the distance \(r\) from the centre. It is decided to build the motorway as two straight east-west sections, together with a semicircular ring road concentric with the town. Calculate the total cost of the motorway as a function of the radius of the ring road, and obtain an equation from which the values of the radius for which the cost is stationary may be found. Describe the cheapest planned route (i) if \(f(r) = k \cdot |20-r|\), and (ii) if \(f(r) = k \cdot |10-r|\), where \(k\) is a constant.