Prove that \(v^2/r\) is the acceleration towards the centre of a circle of radius \(r\) when a particle is describing it with velocity \(v\). A smooth ring of mass \(m\) slides on a wire bent into the form of a circle of radius \(r\) which is made to rotate about a vertical diameter with uniform angular velocity \(\omega\). Find the position of relative rest of the ring on the wire and shew that the pressure between the ring and the wire is then \(m\omega^2r\).
Solve the equations:-
Prove that if \(x_r\) denotes \(x(x-1)(x-2)\dots(x-r+1)\), \[ (x+y)_n = x_n + n x_{n-1}y_1 + \frac{n(n-1)}{2!}x_{n-2}y_2+\dots+nx_1y_{n-1}+y_n. \] Shew that \[ \frac{1}{x+1}+\frac{1}{x+2}+\dots+\frac{1}{x+n} \] \[ = \frac{n}{x+n} + \frac{n(n-1)}{2(x+n)(x+n-1)} + \frac{n(n-1)(n-2)}{3(x+n)(x+n-1)(x+n-2)}+\dots \text{ to } n \text{ terms.} \]
Find the general term in the series \(1+2x+3x^2+8x^3+9x^4+38x^5+\dots\), it being assumed that the relation between successive coefficients (denoting the coefficient of \(x^r\) by \(p_r\)) is of the form \(ap_r+bp_{r+1}+cp_{r+2}+dp_{r+3}=0\). Also find the sum of the series to infinity, when \(x=\frac{1}{4}\).
Shew that the series \(\frac{1}{1^{1+\kappa}}+\frac{1}{2^{1+\kappa}}+\frac{1}{3^{1+\kappa}}+\dots\) converges only if \(\kappa > 0\). Discuss the convergence of the series whose \(n\)th term is \(\frac{n^a}{(n+1)^b}\), where \(a, b\) are given positive numbers.
Shew how to find points representing the sum and the product of two complex numbers whose points are given on the Argand diagram. Shew that the modulus of the arithmetic mean of two complex numbers is greater than the modulus of their geometric mean, if the origin lies inside the rectangular hyperbola whose foci represent the complex numbers.
Shew that \(\cos\frac{A}{2} = \pm\frac{1}{2}\sqrt{1+\sin A} \pm \frac{1}{2}\sqrt{1-\sin A}\), and determine the signs to be taken when \(A=430^\circ\).
If \(A+B+C=\pi\), prove that
The base \(BC\), the angle \(A\) and the height of \(A\) above \(BC\) are given for a triangle \(ABC\). Give rational formulae to determine the other sides and angles. Work out the case where \(BC=18\) feet, the height of \(A\) above \(BC\) is 12 feet, and the angle \(A\) is \(55^\circ\).
Prove that \(r=R(\cos A+\cos B+\cos C-1)\), where \(r, R\) are the radii of the incircle and circumcircle of \(ABC\). Shew that if the line joining the centres of the incircle and nine point circle of a triangle is perpendicular to one of the sides, either the triangle is isosceles or the sides are in arithmetical progression.