A triangle with sides 5, 5, 6 has three circles inscribed in it each touching the other circles and two of the sides. Shew that the radii of the circles are 1, 1, and \(\frac{1}{2}(3-\sqrt{5})\).
By induction or otherwise prove that \[ \frac{1}{n+1} - \frac{m}{n+2} + \frac{mC_2}{n+3} - \frac{mC_3}{n+4} \dots + \frac{(-1)^m}{m+n+1} = \frac{m!n!}{(m+n+1)!}, \] where \(m\) and \(n\) are positive integers.
The mass of a train including the engine is 200 tons and the resistance to motion apart from brakes is 10 lb. weight per ton. The train starts from rest and travels 5 miles in 12 minutes ending at rest. The retardation is double the acceleration and both are uniform and there is a period during which the train runs at its maximum speed of 30 miles per hour. Find (i) the time of getting up full speed; (ii) the force exerted by the brakes; (iii) the rate at which the engine is working 1 minute from the start.
Shew that \[ \frac{\frac{\partial^2 z}{\partial x^2}\frac{\partial^2 z}{\partial y^2} - \left(\frac{\partial^2 z}{\partial x \partial y}\right)^2}{\left(\frac{\partial z}{\partial x}\right)^2 \left(\frac{\partial z}{\partial y}\right)^2} = \frac{\frac{\partial^2 x}{\partial y^2}\frac{\partial^2 x}{\partial z^2} - \left(\frac{\partial^2 x}{\partial y \partial z}\right)^2}{\left(\frac{\partial x}{\partial y}\right)^2 \left(\frac{\partial x}{\partial z}\right)^2}, \] where \(x,y,z\) are variables connected by one relation, which is conceived on the left-hand side as determining \(z\) as a function of \(x\) and \(y\), and on the right-hand side \(x\) as a function of \(y\) and \(z\).
Shew that if \(\alpha + \beta + \gamma = \frac{\pi}{4}\), then \[ (\sin\alpha + \cos\alpha)(\sin\beta + \cos\beta)(\sin\gamma + \cos\gamma) = 2(\sin\alpha\sin\beta\sin\gamma + \cos\alpha\cos\beta\cos\gamma). \]
Assuming that the series \[ c(t) = 1 - \frac{t^2}{2!} + \frac{t^4}{4!} - \dots, \quad s(t) = t - \frac{t^3}{3!} + \frac{t^5}{5!} - \dots \] may be differentiated term by term, prove that the point with rectangular coordinates \(c(t), s(t)\) describes a circle with constant speed as the ``time'' \(t\) varies, and hence deduce that \[ c(t) = \cos t; \quad s(t) = \sin t. \]
Assuming that the resistance to the motion of a train is proportional to the square of the velocity and that if the engine exerts a constant pull the greatest velocity attainable is \(V\), shew that, if when the velocity attained is \(V_1 (
Enunciate the principle of virtual work and explain how to apply it to find the position of equilibrium of a system having one degree of freedom when gravity is the only external force. A pentagon is formed of five equal uniform rods smoothly jointed at their extremities. It hangs with the two upper rods in contact with smooth pegs in the same horizontal line and the lowest rod horizontal. Shew that, if in equilibrium the pentagon is regular, the pegs must divide the rods in the ratio \(2+\sqrt{5}:3\).
If in a triangle \(ABC\) the side \(a\) is increased by a small quantity \(x\) while the other two sides are unaltered, shew that the radius of the circumscribing circle will be increased by \[ \frac{1}{2}x \operatorname{cosec} A \cot B \cot C. \]
Show that, if \[ \frac{1}{1+u}e^{\frac{ux}{1+u}} = P_0(x) + P_1(x)\frac{u}{1!} + P_2(x)\frac{u^2}{2!} + \dots + P_n(x)\frac{u^n}{n!} \dots, \] then \[ P_n(x) = x^n - \frac{n^2}{1!}x^{n-1} + \frac{n^2(n-1)^2}{2!}x^{n-2} - \frac{n^2(n-1)^2(n-2)^2}{3!}x^{n-3} + \dots. \] By putting \(\frac{u}{1+u}=t\), deduce that \[ x^n = P_n(x) + \frac{n^2}{1!}P_{n-1}(x) + \frac{n^2(n-1)^2}{2!}P_{n-2}(x) + \dots. \]