Sum the series \[ 1^3 - 2^3 + 3^3 - 4^3 + \dots - (2n)^3 + (2n+1)^3 \] and \[ \sum_{n=1}^\infty \frac{2.5.8. \dots (3n-1)}{3.6.9. \dots 3n} \left(\frac{1}{3}\right)^n. \]
For what values of \(r\) does the equation \[ x^3 - 3x + r = 0 \] have three distinct real roots? Solve completely the equation \[ 4x^3 - 27a^2(x-a) = 0. \]
Prove that the arithmetic mean of \(n\) positive numbers is greater than their geometric mean, unless the numbers are all equal. Prove that, if \(a, b\) and \(c\) are all positive, \[ (a+b-c)(b+c-a)(c+a-b) \le abc. \]
Factorize the determinants \[ \begin{vmatrix} x & y & x & y \\ y & x & y & x \\ -x & y & x & y \\ y & -x & y & x \end{vmatrix} \quad \text{and} \quad \begin{vmatrix} 1 & x & x^4 \\ 1 & x-y & (x-y)^4 \\ 1 & y & y^4 \end{vmatrix}. \]
The base of a pyramid is a regular hexagon and the slant-faces are six equal isosceles triangles of vertical angle \(\frac{1}{4}\pi\). Show that the angle between the base and any other face is \(\cos^{-1}\{\sqrt{6}-\sqrt{3}\}\) and that the angle between adjacent slant-faces is \(\cos^{-1}(3\sqrt{2}-5)\).
A pair of straight lines \[ ax^2 + 2hxy + by^2 = 0, \quad \dots(1) \] and a point \((\xi, \eta)\) are given. Interpret geometrically the equation \[ ax^2+2hxy+by^2 + \lambda(x^2+y^2-x\xi-y\eta) = 0, \quad \dots(2) \] where \(\lambda\) is a parameter. If \(\lambda\) is given a certain non-zero value, (2) also will represent a pair of straight lines: what will these be? The point \((\xi, \eta)\) is allowed to vary in such a way that the distance between the feet of the perpendiculars from it on the lines (1) is a constant, \(2c\). Show that its locus is \[ (x^2+y^2)(ab-h^2) + c^2\{(a-b)^2+4h^2\} = 0. \]
Show that, if \(P\) and \(P'\) are inverse points with respect to a circle \(S\), any circle through \(P\) and \(P'\) will cut \(S\) orthogonally. A series of circles \(S_1, S_2, \dots\) all pass through the two points \(A, B\); and \(C\) is any point in their plane. A sequence of points \(P_0, P_1, P_2, P_3, \dots\) is defined as follows: \(P_0=C\) and, for \(n \ge 1\), \(P_n\) is the inverse of \(P_{n-1}\) in the circle \(S_n\). Show that the points \(P_0, P_1, P_2, \dots\) are concyclic.
\(P\) is a point on an ellipse whose foci are \(S, S'\). Prove that \(SP, S'P\) make equal angles with the tangent to the ellipse at \(P\). A line through \(S'\) is drawn parallel to \(SP\) to meet the ellipse at \(P'\). If the tangents at \(P\) and \(P'\) meet \(S'P'\) and \(SP\) in \(Q'\) and \(Q\) respectively, prove that \(QQ'\) is parallel to \(PP'\).
Find the equation of the normal and the coordinates of the centre of curvature at the point \((at^2, 2at)\) of the parabola \[ y^2=4ax. \quad \dots(1) \] Show that there are just two real points \(P\) and \(P'\) on (1), for which the corresponding centres of curvature \(C\) and \(C'\) also lie on (1). If the circle of curvature of (1) at \(P\) meets (1) again in the point \(D\), show that the normals to (1) at \(P, C'\) and \(D\) are concurrent.
A circle meets the rectangular hyperbola \(H\) in the points \(A, B, C, D\). Two other circles are drawn, the first meeting \(H\) in \(A, B, P, Q\) and the second meeting \(H\) in \(C, D, R, S\). Prove that the points \(P, Q, R, S\) are concyclic.