Prove the expressions for the area of a triangle (i) \(abc/4R\), (ii) \(r^2 \cot\frac{1}{2}A \cot\frac{1}{2}B \cot\frac{1}{2}C\). \(A', A''\) are the points of contact of the escribed circle of a triangle opposite the angle \(A\) with the two sides meeting in \(A\), and \(B', B''\) and \(C', C''\) have a similar meaning. Prove that the area of the convex hexagon \(A'A''B'B''C'C''\) is \[ \{(a+b+c)(a^2+b^2+c^2)+2abc\}/8R. \]
Express \((a+ib)^{p+iq}\) in the form \(A+iB\) where \(i=\sqrt{-1}\). If \(\sin x = y\cos(x+\alpha)\), expand \(x\) in ascending powers of \(y\).
Prove that two couples of equal and opposite moments in the same plane balance. Three forces \(\lambda a, \lambda b, \lambda c\) act along \(AO, BO, CO\) respectively, where \(O\) is the orthocentre of the triangle \(ABC\). If they are rotated through the same angle \(\theta\) about \(A, B, C\) respectively, shew that they become equivalent to a couple whose moment is \(4\lambda\Delta\sin\theta\) where \(\Delta\) is the area of the triangle \(ABC\).
Prove that any system of forces acting in one plane can in general be reduced to a single force, and find the equation of its line of action referred to any rectangular axes in the plane. A smoothly jointed quadrilateral of rods lie on a smooth horizontal table and is enclosed in a smooth circular hoop which presses tightly at each of the hinges. Prove that the pressures at the hinges are proportional to the sides of the circumscribed quadrilateral which touches the hoop at the hinges and that the stresses in the rods are inversely proportional to their distances from the centre of the hoop.
State the principle of virtual work and prove it for the case of a single lamina acted on by forces in its plane. A smooth ring is fixed above a smooth plane inclined at an angle \(\alpha\) to the horizon. Prove that the rod of least length which passes through the ring and can rest in equilibrium with one end on the plane makes an angle \(\phi\) with the horizontal given by \(\sin(\alpha+2\phi)=3\sin\alpha\).
Shew how to construct geometrically the directions of projection so that a particle projected from a given point \(A\) with given velocity may pass through another given point \(B\). An elastic particle is projected with a given velocity \(\sqrt{2gh}\) from a point \(P\) in a vertical wall to strike another parallel smooth vertical wall at a distance \(a\). After one impact on this wall the particle strikes the first wall at a point \(Q\) vertically above \(P\). Shew that if \(PQ\) is a maximum the angle which the direction of projection of the particle makes with the horizon is \(\tan^{-1}[2he/a(1+e)]\), where \(e\) is the coefficient of restitution between the particle and the wall. Also find \(PQ\).
State the principle of the conservation of linear momentum. A smooth inclined plane of angle \(\alpha\) and of mass \(M\) is free to slide on a smooth horizontal plane. A particle of mass \(m\) is placed on its inclined face and slides down under gravity. Find its acceleration in space and the pressure between it and the plane.
Calculate the loss of kinetic energy when a ball of mass \(m\) moving with velocity \(u\) strikes directly a ball of mass \(m'\) moving with velocity \(u'\). Two equal balls are lying in contact on a smooth table, and a third equal ball, moving along their common tangent strikes them simultaneously. Prove that \(\frac{2}{3}(1-e^2)\) of its kinetic energy is lost by the impact, \(e\) being the coefficient of restitution for each pair of balls.
Simplify the expression \[ \frac{\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{(x+y)^2}{(a+b)^2}}{\frac{x}{a}+\frac{y}{b}-\frac{x+y}{a+b}}. \]
Prove that, if \(n_r\) is the number of combinations of \(n\) things taken \(r\) at a time, \[ \begin{vmatrix} n_r & n_{r+1} & n_{r+2} \\ (n+1)_r & (n+1)_{r+1} & (n+1)_{r+2} \\ (n+2)_r & (n+2)_{r+1} & (n+2)_{r+2} \end{vmatrix} = \frac{n_r(n+1)_r(n+2)_r}{(r+1)_r(r+2)_2}. \]