Let $$f(y) = \int_{-1}^{1} \frac{dx}{2\sqrt{(1-2xy+y^2)}},$$ where the positive value of the square root is taken. Prove that \(f(y) = 1\) if \(|y| \leq 1\). Find the value of \(f(y)\) when \(|y| > 1\). Hence or otherwise prove that if \(|y| < 1\), then $$\int_{y}^{1} \frac{(x-y)dx}{(1-2xy+y^2)^{3/2}} = \int_{-y}^{1} \frac{(x+y)dx}{(1+2xy+y^2)^{3/2}}.$$
A particle \(Q\) of mass \(2m\) is attached to one end of a light elastic string \(PQ\) of length \(2a\) and modulus of elasticity \(\lambda\); a particle \(R\) of mass \(3m\) is attached to the mid-point of the string. The system is then hung in equilibrium from a fixed point \(P\). The particle \(Q\) is given a small downward impulse \(\epsilon\sqrt{\frac{m\lambda}{a}}\). After time \(t\) the ensuing displacements of \(Q\), \(R\) from the equilibrium position are \(x\), \(y\), respectively. Prove that \(\ddot{x} = -3\omega^2(x-y), \quad \ddot{y} = 2\omega^2(x-2y), \quad \text{where } \omega = \sqrt{\frac{\lambda}{6am}}.\) Verify that \(x = \epsilon\left(\frac{3\sqrt{6}}{10}\sin\omega t + \frac{1}{5}\sin\sqrt{6}\omega t\right)\) satisfies the initial conditions. Deduce that this is the correct solution for \(x\), by finding a similar formula for \(y\), which, together with that for \(x\), satisfies the equations of motion and the initial conditions. Is the motion periodic?
A smooth ball of unit mass collides with a second equal ball, which is initially at rest. The coefficient of restitution \(e\) is less than 1. The first ball is aimed so as to suffer the maximum change of direction in the collision. Find this change of direction, and also the proportion of energy which is lost in the collision.
A satellite is planned to have a circular orbit at speed \(v\) and distance \(d\) from the centre of the earth \(O\). It is in fact released at distance \(d(1+\alpha)\), speed \(v(1+\beta)\), and at an angle \(\gamma\) radians to the horizontal, where \(\alpha\), \(\beta\), \(\gamma\) are small inaccuracies. Prove, either by using the differential equation of the orbit or otherwise, that at the lowest point of its orbit the distance of the satellite from \(O\) is, to first order, \(d\{1+2\alpha+2\beta-\sqrt{(1+2\beta)^2+\gamma^2}\}.\) (The differential equation of the orbit referred to polar coordinates \((r, \theta)\) about \(O\) is \(\frac{d^2u}{d\theta^2} + u = \frac{\mu}{h^2}, \text{ where } u = \frac{1}{r}, h \text{ is the angular momentum per unit mass about } O, \text{ and } \mu r^2 \text{ is the force of attraction per unit mass towards } O.)\) Assuming that \(\alpha\), \(\beta\), \(\gamma\) are in absolute value less than \(\frac{1}{10}\), that the earth is 4000 miles in radius, and the atmosphere 200 miles thick, find the minimum height above the earth's surface at which the satellite should be planned to be released in order to be sure of missing the atmosphere. Additional questions on probability and statistics
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:
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:
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 \(3\).