A particle slides down the surface of a smooth fixed sphere of radius \(a\) starting from rest at the highest point. Find where it will leave the sphere, and shew that it will afterwards describe a parabola of latus rectum \(\frac{16}{27}a\), and that it will strike the horizontal plane through the lowest point of the sphere at a distance \(\frac{5(\sqrt{5}+4\sqrt{2})a}{27}\) from the vertical diameter.
A horizontal board is made to perform simple harmonic oscillations horizontally, moving to and fro through a distance 30 inches and making 15 complete oscillations per minute. Find the least value of the coefficient of friction in order that a heavy body placed on the board may not slip.
State and prove the rule for finding the highest common factor of two rational integral functions of \(x\). Find two rational integral functions \(X\) and \(X'\) of \(x\) such that \[ X(x^3+5x^2+7x+4) - X'(x^2+2x+3) = 1. \]
Solve the equation \[ \frac{1}{\sqrt{a+x}-\sqrt{a}} + \frac{1}{\sqrt{a+x}+\sqrt{a}} = \frac{m}{\sqrt{a+x}-\sqrt{a-x}}. \] If \begin{align*} ax^2 &= 1/y+1/z, \\ by^2 &= 1/z-1/x, \\ cz^2 &= 1/x+1/y, \end{align*} prove that \[ abcx^2y^2z^2 = b+c-a, \] hence solve the equations.
Find the conditions that
Prove that \[ \log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots \] when \(|x|<1\). Prove that \[ (1+x)^{1-x} = 1+x+x^2+\frac{1}{2}x^3+\frac{1}{3}x^4+\dots. \]
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 \[ 4n-3\rho:2n=a:r. \]