Problems

(i) A plane figure consisting of points, straight lines and circles is inverted with respect to a circle in the plane; state (without proof) how the most important relations between the elements of the original figure are transferred to the inverse figure. \par (ii) A, B, C, D are four coplanar points, A' is the inverse of A with respect to the circle BCD and B', C', D' are determined similarly; prove, by inversion with respect to any circle whose centre is D, that the six circles BCA', CAB', ABC', ADA', BDB', CDC' have a common point. \par (iii) Expose the fallacy in the following ``argument'':

1. [(a)] ``If two circles touch, their inverses touch'' (true);
2. [(b)] ``concentric circles have double contact and circles having double contact are concentric'' (true);
3. [(c)] ``therefore concentric circles invert into concentric circles'' (untrue).

In a system of generalized homogeneous coordinates \((x,y,z)\) the condition that the lines \(lx+my+nz=0, l'x+m'y+n'z=0\) should be perpendicular is \[ ll'+mm'+2nn' + mn'+m'n+nl'+n'l=0; \] find the envelope (tangential) equation of the circular points at infinity and prove (without using any formulae for areal or trilinear coordinates) that

1. [(i)] the equation of the line at infinity is \(x+y-z=0\);
2. [(ii)] the equation of the circumcircle of the triangle of reference is \[ yz+zx-2xy=0. \]

Find also the equation of the circle with respect to which the triangle of reference is self-polar and interpret the result geometrically.

State (without proof) the rule for expressing the product of two determinants each of the third order as a determinant of the third order. \par If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \[ ax^4+4bx^3+6cx^2+4dx+e=0, \] and \(\theta = \alpha(\beta\gamma+\alpha\delta)\), prove that \[ -a^3 \times \begin{vmatrix} 1 & \theta & \theta^2 \\ 1 & \beta+\gamma & \beta\gamma \end{vmatrix} \begin{vmatrix} 1 & \alpha+\delta & \alpha\delta \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}^2 = \begin{vmatrix} 2a & -4b & 0 \\ -4b & 12c-2\theta & -4d \\ 0 & -4d & 2e \end{vmatrix} \] and deduce that \(\beta\gamma+\alpha\delta, \gamma\alpha+\beta\delta, \alpha\beta+\gamma\delta\) are the roots of the equation in \(y\) \[ (ay-2c)^3 - 4(ae-4bd+3c^2)(ay-2c) + 16 \begin{vmatrix} a & b & c \\ b & c & d \\ c & d & e \end{vmatrix} = 0. \]

Prove that the arithmetic mean of \(n\) positive numbers which are not all equal exceeds their geometric mean. \par Deduce that \[ nx^{\frac{n-1}{2}} < 1+x+x^2+\dots+x^{n-1}, \] so that, if \(0

Explain in precise language what you mean by the statement that \(u_n\) tends to a limit \(l\) as \(n\) tends to infinity, and evaluate \(\lim_{n\to\infty} x^n\) when it exists, considering the special cases which may arise for various values of \(x\). \par Prove that \[ x\sin\alpha+x^2\sin 2\alpha+\dots+x^n\sin n\alpha = \frac{x\sin\alpha}{1-2x\cos\alpha+x^2} - \frac{\sin(n+1)\alpha - x\sin n\alpha}{1-2x\cos\alpha+x^2}x^{n+1}, \] and examine fully the convergence of the series as \(n\) tends to infinity.

P, Q are two polynomials in \(x\) which satisfy the identity \[ \sqrt{P^2-1} = Q\sqrt{x^2-1}. \] Prove the following results:

1. [(i)] P, Q have no common factor.
2. [(ii)] If P is of degree \(n\) in \(x\), Q is of degree \(n-1\) in \(x\).
3. [(iii)] \(\dfrac{dP}{dx}=kQ\), where \(k\) is some constant.

Deduce that \[ \frac{dP}{\sqrt{P^2-1}} = \frac{kdx}{\sqrt{x^2-1}}. \] Find P, Q when \(k=3\), if \(P=0\) when \(x=0\).

Two identical uniform right-angled prisms lie on a horizontal table. Their hypotenuse faces make each an angle \(\alpha\) with the table. The vertical faces are turned away from each other and are at distance \(2a\) apart, the edges opposite to them being in contact. The weight of each prism is W and the coefficient of friction between it and the table is \(\mu\). A horizontal platform of width \(2l\) is laid symmetrically with two parallel edges resting on the planes, the contact being smooth. The load on the platform is gradually increased, the weight of platform and load being represented by \(w\) acting at the middle point of the platform.

