Problems

Two buckets of water each of total mass \(M\) are suspended at the ends of a cord passing over a smooth pulley and are initially at rest. Water begins to leak from a small hole in the side of one of the buckets at a steady slow rate of \(m\) units of mass per second. Establish the equation of motion, and prove that the velocity \(V\) of the bucket when a mass \(M'\) of water has escaped is given by \[ V = \frac{2Mg}{m} \log\frac{2M}{2M-M'} - \frac{gM'}{m}. \]

Solve the equations: \[ \left\{ \begin{array}{l} x+y+z = 1 \\ ax+by+cz=d \\ a^2x+b^2y+c^2z=e, \end{array} \right. \] where \(a, b, c\) are unequal. If \(a=b\), determine sufficient conditions in terms of \(a, b, c, d\) and \(e\), for the existence of a solution. Show that in this case a possible combination of values is \(e=d^2\), and \(d=a\) or \(d=c\).

Prove that if \(a+b+c+d=0\):

1. [(i)] \(\frac{a^5+b^5+c^5+d^5}{5} = \frac{a^3+b^3+c^3+d^3}{3} \times \frac{a^2+b^2+c^2+d^2}{2}\);
2. [(ii)] \(\frac{a^7+b^7+c^7+d^7}{7} = \frac{a^3+b^3+c^3+d^3}{3} \times \left\{ \left(\frac{a^2+b^2+c^2+d^2}{2}\right)^2 - abcd \right\}\).

Prove that the Arithmetic Mean of a number of positive quantities is never less than their Geometric Mean. Show that if \(u+v+w=1\), and \(a \le 1\), where \(u,v,w,a\) are all positive: \[ (\frac{1}{u}-a)(\frac{1}{v}-a)(\frac{1}{w}-a) \ge 27 - 27a + 9a^2 - a^3. \]

Find the coefficient of \(x^n\) in the expansion of \(x^3(1-x)^{-3}\). Hence or otherwise prove that the number of ways in which three non-zero positive integers can be chosen to have as sum a given odd integer \(N\) is the integer nearest to \(N^2/12\). State the number of cases in which two (but not three) of the numbers are equal.

1. [(i)] If \(n \ge 2\), prove that \(y=\sec^{-1}x\) satisfies the equation \[ (x^2-x^n) \frac{d^n y}{dx^n} + \{(3n-4)x^2-n+1\} \frac{d^{n-1}y}{dx^{n-1}} + (n-2)(3n-5)x \frac{d^{n-2}y}{dx^{n-2}} + (n-2)^2(n-3)\frac{d^{n-3}y}{dx^{n-3}} = 0. \]
2. [(ii)] Find the general expression for \(y\) in terms of \(x\) which is such that \[ \frac{d^2 x}{dy^2} = -\frac{d^2 y}{dx^2}. \]

Show that \(2\tan^{-1}e\) is a stationary value for an angle between the tangents drawn at the extremities of a variable focal chord of an ellipse of eccentricity \(e\).

A plane curve is such that the tangent at any point \(P\) is inclined at an angle \((k+1)\theta\) to a fixed line \(Ox\), where \(k\) is a positive constant and \(\theta\) is the angle \(xOP\). The greatest length of \(OP\) is \(a\). Find a polar equation for the curve. Sketch the curves for the cases \(k=2, k=\frac{1}{2}\).

Prove that:

1. [(i)] \(2\pi^3 3^{-\frac{1}{2}} > \int_0^{\pi/3} \sin^{\frac{1}{2}}x \, dx > 2^{\frac{1}{2}} \pi 3^{-1}\),
2. [(ii)] \(2\pi > 12 \int_0^{\pi/4} \tan^{\frac{1}{2}} x \, dx > \pi^{\frac{3}{2}}\).

The arc intercepted on a rectangular hyperbola by a latus rectum is revolved about an asymptote. If \(2a\) is the length of the latus rectum, prove that the area of the surface of revolution generated is \(\pi a^2 (\sqrt{6}+\sinh^{-1}\sqrt{2})\).