Explain briefly the method of mathematical induction and give an illustration of its use. Prove by induction that the sum of \(n\) terms of the series \[ x + (x-2)x + \frac{(x-4)x(x-1)}{2!} + \frac{(x-6)x(x-1)(x-2)}{3!} + \dots \] is \[ \frac{x(x-1)\dots(x-n+1)}{(n-1)!}. \]
An observer sees an aeroplane due N. at an elevation of \(10\frac{1}{2}^\circ\). Two minutes later he sees it N.E. at the same angular elevation. It is known to be going due E. at a speed of 60 m.p.h. Show that it was rising at a rate of 412 feet per minute.
From the points of contact of the inscribed circle with the sides of a triangle perpendiculars are let fall on the line joining the incentre \(I\) to the circumcentre \(S\). Prove that the algebraic sum of these perpendiculars vanishes, and that the algebraic sum of the distances from \(I\) of the feet of these perpendiculars is \(\frac{r}{R}(IS)\), where \(r, R\) are the radii of the inscribed and circumscribed circles.
Express \(\tan n\theta\) in terms of \(\tan\theta\), where \(n\) is an integer. Show that \[ \sum_{s=0}^{2n-1} \tan\left(\theta+\frac{s\pi}{2n}\right) = -2n\cot 2n\theta. \]
Find the equation of the tangent at the point \(\theta\) of the curve \[ x=a(\theta+\sin\theta), \quad y=a(1-\cos\theta), \] and show that if \(p\) is the perpendicular from the origin on the tangent and \(\psi\) the inclination of the tangent to the axis of \(x\), \[ p=2a\psi\sin\psi. \]
Show that the function \(\frac{\sin^2 x}{\sin(x+a)\sin(x+b)}\) (\(0 < a < b < \pi\)) has an infinity of minima equal to 0 and of maxima equal to \(-4\sin a\sin b / \sin^2(a-b)\).
Find the asymptotes of the curve \(x^2y+xy^2 = x^2-4y^2\), and trace it. Find the cubic which has \(x+y=1\) as an asymptote and touches both axes at the origin, the radii of curvature there being 1 and 2 units in length.
Find the integrals: \[ \int \frac{dx}{(x-2)\sqrt{x^2+2x+3}}, \quad \int_0^a x^2(\log x)^2 dx, \quad \int_c^b x^2(x-a)^{\frac{1}{2}}(b-x)^{\frac{1}{2}}dx, \] where \(c\) is a constant. Find the length of the spiral \(r=3\theta\) from \(\theta=0\) to \(\theta=\frac{\pi}{3}\).
Prove geometrically that \(\tan\theta = \csc 2\theta - \cot 2\theta\). If \(\alpha+\beta+\gamma = \frac{\pi}{3}\), prove that \begin{align*} \cos(\beta+\gamma-2\alpha)\sin(\alpha+2\beta) + \cos(\gamma+\alpha-2\beta)\sin(\beta+2\gamma) + \cos(\alpha+\beta-2\gamma)\sin(\gamma+2\alpha) \\ = 4\sin(\alpha+2\beta)\sin(\beta+2\gamma)\sin(\gamma+2\alpha). \end{align*}
If P is the orthocentre of the triangle ABC, O its circumcentre and I its incentre, prove that