Problems

Craps is played between a gambler and a banker as follows. On each throw the gambler throws two dice. On the first throw he wins if the total is 7 or 11, but loses if it is 2, 3 or 12. If the first throw is none of these numbers, he subsequently wins if on some later throw he again scores the same as his first throw, but loses if he scores a 7. Calculate:

1. [(i)] the probability of winning on the first throw;
2. [(ii)] the probability of eventually winning with a 4;
3. [(iii)] the probability of winning.

Define the coefficient of correlation between two variables. The numbers of bacteria present in 10 samples were as follows:

Let \(a\) be a given complex number; prove that there is at least one complex number such that \(z^k = a\). How many solutions does this equation have in general? For what value or values of \(a\) is there an exception to this rule? Prove your statements. Express all the solutions of the equation \[z^2 = 1 + i\] in the form \(z = x + iy\), where \(x\) and \(y\) are real.

When \(x\) is a real number, the notation \([x]\) (the 'integral part' of \(x\)) is used to denote the greatest integer that does not exceed \(x\). Prove the following three statements:

1. [(i)] \([x - y] = [x] - [y] - \epsilon\),

1. [(ii)] \([x] + [x + y] + [y] \leq [2x] + [2y]\);
2. [(iii)] if \(n\) is a positive integer, then \[[x] + \left[x + \frac{1}{n}\right] + \left[x + \frac{2}{n}\right] + \cdots + \left[x + \frac{n-1}{n}\right] = [nx].\]

Find the general solution of the system of equations \begin{align} x + y - az &= b, \\ 3x - 2y - z &= 1, \\ 4x - 3y - z &= 2, \end{align} in each of the three cases: (i) \(a = 1\), \(b = 9\); (ii) \(a = 2\), \(b = -3\); (iii) \(a = 2\), \(b = 0\).

For each positive integer \(n\), let \[u_n = 1 - (n-1) + \frac{(n-2)(n-3)}{2!} - \frac{(n-3)(n-4)(n-5)}{3!} + \cdots\] where the summation stops with the first term that is equal to \(0\). By considering \(u_{n-1} - u_n\) or otherwise, prove that \(u_n\) satisfies a recurrence relation of the form \[au_n + bu_{n+1} + cu_{n+2} = 0,\] and determine the relation. Hence, or otherwise, evaluate \(u_n\) for general \(n\); in particular, show that \(u_n = 0\) whenever \(n - 2\) is a multiple of \(3\).

A navigator wishes to determine the position \(D\) of his ship; he observes three landmarks \(A\), \(B\), \(C\) at his eye level, and measures the angles \(ADB\) and \(BDC\); he then reads off the distances \(AB\) and \(BC\) and the angle \(ABC\) from the chart. Obtain a formula that will enable him in general to calculate the angle \(DAB\) and hence to determine his position. Explain why the method breaks down when \(ABCD\) is a cyclic quadrilateral.

Obtain the general solutions of the trigonometrical equations:

1. [(i)] \(\sin^{-1}(2x) - \sin^{-1}(\sqrt{3}x) = \sin^{-1} x\);
2. [(ii)] \(\sin(\pi \cos x) = \cos(\pi \sin x)\).

Suppose that the functions \(f(x)\) and \(g(x)\) can each be differentiated \(n\) times. Prove that one can write \[\frac{d^n}{dx^n}\{g[f(x)]\} = g'[f(x)]u_1(x) + g''[f(x)]u_2(x) + \cdots + g^{(n)}[f(x)]u_n(x),\] where the functions \(u_k(x)\) depend on \(f(x)\), and on \(n\), but not on \(g(x)\). Show that \(u_k(x)\) is the coefficient of \(s^k\) in the expansion of \[e^{-sf(x)} \frac{d^n}{dx^n} [e^{sf(x)}]\] as a power series in \(s\). Hence, or otherwise, prove that \[u_k(x) = \frac{1}{k!} \sum_{r=0}^k (-1)^{k-r} \binom{k}{r} [f'(x)]^{k-r} \frac{d^n}{dx^n} [f(x)]^r,\] where \(\binom{k}{r}\) denotes the coefficient of \(t^r\) in the binomial expansion of \((1+t)^k\).

Let \(a\) and \(c\) be given real numbers such that \(0 < a < c\); find the least value of \(x\) for which \[a \sec \theta + x \cos \sec \theta \geq c\] for all \(\theta\) between \(0\) and \(\frac{1}{2}\pi\).