1. [(i)] If \(l2\mu W/(\tan\alpha-\mu)\), if \(\tan\alpha > \mu\), but that, if \(\tan\alpha<\mu\), they will not slide however great \(w\) may become.
2. [(ii)] If \(l>a\cos^2\alpha\), shew that the prisms will overturn or slide according as \[ \tan\alpha-\mu < \text{ or } > \frac{l}{3\mu}(\frac{1}{a\cos^2\alpha}-1). \]

Rectangular axes of \(x\) and \(y\) are drawn in a rigid lamina and forces \((X_r, Y_r)\) act at points \((x_r, y_r)\) of the lamina. Write down the conditions of equilibrium. \par If the lamina is now rotated through an angle \(\theta\) in its own plane, the magnitudes and directions of the forces relative to a fixed frame being unaltered and the points of application remaining fixed in the lamina, shew that the forces are still in equilibrium if \[ \Sigma(X_r x_r + Y_r y_r) = 0. \] What is the connection between this result and the fact of the existence of a centre of mass fixed relatively to a rigid lamina?

A locomotive of mass M can exert a pull P. It starts into motion from rest a train of \(n\) trucks, each of mass \(m\). The couplings are loose and inelastic so that each truck moves forward a distance \(a\) before jerking the next truck into motion. A similar coupling connects the engine to the first truck. Shew that the velocity of the train when the last truck has just been started into motion is \(v_n\) given by \[ (M+nm)^2 v_n^2 = Pan\{2M+(n-1)m\}. \]

An engine drives a machine by a belt passing round a flywheel and a light pulley wheel of equal radius on the machine. A constant couple G has to be applied at the pulley wheel to overcome the resistance in the machine. The couple exerted by the engine on the flywheel is variable and equal to \(f(\theta)\), where \(\theta\) is the angle through which the flywheel has turned and \(f(\theta)\) is a function whose values are repeated in each revolution. \par Shew that the angular velocity of the flywheel is also periodic, if \[ G = \frac{1}{2\pi} \int_0^{2\pi} f(\theta)d\theta, \] and that the angular velocity of the flywheel is a maximum or minimum when \(f(\theta)=G\). \par If \(f(\theta)=k|\sin\theta|\), shew that the difference between the greatest and least values of the kinetic energy of the flywheel is \[ 2k\sqrt{1-\left(1-\frac{4}{\pi^2}\right)\left(\frac{2}{\pi}\sin^{-1}\frac{2}{\pi}-\frac{4}{\pi^2}\right)}. \] I'm sorry, the expression in the original paper is very hard to read. A plausible interpretation is: \[ 2k \sqrt{1 - \left(1-\frac{4}{\pi^2}\right)} - \left(\frac{2}{\pi} \sin^{-1}\frac{2}{\pi} - \frac{4}{\pi^2}\right) \] But the provided image seems to show: \[ 2k\sqrt{1 - \left(\frac{4}{\pi^2}\right)\left(\frac{2}{\pi}\sin^{-1}\frac{1}{\pi} - 1\right)^2}. \] The OCR text is `2k\sqrt{(1-(4/\pi^2)) (2/\pi \sin^{-1} 1/\pi - 1)^2}`. Let me re-examine the image. It seems to be: \[ 2k\sqrt{1-\frac{4}{\pi^2}}\left(\frac{2}{\pi}\sin^{-1}\frac{1}{\pi}-1\right). \] However, the mathematical sense of this is questionable. A more likely formula, pieced together from the blurry image, might be: \[ 2k\left(\sqrt{1-\frac{4}{\pi^2}} - \frac{2}{\pi}\cos^{-1}\frac{2}{\pi}\right). \] Given the ambiguity, I'll transcribe the most likely machine-readable text and add a note. The final portion seems to be: \(2k\sqrt{(1-(4/\pi^2))} \left(\frac{2}{\pi}\sin^{-1}\frac{1}{\pi} - 1_i^2\right)\). This is likely incorrect. A more legible version might be \(2k \left(\sqrt{1 - \frac{4}{\pi^2}} - \frac{2}{\pi} \arccos\frac{2}{\pi}\right)\) I will write what is most plausible given the poor quality: \[ 2k \left( \sqrt{1-\frac{4}{\pi^2}} - \frac{2}{\pi} + \frac{2}{\pi} \sin^{-1}\frac{1}{\pi} \right). \] The last part of the image is illegible.