A wedge of mass \(M\) is at rest on a smooth horizontal table, and one of its faces, which is rough, makes an angle \(\alpha\) with the horizontal. A uniform circular cylinder of mass \(m\) and radius \(a\) is held in contact with this face; the axis is horizontal and at a height \(h (>a)\) above the table. The cylinder is then released, and rolls down the inclined surface without slipping. Assuming that the wedge does not topple, show that the velocity of the wedge when the cylinder is about to make contact with the table is \[ \left[ \frac{m \cos\alpha}{m+M} \frac{4g(h-a)(m+M)}{3(m+M)-2m\cos^2\alpha} \right]^{\frac{1}{2}}. \]
Find the equation whose roots are the squares of the roots of the cubic equation \[ x^3 - ax^2 + bx - 1 = 0. \] Find all pairs of values of \(a\) and \(b\) for which the two equations are the same.
A sequence of numbers \(u_0, u_1, u_2, \dots, u_n, \dots\) satisfies the recurrence relation \[ u_{n+1}-2u_n+2u_{n-1}=0 \quad (n=1,2,3,\dots) \] and \(u_0=1, u_1=2\). Show that \(u_{4p}=(-4)^p\), and find expressions for \(u_{4p+1}, u_{4p+2},\) and \(u_{4p+3}\). Show further that \[ 5\sum_1^{4p} u_n^2 = 2^{4p+3}-8. \]
Show that the least sum of money that can be made up of florins (2s.) and half-crowns (2s. 6d.) in precisely 10 ways is £4. 10s., and the greatest sum is £5. 5s. 6d.
An event happens on an average once a year. Show that the chance it will not happen in any particular future year is \(1/e\), where \(e\) is the base of natural logarithms. Find also the chance of it happening twice in any particular future year. \subsubsection*{SECTION B}
The function \(f(t)\) possesses the derivative \(f'(t)\) for all real values of \(t\), and \(f(0)=0\). The relation \[ f(x+y) = \frac{f(x)+f(y)}{1+f(x)f(y)} \] holds for all pairs of real values of the variables \(x\) and \(y\). By differentiating partially with respect to \(x\) and \(y\), or otherwise, determine the form of the function \(f(t)\).
The coordinates \(x, y\) of a plane curve are given in terms of a real parameter \(\lambda\) by the equations \(x=c \log_e(\lambda+\sqrt{1+\lambda^2})\), \(y=c\sqrt{1+\lambda^2}\). Determine the general character and the principal features of the curve, and find the geometrical significance of the parameter \(\lambda\). Deduce that a point on the curve bisects the segment of the normal to the curve at that point intercepted between the centre of curvature and the \(x\) axis.
Obtain power series in increasing integral powers of \(x\) for \(\tan^{-1}x\), and \(\tanh^{-1}x\), where the principal value of the former function is to be taken. Verify from the series obtained that \(\tanh^{-1}ix=i\tan^{-1}x\).
Sketch the curve given by the plane polar equation \(r^3=a^3(1+2\cos\theta)\). Prove that the area enclosed by the curve is \(a^2(\frac{2}{3}\pi+\sqrt{3})\), and the volume enclosed by the surface obtained by revolving the curve about the line \(\theta=0\) is \(6\pi a^3 \sqrt{3}/5\).
\(I(p,q)\) is defined as \[ \int_0^1 x^p(1-x)^q dx, \] where \(p\) and \(q\) are real and non-negative. Show that \[ I(p,q)=I(q,p). \] Obtain a reduction formula for the integral and state any limitations on the values of \(p\) and \(q\) necessary. Prove that if \(p\) and \(q\) are positive integers \[ I(p,q) = p!q!/(p+q+1)! \]