If \(\alpha, \beta, \gamma\) be the distances of the centre of the nine-point circle from the vertices of a triangle, \(p\) its distance from the orthocentre, and \(R\) the radius of the circumscribing circle, then \[ \alpha^2+\beta^2+\gamma^2+p^2 = 3R^2. \]
Show that the equation \[ \{a(x^2-c)-b(y^2-c)\}^2 + 4\{axy+h(y^2-c)\}\{bxy+h(x^2-c)\}=0 \] represents four lines forming the sides of a rhombus.
A rectangular plate has sides ten inches and five inches. If equal squares are cut out at the four corners and if the sides are then turned up so as to form an open box, find the volume of the greatest box so formed.
Trace the curve \[ x = 2a \sin^2 t \cos 2t, \quad y = 2a \sin^2 t \sin 2t. \] Show that the length of its arc is \(8a\), and find the radius of curvature at the origin.
Show that if \[ I_n = \int_0^\infty \frac{dx}{(1+x^2)^n}, \] where \(n\) is a positive integer, then \[ I_{n+1} = \left(1-\frac{1}{2n}\right)I_n. \] and hence evaluate \(I_n\).
Prove that \[ 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n-1}-\log n \] tends to a finite limit, as \(n\to\infty\). If \(p=2^q\), where \(q\) is an integer, prove that \[ \sum_{r=2}^p \frac{1}{r \log r} > \frac{\log(q+1)}{2 \log 2}. \]
Two strings, each of length \(l\), are attached to a ceiling, and the lower ends are attached to a magnet of moment \(M\), length \(l\), and weight \(W\). When the strings are vertical the magnet is in the magnetic meridian but with its north-seeking pole towards the south. Through what angle will it have to turn before it comes to another position of equilibrium? (Assume that the earth's magnetic field exists a couple \(HM \sin\theta\) on the magnet when it makes an angle \(\theta\) with the magnetic meridian.)
The case of a rocket weighs 2 lbs. and the charge 5 lbs. The charge burns at a uniform rate and is completely burnt in 3 seconds, during which time it exerts a constant propulsive force of 20 lb.-wt. If the rocket is fired vertically, find the vertical velocity acquired during the burning of the charge.
A mass is suspended by a light elastic string from a point \(A\) and produces on extension \(k\), the natural length being \(l\). Prove that if the mass is raised through a vertical distance less than \(k\), and is then let go from rest, it makes oscillations of the same period as a simple pendulum of length \(k\). If the mass is raised up to \(A\) and let fall, show that the maximum extension of the string is \(k(1+\sec\alpha)\), where \(\alpha\) is the acute angle given by \(\tan^2\alpha = \frac{2l}{k}\), and that this extension is attained at a time \((\pi-\alpha)\sqrt{\frac{k}{g}}\) after the string first becomes taut.