Express \(\displaystyle\frac{1}{(1-x)(1-x^2)}\) as the sum of three partial fractions, and shew that the coefficient of \(x^n\) in the expansion in ascending powers of \(x\) is \[ \frac{1}{2}\left(n+1+\cos^2\frac{n\pi}{2}\right). \] Shew also that the coefficient of \(x^p y^q\) in the expansion of \((x+y+xy)^n\) is \[ \frac{n!}{(n-p)!(n-q)!(p+q-n)!}. \]
Eliminate \(\theta, \phi\) from the equations \begin{align*} x\cos\frac{\theta-\phi}{2} &= a\cos\theta\cos\frac{\theta+\phi}{2}, \\ y\cos\frac{\theta-\phi}{2} &= b\cos\phi\sin\frac{\theta+\phi}{2}, \\ a^2\cos^2\theta - b^2\cos^2\phi &= c^2. \end{align*}
Find the radius of the circumcircle of a triangle in terms of the sides. Points are taken on the respective sides \(a,b,c\) of a triangle at distances \(p,q,r\) from the middle points of the sides in the same sense round the triangle. Shew that if the perpendiculars to the sides through these points are concurrent, \[ ap+bq+cr=0. \]
Express \(\log(-2)\) and \(\sin^{-1}(2)\) in the form \(a+ib\), where \(a,b\) are real. If \(u = \displaystyle\frac{z^2+az+b}{z^2+a'z+b'}\) where \(a, a', b, b'\) are real, prove that the values of \(z\) for which \(u\) is real lie in the Argand diagram either on the real axis or on a circle whose centre is on the real axis.
Find the roots of the equation \(x^{2n+1}=1\). Prove that if \(\alpha = \pi/(2n+1)\), \[ (1+x)^{2n+1} - (1-x)^{2n+1} = 2(1+x^2\tan^2\alpha)(1+x^2\tan^2 2\alpha)\dots(1+x^2\tan^2 n\alpha). \]
Find the first differential coefficient of \(\tan^{-1}\left(a\tan\frac{x}{2}\right)\), and shew that the \(n\)th differential coefficient of \(e^{ax}\cos bx\) is \((a^2+b^2)^{n/2} e^{ax}\cos(bx+n\alpha)\), where \(\tan\alpha = b/a\). Deduce that the \(n\)th differential coefficient of \(\cos x \cosh x\) is \[ 2^{n/2}\left[\cos x \cosh x \cos\frac{n\pi}{4} - \sin x \sinh x \sin\frac{n\pi}{4}\right], \] and thence that \[ \cos x \cosh x = 1 + \sum_{1}^{\infty}(-1)^n \frac{4^n}{(4n)!} x^{4n}. \]
Shew that if \(f'(a)=0\) and \(f''(a)\) is positive, then \(f(x)\) is a minimum when \(x=a\). Isosceles triangles are described having their vertex at one focus of an ellipse and the other ends of the equal sides on the curve. Find the maxima and minima of the area of such triangles.
Find the equation of the tangent at any point of the curve \(x=f(t), y=F(t)\). Find the equation of the tangent at the point \(\theta\) of the curve \[ x=a(\theta+\sin\theta), \quad y=a(1-\cos\theta), \] and shew that if \(p\) is the perpendicular from the origin on the tangent and \(\psi\) the inclination of the tangent to the axis of \(x\), \[ p=2a\psi\sin\psi. \]
Sketch a few typical applications of the concepts of (i) the `line at infinity,' (ii) the `circular points at infinity,' in pure and analytical geometry. Justify the use of these concepts against any objections you can imagine made to it.
Prove (i) that any rational function can be expressed in the form \[ \Pi(x) + \sum_{\mu}\left\{\frac{A_{\mu,1}}{x-\alpha_{\mu}} + \frac{A_{\mu,2}}{(x-\alpha_{\mu})^2} + \dots + \frac{A_{\mu,r_{\mu}}}{(x-\alpha_{\mu})^{r_{\mu}}}\right\}, \] where \(\Pi(x)\) is a polynomial and the \(\alpha\)'s and \(A\)'s constants, (ii) that any real rational function can be expressed in the form \[ \Pi(x) + \sum_{\mu}\left\{\frac{A_{\mu,1}}{x-\alpha_{\mu}} + \dots + \frac{A_{\mu,r_{\mu}}}{(x-\alpha_{\mu})^{r_{\mu}}}\right\} + \sum_{\nu}\left\{\frac{B_{\nu,1}x+C_{\nu,1}}{x^2+2p_{\nu}x+q_{\nu}} + \dots + \frac{B_{\nu,s_{\nu}}x+C_{\nu,s_{\nu}}}{(x^2+2p_{\nu}x+q_{\nu})^{s_{\nu}}}\right\}, \] where \(\Pi(x)\) is a real polynomial and the \(\alpha\)'s, \(A\)'s, etc. real constants, and (iii) that each of these expressions is unique of its kind.