(i) If \(z_1\) and \(z_2\) are two complex numbers, prove algebraically that \[|z_1|-|z_2| \leq |z_1-z_2| \leq |z_1|+|z_2|.\] Interpret these inequalities on the Argand diagram. (ii) Obtain \(\sqrt{(1+i)}\) in a form \(a+ib\) (for \(a\) and \(b\) real) and show that it is a root of the equation \(z^4 = 2i\). What are the other roots of this equation?
In the differential equations \begin{align*} \frac{d^2y}{dx^2}+p\frac{dy}{dx}+qy &= 0, \quad (A)\\ \frac{d^2y}{dx^2}+p\frac{dy}{dx}+qy &= f(x), \quad (B) \end{align*} \(p\) and \(q\) are constants. Prove that
Evaluate \(\int_1^x (\log_e t)^2\,dt\), for \(x > 0\). Let \(J_n = \log_e(1+\frac{1}{n})\), where \(n\) is a positive integer. By considering an upper bound for \(\int_1^{1+1/n} (\log_e t)^2\,dt\), or otherwise, show that \[J_n^2 - 2(1+\frac{1}{n})J_n + \frac{2}{n} \leq 0.\]
Evaluate \begin{align*} (i)&\int\frac{dx}{x+\sqrt{(2x-1)}} \quad (x > \frac{1}{2});\\ (ii)&\int\frac{dx}{(a^2-x^2)^{3/2}} \quad (|x| < |a|). \end{align*}
Each week, a boy receives pocket money only on condition that he wins two games in a row when playing three successive chess games with his father and mother alternating as opponents. The boy knows that his mother's probability of winning is \(\frac{3}{4}\), but his father's probability of winning is only \(\frac{1}{2}\). To maximise his chance of winning two games in succession, should he play the sequence father-mother-father, or mother-father-mother? Assuming that each week the boy plays the sequence more favourable to him, what is the expected number of weeks between two successive occasions on which he receives pocket money?
There are \(k\) distinguishable pairs of shoes in a dark cupboard. A man draws shoes out, one by one, without replacing them. Assume that each possible order of drawing shoes is equally likely.
The number of accidents occurring in a particular year on the M1 motorway has the Poisson distribution with mean \(\lambda_1\), while the number occurring on the M2 has the Poisson distribution with mean \(\lambda_2\). Assuming that the numbers of accidents occurring on different motorways are independent, prove that the total number of accidents on both motorways has the Poisson distribution with mean \(\lambda_1+\lambda_2\). Given that the total number of accidents on the two motorways is \(n\), find the probability that there were \(k\) accidents on the M1.
Solution: Suppose \(X_1 \sim Pois(\lambda_1), X_2 \sim Pois(\lambda_2)\) \begin{align*} \mathbb{P}(X_1+X_2 = n) &= \sum_{i=0}^n \mathbb{P}(X_1 = i, X_2 = n-i) \\ &= \sum_{i=0}^n \mathbb{P}(X_1 = i)\mathbb{P}(X_2 = n-i) \tag{assuming independent} \\ &= \sum_{i=0}^n e^{-\lambda_1} \frac{\lambda_1^i}{i!} e^{-\lambda_2}\frac{\lambda_2^{n-i}}{(n-i)!} \\ &= e^{-(\lambda_1+\lambda_2)} \sum_{i=0}^n \frac{\lambda_1^i\lambda_2^{n-i}}{i!(n-i)!} \\ &= e^{-(\lambda_1+\lambda_2)} \frac{1}{n!}\sum_{i=0}^n \frac{n!\lambda_1^i\lambda_2^{n-i}}{i!(n-i)!} \\ &= e^{-(\lambda_1+\lambda_2)} \frac{1}{n!}\sum_{i=0}^n \binom{n}{i}\lambda_1^i\lambda_2^{n-i} \\ &= e^{-(\lambda_1+\lambda_2)} \frac{1}{n!}(\lambda_1+\lambda_2)^n \end{align*} Therefore their sum has the same distribution as \(Pois(\lambda_1+\lambda_2)\). \begin{align*} \mathbb{P}(X_1 = k | X_1 + X_2 = n) &= \frac{\mathbb{P}(X_1 = k, X_1+X_2 = n)}{\mathbb{P}(X_1+X_2=n)} \\ &= \frac{e^{\lambda_1+\lambda_2}n! }{(\lambda_1+\lambda_2)^n} \mathbb{P}(X_1 = k, X_2 = n-k) \\ &= \frac{e^{\lambda_1+\lambda_2}n! }{(\lambda_1+\lambda_2)^n} \mathbb{P}(X_1 = k)\mathbb{P}(X_2 = n-k) \\ &= \frac{e^{\lambda_1+\lambda_2}n! }{(\lambda_1+\lambda_2)^n}e^{-\lambda_1} \frac{\lambda_1^k}{k!}e^{-\lambda_2}\frac{\lambda_2^{n-k}}{(n-k)!} \\ &= \binom{n}{k} \frac{\lambda_1^k\lambda_2^{n-k}}{(\lambda_1+\lambda_2)^n} \\ &= \binom{n}{k}p^k(1-p)^{n-k} \end{align*} Where \(p = \frac{\lambda_1}{\lambda_1+\lambda_2}\), ie it is distributed \(Binomial(n, \frac{\lambda_1}{\lambda_1+\lambda_2})\)
A tug-of-war contest is to be held between two colleges. The weights of students in College \(A\) follow a normal distribution with mean 140 lb and standard deviation 8 lb. Thanks to the superiority of its kitchens, the weights of students in College \(B\) follow a normal distribution with mean 150 lb and standard deviation 6 lb. Teams are chosen by selecting \(n\) students at random from each college. How large must \(n\) be in order to ensure that with probability at least 0.9 the combined weight of the College \(B\) team exceeds that of the College \(A\) team by at least 50 lb?
A single stream of cars, each of width \(a\) and exactly in line, is passing along a straight road of breadth \(b\) with speed \(V\). The distance between the rear of each car and the front of the one behind it is \(c\). Show that, if a pedestrian is to cross the road in safety in a straight line making an angle \(\theta\) with the direction of the traffic, then his speed must be not less than \[\frac{Va}{c\sin\theta+a\cos\theta}.\] Show also that if he crosses the road in a straight line with the least possible uniform speed, he does so in time \[\frac{b}{V}\left(\frac{c}{a}+\frac{a}{c}\right).\]
A uniform beam of weight \(W\) stands with one end on a sheet of ice and the other end resting against the smooth vertical side of a heavy chair of weight \(\lambda W\). Show that the maximum inclination of the beam to the vertical is given by \(\tan^{-1} 2\mu\) or \(\tan^{-1} 2\lambda\mu\) according as the chair or the beam is the heavier, the coefficient of friction between the ice and beam, and the ice and chair, being \(\mu\).