Prove that, if \(f(a)=0\) and \(\phi(a)=0\) and if \(f'(a)\), \(\phi'(a)\) do not both vanish, \[ \operatorname{Lt}_{x \to a} \frac{f(x)}{\phi(x)} = \frac{f'(a)}{\phi'(a)}. \] Evaluate \[ \operatorname{Lt}_{x \to n} \frac{1}{(n^2-x^2)^2}\left\{ \frac{n^2+x^2}{nx} - 2\sin\frac{n\pi}{2}\sin\frac{x\pi}{2} \right\} \] where \(n\) is an odd integer.
Prove the formulae for the radius of curvature of a curve \[ \rho = \frac{r dr}{dp} = \frac{\left\{r^2+\left(\frac{dr}{d\theta}\right)^2\right\}^{\frac{3}{2}}}{r^2+2\left(\frac{dr}{d\theta}\right)^2 - r\frac{d^2r}{d\theta^2}}. \] Any point on a curve is taken as pole and the tangent at it is the initial line, prove that the approximate equation of the curve in the neighbourhood of the origin is \(r=2\rho\theta + \frac{4}{3}\rho\frac{d\rho}{ds}\theta^2\), where \(\rho\) and \(\frac{d\rho}{ds}\) are the values at the origin.
Prove that two curves intersect at the same angle as their inverses. \par Shew that if a point is such that when it is taken as the origin of inversion two given circles invert into equal circles, the locus of the point is a circle.
Prove that the series \[ 1^2 + 2^2x + 3^2x^2 + 4^2x^3 + \dots \] is convergent when \(x\) lies between \(-1\) and \(+1\). \par Shew that if \(x=0.9\) the sum to infinity is 54,100.
Put into real partial fractions
Prove that with the usual notation \[ Rr = \frac{abc}{4s} \quad \text{and} \quad r_1+r_2+r_3=r+4R. \] Shew that the radius of the inscribed circle of a triangle whose sides are \(a+x\), \(b+x\) and \(c+x\), where \(x\) is small, is approximately \[ r + \frac{2R-r}{2s}x. \]
Prove that, when \(n\) is a positive integer, \[ \tan n\theta = \frac{n\tan\theta - \frac{1}{3!}n(n-1)(n-2)\tan^3\theta+\dots}{1-\frac{1}{2!}n(n-1)\tan^2\theta+\frac{1}{4!}n(n-1)(n-2)(n-3)\tan^4\theta-\dots}. \] Find an equation whose roots are the tangents of \(\theta, 2\theta, 4\theta, 5\theta, 7\theta\) and \(8\theta\) where \(\theta=20^\circ\), and shew that \(\tan 20^\circ \tan 40^\circ \tan 80^\circ = \sqrt{3}\).
The conic \(ax^2+2hxy+by^2+2gx+2fy+c=0\) is cut by the straight line \(lx+my+1=0\) in \(P\) and \(Q\). Shew that the area of the triangle \(OPQ\), where \(O\) is the origin, is \[ \frac{\{-\left(Al^2+2Hlm+Bm^2+2Gl+2Fm+C\right)\}^{\frac{1}{2}}}{am^2-2hlm+bl^2}, \] where \(A, B, C, F, G, H\) are the minors of \(a, b, c, f, g, h\) in \[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}. \]
A family of conics circumscribe the triangle \(ABC\) and pass through its centroid \(G\). Tangents to one of these conics at \(A, B, C\) cut the opposite sides in \(D, E, F\). Shew that the locus of the centroid of the triangle \(DEF\) is a cubic curve passing through \(G\) and touching the sides of the triangle \(ABC\) at their middle points.
Find the area of a loop of the curve \(y^2=x^2-x^4\). \par Find also the distance from the origin of the centre of gravity of the area included within the loop.