Problems

If the equation \[ x^5 + 5qx^3+5rx^2+t=0 \] has two equal roots, prove that either of them is a root of the quadratic \[ 3rx^2-6q^2x-4qr+t=0. \]

If \(a, b, c\) and \(d\) are all real, and if \((a^2+b^2+c^2)(b^2+c^2+d^2) = (ab+bc+cd)^2\), prove that \(a, b, c\) and \(d\) are in geometrical progression.

Prove that

1. [(i)] \(\dfrac{1}{2^3 \cdot 3!} - \dfrac{1 \cdot 3}{2^4 \cdot 4!} + \dfrac{1 \cdot 3 \cdot 5}{2^5 \cdot 5!} - \dots \text{ to infinity} = \dfrac{23}{24} - \dfrac{2}{3}\sqrt{2}\);
2. [(ii)] \(1+\dfrac{n}{m}+\dfrac{n(n-1)}{m(m-1)}+\dfrac{n(n-1)(n-2)}{m(m-1)(m-2)}+\dots \text{ to } n+1 \text{ terms} = \dfrac{m+1}{m-n+1}\), provided that \(m\) is not less than \(n\).

Prove that the integral part of \((\sqrt{3}+1)^{2n+1}\) is \((\sqrt{3}+1)^{2n+1} - (\sqrt{3}-1)^{2n+1}\). \item[(i)] Solve the equation \[ \sin 2x + \cos 2x + \sin x = \cos x. \] \item[(ii)] If the equation \[ p\cos 4\theta + q\sin 4\theta = r \] has solutions \(\alpha, \beta, \gamma, \delta\) none of which differ by a multiple of \(\pi\), prove that \[ \operatorname{cosec} 2\alpha + \operatorname{cosec} 2\beta + \operatorname{cosec} 2\gamma + \operatorname{cosec} 2\delta = 0. \]

The diagonals \(2a, 2b\) of a rhombus subtend angles \(\theta, \phi\) at a point whose distance from the centre of the rhombus is \(x\); prove that \[ b^2(x^2-a^2)^2 \tan^2\theta + a^2(x^2-b^2)^2 \tan^2\phi = 4a^2b^2x^2. \]

An infinite right circular cone of semi-vertical angle \(\alpha\) cuts a sphere in two circles; the diameter of the larger circle subtends an angle \(2\theta\) at the centre of the sphere. If \(c\) is the radius of the larger circle, prove that the area of the portion of the surface of the sphere lying outside the cone is \[ 4\pi c^2 \cos\alpha \cos(\theta-\alpha)\operatorname{cosec}^2\theta. \] \item[(i)] Sum to \(n\) terms \[ \sin^3\theta + \frac{1}{3}\sin^3 3\theta + \frac{1}{9}\sin^3 9\theta + \frac{1}{27}\sin^3 27\theta + \dots. \] \item[(ii)] Sum to infinity \[ \cos\theta - \frac{1}{2}\cos 3\theta + \frac{1 \cdot 3}{2 \cdot 4}\cos 5\theta - \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\cos 7\theta + \dots. \]

If \(\alpha=\pi/2n\), prove that \[ \frac{\sin 2\alpha \sin 4\alpha \sin 6\alpha \dots \sin(2n-2)\alpha}{\sin\alpha \sin 3\alpha \sin 5\alpha \dots \sin(2n-1)\alpha} = n. \]

(a) Give a geometrical construction for a circle through two given points which intercepts a given length on a given straight line. \item[] (b) \(A, O\) and \(B\) are three fixed points; two circles are drawn, one through \(A\) and \(O\), the other through \(O\) and \(B\), so as to cut at a constant angle. Find the locus of the other point of intersection.

Prove that the two tangents to an ellipse from an external point subtend equal angles at a focus. \(T\) is any point on the tangent to an ellipse at the extremity of its minor axis. Prove that the other tangent from \(T\) to the ellipse also touches the circle through \(T\) and the two foci.

Shew that two conics of a confocal system pass through an arbitrary point of the plane, and that one confocal touches an arbitrary line. Two straight lines meet on the line joining the foci, and two conics of the system are drawn touching the two lines in \(A\) and \(B\) respectively. Shew that the two lines which touch at \(A\) and \(B\) the other confocals through these points also meet on the line joining the foci.