(i) Given that \(\alpha\) and \(\beta\) are the roots of \[ x^2 - px + q = 0, \] form the equation whose roots are \(\alpha^3 - \frac{1}{\beta^3}\), \(\beta^3 - \frac{1}{\alpha^3}\). (ii) Given that the equation \[ x^n - ax^2 + bx - c = 0, \quad (c \neq 0, n > 2), \] has a thrice repeated root \(\xi\), establish the relations \[ \xi = \frac{(n-1)b}{2(n-2)a} = \frac{2nc}{(n-1)b}, \] and \[ \xi^n = \frac{2c}{(n-1)(n-2)}. \]
Find the sum of the series \[ x+2^2x^2+3^2x^3+4^2x^4+\dots+n^2x^n. \] Hence, or otherwise, evaluate \[ S_n = 1-2^2+3^2-\dots+(-1)^{n-1}n^2. \]
Given that \(a_r, b_r\) and \(c_r\) are all real and positive numbers for \(r=1, 2, \dots, n\), and that \[ a_r^2 = b_r^2+c_r^2, \quad r=1, 2, \dots, n, \] \[ A_n = \sum_{r=1}^n a_r, \quad B_n = \sum_{r=1}^n b_r, \quad C_n = \sum_{r=1}^n c_r, \] prove by induction, or otherwise, that for \(n \ge 1\), \[ A_n^2 \ge B_n^2 + C_n^2. \]
Solve completely the equation \[ \sin 3x = \cos 4x. \] Hence, or otherwise, find all the roots of \[ 8y^4+4y^3-8y^2-3y+1=0. \]
A tetrahedron \(ABCD\) has edges of lengths \(AB=AC=AD=a\), and \(BC=CD=DB=b\). A sphere is inscribed in this tetrahedron so that it touches the four faces. Find the radius of this sphere in terms of \(a\) and \(b\).
The normal at the point \(P\) on the parabola \(y^2=4ax\) meets the parabola again in \(Q\), and \(R\) is the pole of \(PQ\). The chord through \(P\) and the focus \(S\) meets the parabola again in \(T\). Prove that \(RT\) is parallel to the axis of the parabola, and also that \(PR\) is bisected by the directrix. Find the locus of \(R\) as \(P\) varies on the given parabola.
The polar of the point \(P\) with respect to the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] meets the ellipse in the points \(Q\) and \(R\). Given that \(QR\) is of constant length \(2c\), prove that the locus of \(P\) is \[ \left(\frac{x^2}{a^2} + \frac{y^2}{b^2} - 1\right)\left(\frac{a^2y^2}{b^4} + \frac{b^2x^2}{a^4}\right) = c^2 \left(\frac{x^2}{a^4} + \frac{y^2}{b^4}\right). \]
Prove that, in general, one conic of the confocal system \[ \frac{x^2}{a+\lambda} + \frac{y^2}{b+\lambda} = 1, \] where \(\lambda\) is a parameter, touches the line \(x \cos \alpha + y \sin \alpha = p\), and find the co-ordinates of the point of contact. Find also the equation of the conic of the confocal system which has the given line as a normal.
Prove that the inverse of a circle is either a circle or a straight line. Prove also that the angle at which two curves cut is unaltered by inversion. Given a coaxial system of circles intersecting in two real points, prove that there is at least one circle of the system orthogonal to a given circle, and discuss the conditions that there should be more than one such circle of the system. Prove also that there are either none, one, or two real circles of the system which touch a given circle, and discuss the conditions for each case.
Prove Desargues' theorem, that if two triangles in the same plane are in perspective from a point then their corresponding sides intersect in three collinear points (the axis of perspective). Prove that if three triangles in the same plane are in perspective, then the three axes of perspective of the triangles taken in pairs meet in a point. (The converse of Desargues' Theorem may be assumed.)