If \(\alpha, \beta\) are the values of \(x\) which satisfy \[ x^2y^2+1+a(x^2+y^2)+bxy=0 \] for any given value of \(y\), show that \[ \alpha^2\beta^2+1+A(\alpha^2+\beta^2)+B\alpha\beta=0, \] where \[ A = \frac{(1-a^2)^2}{ab^2}, \quad B = \frac{2(1-a^2)^2-b^2(1+a^2)}{ab^2}. \]
Given \(n\) letters, \(a,b,c \dots\) find the number of homogeneous products of \(r\) dimensions which can be formed of these letters and their powers. In how many ways can a batsman make 14 runs in six balls, not scoring more than 4 runs off any ball?
Find the coefficient of \(x^n\) in the expansion of \(\frac{3-x}{(2-x)(1-x)^2}\) in powers of \(x\). Find the sum of the series \(\sum_{n=2}^\infty \frac{n^2x^n}{n^2-1}\), when convergent.
Prove the law of formation of a convergent of the continued fraction \[ a_1 + \frac{1}{a_2+} \frac{1}{a_3+} \dots \] from the two convergents next preceding it. If \(p_n/q_n\) is the \(n\)th convergent of \(\frac{1}{2+} \frac{1}{3+} \frac{1}{4+} \dots\), prove that \(2p_{2n}=3q_{2n-1}\); also find \(p_9\).
Find the sum of the series \[ 1-x+u_2x^2+u_3x^3+\dots+u_nx^n+\dots, \] where \(n^2u_n+(2n-1)u_{n-1}+u_{n-2}=0\). Find \(u_n\) in terms of \(n\).
Prove geometrically that \(\tan(A+B)(1-\tan A\tan B) = \tan A+\tan B\), assuming that \(A+B<\pi/2\). If \(A+B+C=\pi\), prove that \[ \Sigma \tan A\cot B\cot C = \Sigma \tan A - 2\Sigma \cot A. \] Also if \(\tan\gamma = \frac{n\sin\alpha\cos\alpha}{1-n\sin^2\alpha}\), prove that \(\tan(\alpha-\gamma)=(1-n)\tan\alpha\).
With the usual notation prove that \[ r = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] If the perpendiculars from the vertices \(A,B,C\) of a triangle on the opposite sides be produced to \(A'B'C'\) so that \(AA'=BC, BB'=CA, CC'=AB\), show that the area of \(A'B'C'\) is \[ 4R^2(2\sin A\sin B\sin C - \cos A\cos B\cos C - 1). \]
Find the lengths of the diagonals of a quadrilateral inscribed in a circle, in terms of the sides. If a quadrilateral \(ABCD\) be inscribed in a circle and \(AB, DC\) meet in \(E\), and \(BC, AD\) in \(F\) and if \(AC, BD\) meet \(EF\) in \(G\) and \(H\), prove that \[ GH/EF = 2abcd(b^2-d^2)(a^2-c^2)/(a^2d^2-b^2c^2)(a^2b^2-c^2d^2), \] where \(a,b,c,d\) are the lengths of \(AB,BC,CD,DA\).
Prove that if \(x^2<1\), \[ \frac{\sin\theta}{1-2x\cos\theta+x^2} = \sin\theta+x\sin 2\theta+x^2\sin 3\theta+\dots. \] Express \(\frac{\sin n\theta}{\sin\theta}\) in a series of descending powers of \(\cos\theta\).
Sum to \(n\) terms the series whose \(r\)th term is