Define the instantaneous centre for a lamina moving in its own plane, and show that the motion of the lamina can be reproduced by the rolling of the locus of \(I\) in the lamina (body centrode) on the locus of \(I\) in space (space centrode). A straight rod \(AB\) is constrained to move with its ends \(A, B\), respectively in two intersecting straight guide rails. Determine the body and space centrode for the motion of the rod.
Define a determinant (of any order), and from your definition prove that the value of a determinant is unaltered if to the elements of any column are added any multiple of the corresponding elements of another column. If \(x_1, x_2, x_3\) are the roots of the equation \(x^3 = px + q\), show that \[ \begin{vmatrix} x_1^4 & x_1^3 & 1 \\ x_2^4 & x_2^3 & 1 \\ x_3^4 & x_3^3 & 1 \end{vmatrix} = p^2 \begin{vmatrix} x_1^2 & x_1 & 1 \\ x_2^2 & x_2 & 1 \\ x_3^2 & x_3 & 1 \end{vmatrix}. \] Show also that \[ \begin{vmatrix} x_1^6 & x_1^5 & x_1 \\ x_2^6 & x_2^5 & x_2 \\ x_3^6 & x_3^5 & x_3 \end{vmatrix} = q(q^2 - p^3) \begin{vmatrix} x_1^2 & x_1 & 1 \\ x_2^2 & x_2 & 1 \\ x_3^2 & x_3 & 1 \end{vmatrix}. \]
Investigate for what values of \(\lambda, \mu\) the simultaneous equations \begin{align*} x + y + z &= 6, \\ x + 2y + 3z &= 10, \\ x + 2y + \lambda z &= \mu \end{align*} have (i) no solution, (ii) a unique solution, (iii) an infinite number of solutions. In case (iii) give the general solution.
Starting from any (stated) definition of the natural logarithm of a positive number \(x\), prove that
Prove that, if \(a\) is real, the equation \[ e^x = x + a \] has two real roots if \(a\) is greater than 1 and no real roots if \(a\) is less than 1. Prove that the equation has no root of the form \(iv\), where \(v\) is real and not zero, and that, if \(u + iv\) is a complex root, \(u\) is positive.
A point \(P\) in a plane has the complex coordinate \(z (= x + iy)\) in relation to an origin \(O\) in the plane. Show that, if \(M\) is positive, and \(0 \le \alpha < 2\pi\), the point \(Q\) given by \(Me^{i\alpha}z\) is so situated that the angle \(POQ\) is equal to \(\alpha\), and the length \(OQ\) is \(M\) times the length \(OP\). Points in the plane of \(Z\) are related to points in the plane of \(z\) by the relation \(Z = \lambda z + \mu\), where \(\lambda \ne 0\) and \(\lambda\) and \(\mu\) are constants. If \(Z_1\) corresponds to \(z_1\) and \(Z_2\) corresponds to \(z_2\), prove that to any point on the line joining \(z_1\) and \(z_2\) there corresponds a point on the line joining \(Z_1\) and \(Z_2\). If the relation between \(Z\) and \(z\) is of the form \(Z = \lambda z^2 + \mu\), where \(\lambda \ne 0\) and \(\lambda\) and \(\mu\) are constants, show that to a straight line in the \(z\) plane there corresponds a parabola in the \(Z\) plane.
\(\alpha, \beta, \gamma\) and \(\alpha', \beta', \gamma'\) are the sides of two triangles circumscribed to a circular cone with vertex \(O\). Points \(P, Q, R, P', Q', R'\) are taken so that \(OP\) meets \(\beta\) and \(\gamma'\), \(OQ\) meets \(\gamma\) and \(\alpha'\), \(OR\) meets \(\alpha\) and \(\beta'\), \(OP'\) meets \(\gamma\) and \(\beta'\), \(OQ'\) meets \(\alpha\) and \(\gamma'\), \(OR'\) meets \(\beta\) and \(\alpha'\). Prove that a line can be drawn through \(O\) to meet \(PP'\), \(QQ'\) and \(RR'\).
Prove that there is a conic \(S'\) passing through two given points \(P_1\) and \(P_2\) and the four points of contact of the tangents from \(P_1\) and \(P_2\) to a given conic \(S\). Show that \(P_1P_2\) has the same pole with respect to both \(S\) and \(S'\). Show that, if a conic \(S'\) passes through given points \(P_0\) and \(P_1\) and the points of contact of the tangents from \(P_1\) to a given conic \(S\), then the common chord containing the other points of intersection of \(S\) and \(S'\) passes through the intersection of \(P_0P_1\) and the polar of \(P_0\) with respect to \(S\). Restate this theorem as a theorem of metrical geometry for the case when \(S\) is a circle and \(P_1\) its centre.
Explain how to apply theorems of projective geometry to parallel lines, to circles, to right angles and to the foci of conics. Justify your statements. Illustrate by stating, without proof, the projective generalisations of the following theorems: \begin{quote} "The locus of the point of intersection of perpendicular tangents to a conic is a circle concentric with the conic." \end{quote} \begin{quote} "The lines joining a point \(P\) of an ellipse to the foci are equally inclined to the tangent at \(P\)." \end{quote}
Explain what is meant by a conservative co-planar field of force. A particle moves under a force whose components at the point \((x, y)\) are \((\lambda x + \mu y, \nu x)\). Find the relation among the constants \(\lambda, \mu, \nu\) in order that the field may be conservative. When this condition is satisfied, find the potential energy.