If \(\alpha\) is a root of \(ax^2+2bx+c=0\) and \(\beta\) a root of \(a'x^2+2b'x+c'=0\), find the equation whose roots are the different values of \(\alpha/\beta\).
If \(p_r/q_r\) is the \(r\)th convergent to the continued fraction \[ \frac{1}{a+} \frac{1}{b+} \frac{1}{a+} \frac{1}{b+} \dots, \] shew that \(p_{2n+1}=q_{2n}\) and \(ap_{2n}=bq_{2n-1}\).
Prove that, if \(n\) is integral, \[ \sin n\theta = \cos^n\theta \left\{n\tan\theta - \frac{n(n-1)(n-2)}{3!}\tan^3\theta + \dots\right\}. \] Find the roots of the equation \[ (1+x^2)^4 = 8x(1-7x^2+7x^4-x^6), \] and shew that the roots of the equation \(1-4x-6x^2+4x^3+x^4=0\) are the values of \(\tan(4r+1)\pi/16\), (\(r=0, 1, 2, 3\)).
Find the \(n\)th differential coefficients of \(\tan^{-1}x\) and \(x e^x \cos x\) with respect to \(x\).
Shew how to find the stationary values of a function \(f(x)\) and how to discriminate between the maxima and minima. Prove that, if \(Y\) is the foot of the perpendicular from the centre of an ellipse upon the tangent at \(P\), the maximum length of \(PY\) is \(a-b\).
Prove the formula for radius of curvature \(\rho = r \frac{dr}{dp}\). In the curve \(r^n=a^n\cos n\theta\) the angle between the radius vector to \((r, \theta)\) and the radius to the centre of curvature at that point is \(\psi\). Prove that \(n\tan\psi = \tan n\theta\) and that the distance of the centre of curvature from the origin is \(nr \sec\psi/(n+1)\).
If \(z^2 = (y^2-nx)^2\), verify that \[ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x} \quad \text{and} \quad \frac{\partial z}{\partial x}\frac{\partial z}{\partial y} - \frac{\partial z}{\partial x \partial y}\left(x\frac{\partial z}{\partial y}\right) = \frac{9}{4}n^2. \]
The normals from a point to the cubic \(ay^2=x^3\) make angles with the axis of \(x\) whose sum is \(\alpha\). Shew that the locus of the point is \(y=(x+\frac{2}{3}a)\tan\alpha\).
Find formulae of reduction for \(\int \sin^n x dx\) and \(\int (ax^2+2bx+c)^{-n}dx\).
Shew that the area of a closed curve is \(\frac{1}{2}\int(xdy-ydx)\) taken round the curve. Prove that the area of a loop of the curve \(a^2y^2=4x^2(a^2-x^2)\) is \(\frac{4}{3}a^2\).