Determine the following:
Define the radius of curvature \(\rho\) at a point \(P\) of a plane curve and interpret its sign. Shew that the circle which passes through \(P\) and has there the same \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) as the curve has radius \(\rho\). Hence or otherwise shew that the radius of curvature at the origin of \[ x^4+y^4+x^3+y^3=2a(x+y) \] is \(a\sqrt{2}\).
Define the envelope of a system of curves and shew how it may be found. Prove that the tangents to a curve envelope the curve itself. Find the envelope of a system of coaxal ellipses all of the same area.
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 \(W\cot 18^\circ\).