Find the condition that, if two straight lines are represented by the general equation of the second degree in areal coordinates, the two straight lines shall be parallel.
Prove that any two conics which intersect in four real points have a common self-conjugate triangle. Prove that the polars with respect to two conics of a point on a given straight line intersect on a conic circumscribing the triangle to which the two conics are self-conjugate.
Prove that, if \(a+b+c=0\), and no two of \(a, b, c\) are equal, constants \(A, B, C\) can be found to make the equation \[ A(x-a)^3 + B(x-b)^3 + C(x-c)^3 + x = 0 \] true for all values of \(x\), and determine their values.
Solve the equations:
Shew by induction or otherwise that the sum of \(n\) terms of the series \[ 1 + \frac{n-1}{n-\frac{1}{2}} + \frac{(n-1)(n-2)}{(n-\frac{1}{2})(n-\frac{3}{2})} + \dots \text{ is } 2n-1. \]
Prove that when \(x\) is increased without limit the expression \((1+1/x)^x\) has a finite limit. Prove that \[ \frac{1}{3} + \frac{x}{4.1} + \frac{x^2}{5.1.2} + \frac{x^3}{6.1.2.3} + \dots = \frac{1}{x^2}\{e^x(x^2-2x+2)-2\}. \]
Explain how \(\sqrt{13}\) can be expanded as a simple continued fraction. Shew that, if \(p_n/q_n\) is the \(n\)th convergent, \(p_4/q_4=48/13\); and prove the relations \[ p_{2n+2}=10p_{2n}-p_{2n-2}, \quad q_{2n+2}=10q_{2n}-q_{2n-2}. \]
Prove that \[ \cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta, \] where \(\alpha, \beta\) are angles of any magnitude. Express \[ \cos^2 2\alpha + \cos^2 2\beta + \cos^2 2\gamma + 2\cos 2\alpha \cos 2\beta \cos 2\gamma - 1 \] as a product of four cosines.
Express the area of a triangle (1) symmetrically in terms of \(R\) the circumradius and the angles, (2) in terms of \(R\) and \(a, b\), two of the sides.
In the triangle \(ABC\), \(A=60^\circ\), \(b-c=4\), and the perpendicular distance of \(A\) from \(BC\) is 11. Prove that \(\sin\frac{1}{2}(B-C)=\frac{1}{4}\), and find the sides.