Prove that, if \(bc+ca+ab=0\), then \[ \Sigma a^5 = \Sigma(a^2)\{\Sigma(a^3)+2abc\}. \]
Prove that, if \(x\) is large, \[ \left(1+\frac{1}{x}\right)^{x+\frac{1}{2}} = e\left(1+\frac{1}{12x^2}+\dots\right). \]
Prove that, if \(x\) denote any convergent of the continued fraction \[ \frac{1}{a+} \frac{1}{b+} \frac{1}{a+} \frac{1}{b+} \dots, \] then the next convergent is \(a+a/bx\).
Prove that, if the incircle of a triangle passes through the circumcentre, then \[ \cos A + \cos B + \cos C = \sqrt{2}. \]
A uniform heavy beam rests across and at right angles to two horizontal rails which support the beam at its points of trisection. Shew that, if the coefficient of friction is the same at both contacts and a gradually increasing force is applied to the beam at one end parallel to the rails, equilibrium will be broken by sliding on the nearer rail only.
A uniform hemisphere of weight \(W\) and radius \(a\) is placed symmetrically on top of a fixed sphere of radius \(b\), the curved surfaces being in contact and sufficiently rough to prevent sliding. Shew that if the hemisphere is rolled through a small angle \(\theta\) the gain of potential energy is approximately \[ \frac{1}{2}W(3b-5a)a\theta^2/(a+b), \] and deduce the condition for stability.
A weight of 200 lb. hanging from a rope is raised by a force which starts at 300 lb. and decreases uniformly by 1 lb. for every foot the weight is lifted. Find the velocity when the weight has risen 40 feet.
A body is to be projected with given velocity from \(P\) so as to pass through \(Q\). Prove that the product of the two possible times of flight is \(2PQ/g\).
A particle of mass \(m\) is placed on a smooth wedge of mass \(M\) and slope \(\alpha\), resting on a smooth horizontal plane. When the velocity of the wedge is \(V\) it encounters a fixed obstacle which reduces it to rest. Shew that the velocity of the particle relative to the wedge is reduced in the ratio \[ M+m\sin^2\alpha : M+m. \]
\(AB, AC\) are two given straight lines and \(P\) is a given point in their plane. Shew how to draw a line through \(P\) so that \(PB=PC\).