(i) Prove that \(x=2\sin 10^\circ\) is a root of the equation \(x^3-3x+1=0\), and find the other two roots. (ii) If \(c = \cos^2\theta - \frac{1}{3}\cos^3\theta\cos 3\theta + \frac{1}{5}\cos^5\theta\cos 5\theta - \dots\) to infinity, prove that \[ \tan 2c = 2\cot^2\theta. \]
(i) If \(x+iy = a\cos(u+iv)+ib\sin(u+iv)\), where \(x,y,u,v,a\) and \(b\) are real quantities, and \(i\) denotes \(\sqrt{-1}\), shew that \[ \frac{x^2}{\cos^2 u} - \frac{y^2}{\sin^2 u} = a^2-b^2. \] (ii) A point \(A\) is taken within a circle of radius \(a\), at a distance \(b\) from the centre, and points \(P_1, P_2, \dots P_n\) are taken on the circumference so that \(P_1P_2, P_2P_3, \dots P_nP_1\) subtend equal angles at \(A\). Prove that \[ AP_1+AP_2+\dots+AP_n=(a^2-b^2)(AP_1^{-1}+AP_2^{-1}+\dots+AP_n^{-1}). \]
If perpendiculars are drawn from the orthocentre of a triangle \(ABC\) on the bisectors of the angle \(A\), shew that their feet are collinear with the middle point of \(BC\).
\(AA'\) is the major axis of an ellipse of which \(S, S'\) are the foci and \(P\) is any point on the curve. \(AR, A'R'\) are drawn parallel to \(SP, S'P\) respectively to meet the tangent at \(P\) in \(R, R'\). Prove that \[ AR+A'R' = AA'. \]
Shew that if \(AOA', BOB', COC'\) are chords of a conic, and \(P\) is any point on the conic, then the points of intersection of the straight lines \(BC, PA'\), of \(CA, PB'\), and of \(AB, PC'\) lie on a straight line through \(O\). What is the theorem obtained by reciprocating with respect to an arbitrary point?
Shew that the poles of a given line with respect to the conics touching four given lines lie on a straight line.
\(A_1, A_2, \dots, A_n\) are \(n\) points whose coordinates are \((x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\). \(A_1A_2\) is bisected at \(B_1\), \(B_1A_3\) is divided at \(B_2\) so that \(2B_1B_2 = B_2A_3\), \(B_2A_4\) is divided at \(B_3\) so that \(3B_2B_3 = B_3A_4\), and so on. Find the coordinates of \(B_{n-1}\).
Prove that if \(\lambda, \mu, \nu\) are such that \[ \lambda(ax^2+2hxy+by^2+2x) + \mu(a'x^2+2h'xy+b'y^2+2y) + 2\nu xy = 0 \] represents two straight lines, one of the lines passes through the origin, and the other touches the conic \[ (ax+b'y+2)^2 = 4a'bxy. \]
Prove that the locus of middle points of chords of the conic \(\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1\) which touch the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is the curve \[ \left\{\frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda}\right\}^2 = \frac{a^2x^2}{(a^2+\lambda)^3} + \frac{b^2y^2}{(b^2+\lambda)^3}. \]
A tangent is drawn at any point \(P\) of an ellipse cutting the axes \(CA, CB\) in \(M, N\) and the rectangle \(CMC'N\) is completed. If \(R\) be the foot of the perpendicular let fall from \(C\), the centre, upon \(CP\), shew that \(CR\) passes through the centre of curvature at \(P\) and that the locus of \(R\) is \[ (x^2+y^2)^2 = \{(ax)^{2/3}+(by)^{2/3}\} \{(ax)^{2/3}-(by)^{2/3}\}^2, \] \(CA, CB\) being the axes of \(x,y\) and \(2a, 2b\) the major and minor axes of the ellipse.