In the jointed frame of light rods shewn below, equal and opposite forces are applied at \(A\) and \(B\) in the line \(AB\). Draw a force diagram for the frame, and state which members are in compression and which in tension. [Diagram of a jointed frame of light rods is shown, with vertices A, B, C, D, E, F and various angles specified.]
Shew that the centres of the squares described on the hypotenuse of a right-angled triangle are each equally distant from the two sides containing the right angle, and determine the lengths of these distances in terms of the sides.
Give an account of the process by which the highest common factor of two polynomials \(f(x)\) and \(\phi(x)\) is obtained, and indicate the principles on which the process depends. Prove that, if \(X_1, X_2, X_3\) are the functions of \(x\) used as divisors at three successive stages of the process, a value of \(x\) which makes \(X_2\) vanish will make \(X_1=X_3\).
Forces of 1, 2, 3, 4, 5, 6 pounds weight respectively act at the corners of a regular hexagon inscribed in a circle of radius \(R\), in directions perpendicular to the plane of the circle and in the same sense. Shew that the resultant cuts the plane at a distance \(\dfrac{2R}{7}\) from the centre.
A sphere rolls on a parabolic wire with which it is in contact at two points; shew that the locus of the centre of the sphere is a parabola.
Prove that if \(u\) is a complex number, and \(m\) and \(n\) are positive integers prime to one another, \(u^{m/n}\) has \(n\) values. Illustrate by a diagram. Shew that, if \[ r^2 = a^2+b^2+c^2, \quad \text{and} \quad \frac{b+ic}{r+a}=z, \] then \[ \frac{c+ia}{r+b} = i\frac{1-z}{1+z}, \quad \text{and} \quad \frac{a+ib}{r+c} = \frac{1+iz}{1-iz}. \] \item[] \hspace{1cm} Or Shew that when \(n\) is an integer \(\cos n\theta\) and \(\sin n\theta/\sin\theta\) are polynomials in \(\cos\theta\). Deduce that \[ \cos n\theta = \prod_{r=1}^{n} \left\{1 - \frac{\sin^2 \theta/2}{\sin^2 (2r-1)\pi/4n}\right\}, \] and shew that \[ \frac{\cos n\theta - \cos n\alpha}{1 - \cos n\alpha} = \prod_{1}^{n} \left\{1 - \frac{\sin^2\theta/2}{\sin^2(r\pi/n + \alpha/2)}\right\}. \]
The distance between the axles of a railway truck is \(d\) feet, and the centre of gravity is halfway between them and at a perpendicular distance \(h\) feet from the rails. With the lower wheels locked it is found that the greatest incline upon which the truck can rest is \(\alpha\). Prove that the coefficient of sliding friction between the wheels and the rails is given by \(\mu = \dfrac{2d \tan\alpha}{d+2h\tan\alpha}\).
Shew that the square of any even number \(2n\) is equal to the sum of \(n\) terms of a series of integers in Arithmetical Progression; and that the square of any odd number \(2n+1\) exceeds by unity the sum of \(n\) terms of another such progression.
State and prove Pascal's and Brianchon's theorems. Discuss various limiting cases in which one or more pairs of the six points or of the six lines coincide. \item[] \hspace{1cm} Or A conic may be defined (a) as the projection of a circle, (b) as the locus of the intersection of corresponding lines of two homographic pencils. Shew that the two definitions are equivalent.
Three equal uniform rods \(OA, OB, OC\) freely jointed at \(O\) form a tripod with the feet \(A, B, C\) symmetrically placed on a smooth horizontal floor. The weight of each rod is \(W\). At three points, one-third of the way down \(OA, OB, OC\) respectively, are attached strings of equal length, one-sixth that of a rod, and their other ends are fastened together and support a weight \(w\). Prove by the principle of virtual work, or otherwise, that the system is in equilibrium when each rod makes with the vertical an angle \(\theta\) given by \(\sin^2\theta = \dfrac{3W(9W+8w)}{4(9W+2w)(3W+2w)}\).