Find the equation whose roots are the squares of the reciprocals of the roots of the equation \[ ax^2+2bx+c=0. \]
Prove that \[ {}_nC_r = \frac{n!}{r! (n-r)!}. \] In a certain examination 20 papers are set. These are divided into five groups A, B, C, D, E: A, B, C contain five papers each, D and E contain three and two respectively. A candidate may take the examination in one of the following ways:
Prove that \(e\) is an incommensurable number, and that \(e^x\) tends to infinity with \(x\) more rapidly than any power of \(x\).
Show that any surd can be converted into a continued fraction and prove that if \(a\) is positive \[ m + \cfrac{a}{m + \cfrac{a}{m + \dots}} = \frac{\sqrt{m^2+4a}+m}{2}. \]
Prove that \[ \sum_{s=1}^{n-1} \sin \frac{s\pi}{n} = \cot\frac{\pi}{2n}. \] Hence calculate \(\cot 7^\circ 30'\) in terms of the trigonometrical ratios of 45\(^\circ\) and 30\(^\circ\).
From both ends of a measured base \(AB\) the bearings \(CAB, CBA, C'AB, C'BA\) of two points \(C, C'\) are measured; the four points \(C, C', A, B\) lie in a horizontal plane. Find \(CC'\) in terms of the measured quantities. If \(AB=2\) miles, \(CAB=CBA=45^\circ\), \(C'AB=30^\circ\) and \(C'BA=60^\circ\), find \(CC'\).
For \(n\) any integer prove that \[ \cos n\theta + i\sin n\theta = (\cos\theta+i\sin\theta)^n. \] Prove that \[ \tan n\theta = \frac{n\tan\theta - {}_nC_3 \tan^3\theta + {}_nC_5\tan^5\theta - \dots}{1 - {}_nC_2 \tan^2\theta + {}_nC_4\tan^4\theta - \dots}. \]
Prove that \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}. \] Find \[ \frac{d^2(\sin^{-1}x)}{dx^2}, \quad \frac{d^3(\cos^2x\sin x)}{dx^3}. \]
For \(m=-2\) and \(m=-\frac{1}{2}\) show that the curve \(r^m = a^m \cos m\theta\) becomes a rectangular hyperbola and a parabola respectively. In the curve \(r^m = a^m \cos m\theta\) show that \(r^{m+1}=a^m p\), where \(p\) is the perpendicular on the tangent at \((r, \theta)\).
\(P\) is a point near the origin on the curve \(y=x^2\). If \(\rho\) is the radius of curvature at \(P\) and if \(R\) is the radius of the circle through \(P\) touching the curve \(y=x^2\) at the origin, show that as \(P\) approaches the origin \(\text{Lt}\left(\frac{\rho}{R}\right) = \frac{1}{2}\).