Continued fractions and their use for approximation to irrational numbers.
The determination of the nature and position of the quadric represented by the general equation of the second degree \[ (a, b, c, d, f, g, h, u, v, w \gamma x, y, z, 1)^2 = 0. \]
Give a proof of Cauchy's Theorem. Indicate some of its simplest applications (as for example in the theory of doubly periodic functions).
Fourier's Series.
Show that the eliminant of the equations \begin{align*} x+y+z &= 0 \\ \frac{x^2}{a} + \frac{y^2}{b} + \frac{z^2}{c} &= 0 \\ ayz + bzx + cxy &= 0 \end{align*} is \((b+c)(c+a)(a+b) = a^3+b^3+c^3+5abc\).
Show that if \(x^n + a_1 x^{n-1} + \dots + a_n = 0\), where the \(a\)'s are rational numbers, then any polynomial in \(x\) with rational coefficients can be expressed as a polynomial of degree not greater than \((n-1)\) with rational coefficients. Show that if \(x^4 + 4ax^3 + 6bx^2 + 4cx + d = 0\) and \(y=x^2+2ax\), then \(y\) satisfies a quadratic equation provided \(c=3ab-2a^3\) and that the values of \(y\) are real if \(c^2 \geq a^3d\).
\(A_1, A_2, \dots, A_n\) are \(n\) places in succession through which a road passes and \(P, Q\) are places on opposite sides of the road. From each of \(A_1, A_2, \dots, A_n\) a road passes to \(P\) and a road to \(Q\). Find the number of ways of going from \(P\) to \(Q\) by the roads, and also the number of ways of going from \(P\) to \(Q\) without going through \(A_r\), where \(r\) is not \(1\) or \(n\).
Find \(a, b, c, d\) so that the coefficient of \(x^n\) in the expansion of \[ \frac{a+bx+cx^2+dx^3}{(1-x)^4} \] may be \(n^3\). Find the coefficient of \(x^n\) in the expansion of \((1+\lambda x+x^2)^n\), where \(n\) is a positive integer; and if \(\lambda=2\cos\theta\), deduce that \[ c_0^2 + c_1^2 \cos 2\theta + \dots + c_n^2 \cos 2n\theta = n! \cos n\theta \left\{\frac{(2\cos\theta)^n}{n!} + \frac{(2\cos\theta)^{n-2}}{1! (n-2)!} + \frac{(2\cos\theta)^{n-4}}{2! (n-4)!} + \dots \right\}, \] where the indices of \(\cos\theta\) are positive numbers or zero, and \((1+x)^n = c_0+c_1x+\dots+c_nx^n\).
Two equilateral triangles \(ABC, ABD\) have a side \(AB\) common and their planes at right angles. Find (i) the acute angle between \(AC, AD\) and (ii) the distance between the mid-points of \(AC, BD\).
Find the sums of the infinite series