The function \(B(x, y)\) is defined by the equation, \[B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt,\] for positive \(x\) and \(y\). Show that
Sketch the curve whose equation in polar coordinates is \[r = 1 - \frac{5}{6} \sin \theta.\] Find the range of real values of \(b\) for which the simultaneous equations \[(x^2 + y^2 + \frac{5}{6}y)^2 = x^2 + y^2\] \[y = b\] have a real solution.
Show that \[x^2 y'' + 2x(x+2)y' + 2(x+1)^2 y = e^{-x}\] can be transformed to a second order linear differential equation with constant coefficients by using the substitution \(y = zx^n\) for a suitable value of \(n\). Find the general solution of the original equation.
The number of messages to be sent by carrier pigeon during a week is a random variable whose distribution is Poisson with parameter \(\mu\). Each bird released with a message has probability \(p\) of evading predatory hawks and arriving at its destination. At the end of each week, all birds which arrived are returned safely to the starting point. Assuming no shortage of pigeons, find the distribution of the net loss of birds per week, and hence write down its mean and variance. Suppose \(\mu = 50\). What is the smallest number of pigeons that needs to be available in order that one may be 99\% confident that all messages to be sent during a week can be carried?
A uniform solid sphere of radius \(r\) and mass \(m\) is drawn slowly and without slipping up a flight of steps by a horizontal force which is always applied to the highest point of the sphere and is always perpendicular to the vertical planes which form the faces of the steps. If each step is of height \(\frac{1}{2}r\) (and of breadth greater than \(r\)), prove that the coefficient of friction between the sphere and the edge of the steps must exceed \(\tan (\pi/6)\), and find the maximum horizontal force throughout the movement.
The annual frisbee-throwing competition between Oxford and Cambridge mathematicians takes place on an infinitely large horizontal plain in perfect weather. According to the rules, each frisbee must be a flat thin disc, so that the air resistance to motion parallel to the plane of the disc is negligible, but resistance to motion perpendicular to the disc is exceedingly large. It must be thrown from ground level with a given speed \(V\) and at a given inclination \(\alpha (>0)\) to the horizontal (i.e the axis of the disc makes an angle \(\alpha\) to the vertical). The angle \(\beta\) between its initial velocity and a horizontal line in the plane of the disc may be chosen freely. As they throw, the competitors give the frisbee a spin about its axis, and the resulting gyroscopic effect is such that the direction of the axis is constant during flight. The longest throw wins. Show that the competitor who chooses \(\beta\) closest to \(\pi/4\) wins.
A uniform rod \(BC\) is suspended from a fixed point \(A\) by stretched springs \(AB\), \(AC\). The springs are of different lengths but the ratio of tension to extension is the same constant \(k\) for each. The rod is not hanging vertically. Show that the ratio of sums of the stretched springs is equal to the ratio of the lengths of the unstretched springs.
A tennis player serves from height \(H\) above the ground, hitting the ball with speed \(v\) at an angle \(\alpha\) below the horizontal. The ball just clears the net of height \(h\) at distance \(a\) from the server and hits the ground a further distance \(b\) beyond the net. Show that \[v^2 = \frac{g(a+b)^2 (1 + \tan^2 \alpha)}{2[H-(a+b)\tan \alpha]}\] with \[\tan \alpha = \frac{(2a+b)H}{a(a+b)} - \frac{(a+b)h}{ab}\] What restriction must be imposed on \(H\) in terms of \(a\), \(b\) and \(h\) for such a serve to be possible?
An ellipse is given by \(x = a\cos\theta, y = b\sin\theta\), where \(a\) and \(b\) are positive.
For the equation \[2x \frac{d^2y}{dx^2} + \frac{dy}{dx} + \frac{1}{2}y = 0, \quad x > 0,\] look for a solution in the form \[y = \sum_{r=0}^{\infty} a_r x^{r+\frac{1}{2}},\] and find a relation between successive coefficients in the sequence \(\{a_r\}\). Hence express this solution in terms of simple functions.