Prove that, if \(N\) and \(n\) are nearly equal, then \[ \left(\frac{N}{n}\right)^{1/3} = \frac{N+n}{n + \frac{1}{3}\frac{N+n}{4n-N}} \text{ approximately,} \] the error being approximately \(\frac{7}{81}\left(\frac{N-n}{n}\right)^3\).
Prove that \[ \frac{1}{0!2n!} - \frac{1}{1!3!2n-1!} + \frac{1}{2!4!2n-2!} - \dots + (-)^{n+1} \frac{1}{n-1!n+1!} = \frac{1}{n-1!n+1!2n!}. \]
Eliminate \(x, y\) from the equations \[ \tan x + \tan y = a, \quad \sec x + \sec y = b, \quad \sin x + \sin y = c. \]
A quadrilateral is such that one circle can be described about it and another can be inscribed in it. Shew that the radius of the former is \[ \frac{1}{4}\{(ab+cd)(bc+da)(ca+bd)/abcd\}^{1/2}, \] where \(a, b, c, d\) are the lengths of the sides.
A regular hexagon \(ABCDEF\) is formed of six equal uniform heavy rods freely jointed to each other at their ends. It is suspended freely from the angular point \(A\) and the regular hexagonal form is maintained by a light horizontal rod \(PQ\) freely jointed to a point \(P\) in \(BC\) and to a point \(Q\) in \(FE\). Prove that \[ BP:PC = FQ:QE = 1:5. \]
A uniform heavy wire is bent into the form of an ellipse of semi-axes \(a\) and \(b\). It is hung over a rough peg so as to rest in a vertical plane. Prove that the wire will rest in equilibrium with any point in contact with the peg if the coefficient of friction between the peg and the wire is equal to \((a^2-b^2)/2ab\).
An imperfectly elastic particle is projected with velocity \(V\) from a point in a smooth inclined plane of angle \(\alpha\) in a vertical plane containing the line of greatest slope through the point of projection. The velocity of the particle parallel to the plane vanishes at the same instant as the particle ceases to rebound perpendicularly to the plane. Prove that the range on the plane is \[ 2V^2 \sin\alpha/g\{4\sin^2\alpha + (1-e)^2\cos^2\alpha\}, \] where \(e\) is the coefficient of restitution between the particle and the plane.
A uniform hemisphere of given mass rests on a smooth horizontal plane and a smooth perfectly elastic particle of mass equal to that of the hemisphere is dropped vertically so as to strike the hemisphere with a given velocity. Shew that in order that the velocity of the hemisphere after impact may be a maximum, the point of impact must be at an angular distance \(\sin^{-1}(1/\sqrt{3})\) from the highest point of the hemisphere.
If a particle slide along a chord of a circle under the action of an attractive force varying as the distance from a given point \(O\), prove that the time will be the same of describing from rest all chords which terminate at an extremity of the diameter of the circle through \(O\).
Find the values of \(\cos 15^\circ\) and \(\cos 18^\circ\) without using tables. If \[ \tan\frac{\theta}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{\phi}{2}, \] prove that \[ (1+e\cos\theta)(1-e\cos\phi) = 1-e^2. \]