Two small smooth pegs, in the same horizontal, are fixed vertically beneath a smooth horizontal wire. One end of each of two uniform rods of the same weight per unit length is smoothly hinged to a small ring which can slide on the wire. If there is a position of equilibrium in which the rods rest, one on each of the two pegs, making angles \(\tan^{-1}\frac{1}{2}\) and \(\tan^{-1}\frac{1}{3}\) with the horizontal respectively, find the ratio of their lengths.
A conic is given by the general Cartesian equation. Shew how to determine the position and magnitude of its principal axes. Find conditions that the conic should be a parabola, or one of the degenerate forms. Prove that the conic \[ x^2 - y^2 + 2\sqrt{-1} xy = 1 \] satisfies the conditions for both a parabola and a rectangular hyperbola, and interpret this result.
If \[ (1 + x)^n = a_0 + a_1x + a_2x^2 + \dots + a_nx^n, \] prove that \[ a_0^2 + a_1^2 + a_2^2 + \dots + a_n^2 = \frac{(2n)!}{n!n!}, \] and find the value of \[ a_0^2 - a_1^2 + a_2^2 - \dots + (-1)^n a_n^2. \]
Prove that, if \(a\) and \(b\) are positive integers which have no common factor, integers \(A\) and \(B\), positive or negative, can be found such that \(Aa + Bb = 1\). If the positive integers \(a_1, a_2, \dots, a_n\) have the greatest common divisor \(g\), prove that integers \(A_1, A_2, \dots, A_n\), positive, negative, or zero, can be found, such that \[ g = A_1a_1 + A_2a_2 + \dots + A_na_n. \]
The framework of smoothly jointed bars shown in the figure is freely supported at \(A\) and hinged to a support at \(B\). It is acted on by a force of 1 ton weight as indicated. Find graphically the reactions on the supports at \(A\) and \(B\) and the stresses in each bar. Indicate which bars are in tension.
Investigate the possible forms of the graph \[ y = \frac{x+a}{x^2+b}, \] for different values, positive and negative, of \(a\) and \(b\).
Prove that \(\log_e \{\log_e (1+x)^{1/x}\} = -\frac{1}{2}x + \frac{5}{24}x^2 - \frac{1}{8}x^3 - \dots\). Find, without using tables, the value of \(\log_e (\log_e 1.01)\) correct to five places of decimals, having given \(\log_e 10 = 2 \cdot 302585\).
Prove that, if the equation \[ (a + \cos\theta) \cos(\theta-\gamma) = b \] is satisfied by \(\theta_1, \theta_2, \theta_3, \theta_4\), four different values of \(\theta\) which lie between 0 and \(2\pi\), \begin{align*} \cos\theta_1 + \cos\theta_2 + \cos\theta_3 + \cos\theta_4 &= -2a, \\ \sin\theta_1 + \sin\theta_2 + \sin\theta_3 + \sin\theta_4 &= 0, \\ \theta_1 + \theta_2 + \theta_3 + \theta_4 - 2\gamma &= 0 \text{ (or } 2r\pi). \end{align*}
Out of a hollow shell bounded by concentric spherical surfaces a hollow ring is cut by two parallel planes. Show that the centres of gravity of the volume of the ring and of its whole surface, plane and curved, are coincident.
Prove that the series \[ 1 + \frac{1}{2^a} + \frac{1}{3^a} + \frac{1}{4^a} + \dots + \frac{1}{n^a} + \dots \] is divergent if \(a = 1\) or \(a < 1\), and convergent if \(a > 1\). Prove in another way that the series \[ S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}+\dots \] is divergent by shewing that the sum of each group of fractions whose denominators contain the same number of digits exceeds \(9/10\). Deduce that, if all fractions in which the figure 0 occurs are omitted from the series \(S\), the sum of those fractions that remain whose denominators contain \(r\) digits is less than \(9/10^{r-1}\), and that the fractions that remain form a convergent series. Shew also that, if all fractions are omitted from \(S\) in which any other digit than 0, e.g. 7, occurs, the fractions that remain form a convergent series.