A bead of mass \(m\) slides on a smooth wire in the form of the parabola \(x^2=4ay\), which is fixed with its axis vertical and vertex downwards. The bead is released from rest. Prove that during the subsequent motion the horizontal displacement \(x\) of the bead from the axis satisfies the equation \[ \frac{1}{4a^2}\dot{x}^2(4a^2+x^2) = ag(c^2-x^2), \] where \(c\) is the initial value of \(x\). Find the reaction of the wire on the bead when the bead is at the lowest point of the wire. If \(c\) is small, what is the period of the resulting small oscillation?
The pendulum of a clock consists of a uniform rod \(AB\), of length \(2a\) and mass \(M\), freely suspended from the end \(A\), and a particle of mass \(\frac{1}{3}M\) is attached to the rod at a distance \(a(1+x)\) from \(A\). Prove that the period of a small oscillation about the position of stable equilibrium is \[ 2\pi\sqrt{\left(\frac{5+2x+x^2}{4+x}\right)\frac{a}{g}}. \] Prove that, if \(x\) is small, the period is approximately \[ \frac{2\pi}{n}\left(1+\frac{3}{40}x^2\right), \] where \(n^2=4g/(5a)\). If the clock keeps good time when \(x=0\), how many seconds will it lose in a day when \(x=1/1000\)?
Find a condition in terms of \(a_0, a_1, a_2, a_3\) that the cubic equation \[ a_0x^3+3a_1x^2+3a_2x+a_3=0 \] should have two repeated roots. Hence or otherwise show that the turning values of the polynomial \[ a_0x^3+3a_1x^2+3a_2x+a_3 \] are the roots of the equation \[ a_0^2 y^2 - 2y(a_0^2a_3 - 3a_0a_1a_2 + 2a_1^3) + (a_0a_2-a_1^2)^2 - 4(a_1a_3-a_2^2)(a_0a_2-a_1^2)=0. \]
In a recurring series of terms \(u_0, u_1, u_2, \dots u_n, \dots\) the recurrence relation \[ u_{n+2} - 2u_{n+1}\cosh\theta + u_n = 0 \] is satisfied for \(n \ge 0\). Prove that \(u_r, u_{n+r}, u_{2n+r}\) will satisfy the relation \[ u_{2n+r} - 2u_{n+r}\cosh n\theta + u_r = 0 \] where \(r \ge 0\). Show further that if \(u_{r_0}=0\), then \(u_n\) is proportional to \(\sinh(n-r_0)\theta\).
If \(pu+qv+rw=1\), where \(p, q, r, u, v, w\) are all positive quantities, prove that \[ \frac{p}{u} + \frac{q}{v} + \frac{r}{w} \ge (p+q+r)^2. \] Prove further that if \(p, q, r\) are integers, \[ u^{-p}+v^{-q}+w^{-r} \ge 3(p+q+r)^{\frac{p+q+r}{3}}. \] (It may be assumed that the arithmetic mean of a number of positive quantities is never less than the geometric mean.)
A number \(p\) of objects are put at random in \(n\) different cells. Prove that the chance that \(k\) objects are in any particular cell is \({}^p C_k \frac{(n-1)^{p-k}}{n^p}\), where \(k \le p\). Prove also that the number of ways in which \(p\) like objects can be put in the \(n\) cells so that no cell is empty is \({}^{p-1}C_{n-1}\), where \(p>n\). \subsubsection*{SECTION B}
The functions \(\phi(t)\) and \(\psi(t)\) possess derivatives \(\phi'(t)\) and \(\psi'(t)\) for all real values of \(t\), and \(\psi(0)=1\). If the relation \[ \phi(x^2+y^2) = \psi(x)\psi(y) \] holds for all pairs of values of the variables \(x\) and \(y\), determine the forms of the functions \(\phi(t)\) and \(\psi(t)\).
Prove that, if \(0 < x < 1\), then \[ \pi < \frac{\sin\pi x}{x(1-x)} < 4. \] Sketch the graph of the function in this range of \(x\).
Establish Leibniz' theorem for the \(n\)th derivative of the product of two functions. If \(f=(px+q)/(x^2+2bx+c)\), prove that \[ (x^2+2bx+c)f_{n+2} + 2(n+2)(x+b)f_{n+1} + (n+1)(n+2)f_n = 0. \]
Prove that \[ \int_0^\pi x f(\sin x) \, dx = \frac{\pi}{2} \int_0^\pi f(\sin x) \, dx. \] Hence, or otherwise, evaluate the integral \[ \int_0^\pi \frac{x\sin x}{1+\cos^2 x} \, dx. \]