Determine the asymptotes of the curve \[ r\cos 3\theta = a \] and sketch the curve.
Prove that \[ 2\tan^{-1}\left(\tan\frac{\theta}{2}\tan\frac{\phi}{2}\right) = \cos^{-1}\left(\frac{\cos\theta+\cos\phi}{1+\cos\theta\cos\phi}\right). \] Eliminate \(\theta, \phi\) from \[ a\cos\theta+b\sin\theta+c=0, \quad a'\cos\phi+b'\sin\phi+c'=0, \quad \theta+\phi=\alpha. \]
In a triangle prove that \[ \text{(i) } a = \frac{r_1(r_2+r_3)}{\sqrt{r_2r_3+r_3r_1+r_1r_2}}, \quad \text{(ii) } \sin\frac{A}{2} = \frac{r_1}{\sqrt{(r_1+r_2)(r_1+r_3)}}. \] The circles escribed to the sides \(AB, AC\) of a triangle \(ABC\) touch \(BC\) at \(P\) and \(Q\) respectively. Prove that \(\tan APB \tan AQC = 4s(s-a)/(b+c)^2\).
Express \(\tan n\theta\) in terms of \(\tan\theta\). Prove that \[ \text{(i) } \sum_{r=0}^{n-1} \tan(\theta+r\pi/n) = -n\cot n\theta, \quad \text{(ii) } \sum_{r=0}^{n-1} \cot(\theta+r\pi/n) = n\cot n\theta, \] if \(n\) is even and find the value of each when \(n\) is odd.
Express \(\frac{x}{(x+1)^2 - (1-x)^2}\) in the form \(\sum_{r=1}^{r=3} \frac{a_r}{x^2+\tan^2 r\pi/7}\), and find the values of \(a_1, a_2, a_3\).
Prove that the resultant of forces \(\lambda.OA\) and \(\mu.OB\) is \((\lambda+\mu)OG\), where \(G\) is the centre of inertia of masses \(\lambda\) at \(A\) and \(\mu\) at \(B\). A quadrilateral \(ABCD\) is inscribed in a circle, centre \(O\); and forces proportional to the areas of the triangles \(BCD, CDA, DAB\) and \(ABC\) act along \(OA, BO, OC,\) and \(DO\) respectively; shew that they are in equilibrium.
State the principle of virtual work; and explain how it may be applied to determine the stresses in the rods of a smoothly jointed framework in equilibrium under the action of given external forces. A regular pentagon \(ABCDE\) is formed of five uniform rods each of weight \(W\) freely jointed at their extremities. It is freely suspended from \(A\) and is maintained in its regular pentagonal form by a light rod joining \(B\) and \(E\). Prove that the stress in this rod is \(W\cot 18^\circ\).
A particle is projected freely under gravity: prove that its path is a parabola and that its velocity at any point is that due to a fall from rest at the level of the directrix. A stone is projected from a point \(P\) on the ground over a house so as just to clear the tops of the walls and the ridge of the roof; the breadth of the house is \(2a\), the height of each wall \(l\), and the height of the ridge \(h+l\). Find the position of \(P\) and the velocity of projection.
State the principle of the conservation of linear momentum. A particle of mass \(m\) lies on a smooth horizontal plane and is connected by smooth light inextensible strings, both taut, with particles of masses \(m'\) and \(m''\) also lying on the plane, the angle between the strings being \(2\alpha\). A blow is given to \(m\) in a direction bisecting the angle \(2\alpha\) so as to jerk the other masses into motion. Shew that the mass \(m\) begins to move in a direction \(\tan^{-1}\left(\frac{(m'-m'')\sin\alpha\cos\alpha}{m+(m'+m'')\sin^2\alpha}\right)\) with the bisector of the angle between the strings. Also find the kinetic energy of the system.
Two small heavy rings of masses \(m, m'\) are connected by a light rod, and slide upon a smooth vertical circular wire of radius \(a\), the rod subtending an angle \(\alpha\) at the centre; prove that the motion is the same as that of a simple pendulum of length \[ a(m+m')(m^2+2mm'\cos\alpha+m'^2)^{-1/2}, \] and find the pressures of the rings on the wire in any position if the system started from rest when the rod was vertical.