Shew how to determine the asymptotes of an algebraic curve, including the cases in which the curve has (i) asymptotes parallel to an axis, (ii) a pair of parallel asymptotes. Find the asymptotes of the curve \[ 2x(x+y)^2 - (x+y)^2(x-2y)+x=0. \]
Define the curvature at any point of a curve, and obtain that of the curve \(x=f(t), y=F(t)\) at the point \(t\). Find the radius of curvature at any point of the curve \[ x^2y = x^3+y^3. \]
Integrate with respect to \(x\) \[ \frac{1}{1+x+x^2}, \quad \frac{1}{(x+1)\sqrt{x^2+x+2}}, \quad x\sin 2x\sin 3x. \] By means of the substitution \[ \tan\tfrac{1}{2}\theta = \sqrt{\frac{1+e}{1-e}}\tan\tfrac{1}{2}\phi, \] evaluate \[ \int_0^\pi \frac{d\theta}{(1+e\cos\theta)^2}. \]
Explain the method of integration by parts, and shew that if \(\int \phi(x)\,dx\) is known then \(\int \phi^{-1}(x)\,dx\) can be found, where \(\phi^{-1}(x)\) is the inverse function corresponding to \(\phi(x)\).
Find formulae of reduction for \[ \int \sin^n x\,dx, \quad \int x(1+x^2)^n\,dx, \] where \(n\) is positive. If \[ f(m,n) = \int_0^{\frac{\pi}{2}} \cos^m x \cos nx\,dx, \] prove that \[ (m+n)f(m,n)=mf(m-1,n-1). \]
Find the area of a loop of the curve \[ (x^2+4y^2)^2 = x^2-9y^2. \]
If \begin{align*} a(x+y+b)+x^2y^2+bxy(x+y) &= 0, \\ a(z+x+b)+z^2x^2+bzx(z+x) &= 0, \end{align*} and \(y\) and \(z\) are unequal, prove that \[ a(y+z+b)+y^2z^2+byz(y+z)=0. \]
Prove that if \(0
Prove that the number of ways in which \(n\) different letters can be arranged in a row is \(n!\). Prove also that when they are arranged in a row in any way they can be divided up into \(r\) words in \(^{n-1}C_{r-1}\) ways, so that the total number of sentences of \(r\) words and \(n\) letters which can be formed from them is \[ n!(n-1)!/(n-r)!(r-1)!. \]
A coin is to be tossed twice; what is the chance that heads will turn up at least once? Point out the error in the following solution, given by D'Alembert: "Only three different events are possible; (i) heads the first time, which makes it unnecessary to toss again; (ii) tails the first time and heads the second, (iii) tails both times. Of these three events two are favourable; therefore the required chance is \(\frac{2}{3}\)."