A conic passes through four fixed points A, B, C and X. The tangents at A, B, C meet BC, CA, AB respectively in P, Q, R. Shew that P, Q, R are collinear and that the envelope of the straight line PQR is a conic which touches BC, CA, AB where they are met by AX, BX, CX respectively.
A uniform isosceles triangle lamina is supported vertically with its vertex downwards upon two smooth pegs in a horizontal line. Prove that, if \(p\) is the distance of the centre of gravity from the vertex, \(\alpha\) the vertical angle and \(q\) the distance between the pegs, there will be one or three positions of equilibrium according as \(p\) is \(>\) or \(< 2q\text{cosec}^2\frac{1}{2}\alpha\). (Note: Transcribed from image, differs from OCR.)
The axis of a fixed circular cylinder of radius \(a\) is horizontal; from a point in the horizontal plane containing the axis, at a distance \(b\) from it, a perfectly elastic particle is projected so as to strike the cylinder and return through the point of projection. Shew that it must strike the cylinder at a point at which the radius through it makes an angle with the horizon given by \(a\cos\theta = b\cos 2\theta\).
Two equal particles, of mass \(m\), connected by an elastic string, of natural length \(a\), are placed on a smooth horizontal table at a distance \(a\) from one another. One of the particles is projected with a velocity \(v\) in the direction of the string produced; shew that in the subsequent motion the greatest extension of the string is \(v\sqrt{am/2\lambda}\), where \(\lambda\) is the coefficient of elasticity of the string. (Note: This seems to refer to Hooke's constant, not coefficient of elasticity.)
Shew that the tangent to an ellipse at any point P is the polar, with regard to the confocal hyperbola which passes through P, of the centre of curvature at the point P of the ellipse.
The separation and approximate calculation of the real roots of algebraic equations.
Discuss the general equation of the second degree in three dimensions, obtaining the necessary conditions for the various types of quadrics and degenerate quadrics.
Moving axes as applied to the geometry of curves and surfaces.
The uniform convergence of series.
The theory of Riemann integration.