Problems

\(z, w, a\) are complex numbers and \(a\) lies inside the unit circle in the Argand diagram and \[ w = \frac{z-a}{1-\bar{a}z} \] (where \(a\bar{a}=|a|^2\)). Shew that \(|z|<1\) implies \(|w|<1\) and conversely.

Find the values of \(x\) which give maxima and minima of \[ \sin x + \frac{1}{3}\sin 3x + \frac{1}{5}\sin 5x. \] Distinguish between the maxima and minima.

Prove that the coefficient of \(x^n\) in the expansion of \[ \frac{a}{ax^2-2bx+c} \] in ascending powers of \(x\), where \(ac>b^2\), is \[ \left(\frac{a}{c}\right)^{\frac{n+1}{2}} \frac{\sin(n+1)\theta}{\sin\theta}, \] where \(\sqrt{(ac)}\cos\theta=b\).

1. Obtain a reduction formula for \[ \int (\sec x)^n \,dx. \]
2. Shew that, if \(m\) and \(n\) are positive integers and \(n \ge 2\), then \[ \int_0^\infty \frac{x^m}{(1+x)^{m+n}} \,dx = \frac{m!(n-2)!}{(m+n-1)!}. \]

Evaluate

1. \(\displaystyle \int_0^a \frac{dx}{x+\sqrt{(a^2-x^2)}}\),
2. \(\displaystyle \int_0^{2\pi} \frac{\sin^2\theta \,d\theta}{a-b\cos\theta}\), where \(a>b>0\),
3. \(\displaystyle \int_1^\infty \frac{dx}{(1+x)\sqrt{x}}\).

Define the area of the surface of a body formed by the revolution of a curve about a straight line in its plane. A circular arc revolves about its chord; prove that the area of the surface generated is \(4\pi a^2(\sin\alpha-\alpha\cos\alpha)\), where \(a\) is the radius and \(2\alpha (<\pi)\) is the angular measure of the arc.

\(AB\) is a diameter and \(P\) any point of a circle \(S\). The tangent to \(S\) at \(P\) meets \(AB\) produced in \(T\). \(L\) is the mid-point of \(TP\) and \(H\) is the foot of the perpendicular from \(T\) on \(AP\) produced. Prove that \(HL\) is perpendicular to \(AB\).

Three points \(L, A, B\) are taken on a circle \(S\), and \(O\) is the mid-point of \(AB\). Prove that the tangents at \(A, B\) to the circles \(AOL, BOL\) meet in a point \(M\) of \(S\), and that the tangents at \(A, B\) to the circles \(AOM, BOM\) meet in \(L\).

Discuss briefly the process of inversion with respect to a circle. \(P_1, P_2\) are the points of contact of a common tangent of two circles \(C_1, C_2\) and \(L, M\) are the limiting points of the coaxal system determined by \(C_1, C_2\). \(P_1L\) meets \(C_1\) again in \(Q_1\), and \(P_2L\) meets \(C_2\) again in \(Q_2\). By inversion with respect to \(L\) or otherwise, prove that \(Q_1Q_2\) is a common tangent of \(C_1, C_2\).

Two variable points \(P(x,0)\) and \(P'(x',0)\) on the line \(y=0\) have their coordinates connected by the relation \[ axx' + bx + cx' + d = 0, \] where \(a, b, c, d\) are constant and \(ad \neq bc\). Show that there exist in general two points \(A\) and \(B\) on the line such that the cross-ratio \((APBP')\) has a constant value \(k\). Investigate the cases \(k=-1, 1\).