In two dimensions, show that the relation \[\mathbf{l.m} = l_1m_1+l_2m_2\] is equivalent to \[\mathbf{l.m} = lm\cos\theta,\] where \((l_1, l_2)\) are the Cartesian components of \(\mathbf{l}\), \(l\) is the length of \(\mathbf{l}\), and \(\theta\) is the angle between \(\mathbf{l}\) and \(\mathbf{m}\). Using vectors, prove the triangle inequality, that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
\(A\), \(B\), \(C\) and \(D\) are complex numbers. Describe the set of points in the complex plane that satisfy the equation \[Az\bar{z}+Bz+C\bar{z}+D = 0.\] You should distinguish carefully between the cases that arise, and you may find it helpful to consider first the case \(A = 0\).
(i) Stones are thrown at random into \(n\) tin cans. Let \(P(m)\) be the probability that all the tin cans contain at least one stone after \(m\) throws. Show that \[n\left(1-\frac{1}{n}\right)^m \geq 1-P(m) \geq \left(1-\frac{1}{n}\right)^m.\] (ii) Two points \(x\) and \(y\) are chosen at random in the interval \(0 \leq t \leq 1\). What is the probability that \(|x - y| \geq 1/5\)?
A coin which has the probability \(p\) of falling heads is tossed repeatedly until exactly \(k\) heads have been obtained. Show that the probability that this requires \(n\) tosses is \[\binom{n-1}{k-1}p^k(1-p)^{n-k} \quad (n = k, k+1, \ldots).\] Show that this probability is the coefficient of \(z^n\) in the expansion of \[\left(\frac{pz}{1-(1-p)z}\right)^k.\] By differentiating this series, or otherwise, deduce the mean of \(n\).
Do you think that the following deductions are correct? Explain your reasons simply but clearly. (i) The average age at death of generals is considerably higher than that for the whole population. This shows that generals take care not to expose themselves to danger. (ii) I have tossed this coin twice and it came down heads each time. Therefore it is probably an unfair coin. (iii) I have tossed this coin 1000 times and it came down heads 276 times. Therefore it is probably an unfair coin.
Copies of a daily newspaper, which appears six times a week, are examined for misprints over a long period. It is discovered that the probability of there being one or more misprints in a given issue is \(\frac{1}{3}\). What is the most likely number of misprints in a week? What assumptions have you made? Find an expression for the probability that there will be fewer than the most likely number in a week. [\(\log_e (3/2)\) is approximately 0.4055.]
Solution: Assuming misprints are independent and occur at a constant average rate, then they are distributed as a Poisson random variable. The sum of Poisson random variables is Poisson, so the distribution of number of misprints in a week is \(Pois(2)\). The mode of a \(Pois(\lambda)\) is \(\lfloor \lambda \rfloor\), so the most likely is \(2\). Let \(X \sim Pois(2)\), then \begin{align*} \mathbb{P}(X < 2) &= \mathbb{P}(X = 0) + \mathbb{P}(X = 1) \\ &= e^{-2}\left ( \frac{2^0}{0!} + \frac{2^1}{1!}\right) \\ &= \frac{3}{e^2} \end{align*}
A particle moves in a straight line under a force \(F\), its mass increasing by picking up matter whose previous velocity was \(u\). If the mass and velocity of the particle at time \(t\) are \(m\) and \(v\), respectively, show that \[\frac{d}{dt}(mv) - \frac{dm}{dt}u = F.\] A particle whose mass at time \(t\) is \(m_0(1 + at)\) is projected vertically upwards under gravity at time \(t = 0\) with velocity \(V\), the added mass being picked up from rest. Show that it rises to a height \[\frac{g+2aV}{4a^2}\log\left(1+\frac{2aV}{g}\right) - \frac{V}{2a}.\]
A particle is projected horizontally from a point \(A\) on a vertical wall directly towards a parallel wall, which is a distance \(d\) away. The particle strikes the ground, which is horizontal, at \(B\), a distance \(b\) from the first wall, before bouncing on to hit the parallel wall at \(C\). It then rebounds towards the first wall. Assuming all impacts are perfectly elastic, find the condition on \(b/d\) for the particle to hit the first wall again before it hits the ground a second time. If this condition is satisfied and the particle hits this wall at a height \(\frac{1}{2}h\) above the ground, where \(h\) is the height of \(A\), calculate the height of \(C\) as a fraction of \(h\).
A bead of mass \(m\) slides down a rough wire in the shape of a circle. The wire is fixed with its plane vertical and the coefficient of friction between the bead and the wire is \(\mu\). Show that the reaction \(R\) between the bead and the wire satisfies \[\frac{dR}{d\theta} - 2\mu R + 3mg\sin\theta = 0,\] where \(\theta\) is the angle the radius to the bead makes with the downward vertical. Show that this equation is satisfied by \[R = A\cos\theta + B\sin\theta + Ce^{2\mu\theta},\] where \(C\) is arbitrary and \(A\) and \(B\) are to be determined. If the bead is released from rest at the same level as the centre of the circle and comes to rest at its lowest point show that \[(1 - 2\mu^2)e^{\mu\pi} = 3\mu.\]
Four identical spheres rest in a pile on a table, three touching each other and the fourth symmetrically on top. Let \(\alpha\) be the angle between the vertical and any nonhorizontal line-of-centres (\(\sin \alpha = 1/\sqrt{3}\)). Show that the spheres will stay in place without slipping provided that (a) the coefficient of limiting friction between two spheres is greater than \(\tan(\frac{1}{2}\alpha)\), and (b) the coefficient of limiting friction between a sphere and the table is greater than \(\frac{1}{3}\tan(\frac{1}{2}\alpha)\).