Sum the series \[ \frac{1}{1.2.4} + \frac{1}{2.3.5} + \frac{1}{3.4.6} + \dots \text{ to } n \text{ terms.} \] Given that \[ 2+5x+13x^2+35x^3+\dots \] is a recurring series with a scale of relation of the form \(1+ax+bx^2\), find \(a\) and \(b\) and also find the sum of the series to \(n\) terms.
Find from first principles the differential coefficients of \(x^n\) and \(\cos^{-1}x\). Find the \(n\)th differential coefficient of \[ \frac{1}{x^2-3x+2}. \]
If \(f(x,y)=0\), prove that \[ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}\frac{dy}{dx}=0. \] If \[ ax^2+2hxy+by^2=1, \] prove that \[ x\left(\frac{d^2y}{dx^2}\right)^{\frac{1}{3}} + y\left(\frac{d^2x}{dy^2}\right)^{\frac{1}{3}} + (ab-h^2)^{\frac{1}{3}} = 0. \]
Prove the formulae for the radius of curvature \(\rho\) of a curve \[ \text{(i) } r\frac{dr}{dp}, \quad \text{(ii) } \frac{\{r^2+\left(\frac{dr}{d\theta}\right)^2\}^{\frac{3}{2}}}{r^2+2\left(\frac{dr}{d\theta}\right)^2 - r\frac{d^2r}{d\theta^2}}. \] If \(n\) is the length of the normal at a point of the curve \(r=a(1+\cos\theta)\) intercepted between the curve and the initial line, prove that \[ 4n-3\rho:2n=a:r. \]
Evaluate the integrals \[ \int\frac{dx}{(x^2+a^2)^2}, \quad \int\frac{dx}{(x^2-1)\sqrt{x^2+x-1}}, \quad \int_0^{\frac{\pi}{2}}\frac{dx}{\sqrt{2x+1}-\sqrt{x+2}}, \quad \int_0^{\frac{\pi}{2}}\sin^{2n}xdx. \] If \[ u_n = \int_0^{\frac{\pi}{m}} x^n\sin mxdx, \] prove that \[ u_n = \frac{n\pi^{n-1}}{m^2 2^{n-1}} - \frac{n(n-1)}{m^2}u_{n-2} \] if \(m\) is an integer of the form \(4r+1\).
Trace the curve \[ y^2(1+x^2)-4y+1=0, \] and find its area.
Draw a diagram to illustrate the truth of the algebraical identity \[ (a-b)(a+b) = a^2-b^2. \] \item[*3.] Prove that the angle at the centre of a circle is double any angle at the circumference standing on the same arc.
Prove that the internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the bisected angle.
A milkman buys eggs at 10 for 3s. and sells them at \(4\frac{1}{2}\)d. each; what is his profit per cent.?
A man sells out £800 of Swedish Bonds at \(118\frac{1}{4}\) and reinvests the proceeds in five per cent. National War Bonds at par; what income does he now receive?