Problems

(i) Explain fully what is meant by an involution of pairs of points on a straight line and prove that \[ BC' \cdot CA' \cdot AB' + B'C \cdot C'A \cdot A'B = 0 \] is a necessary and sufficient condition that the three pairs \((A, A')\), \((B, B')\), \((C, C')\) should belong to an involution range. \par State the corresponding condition that three pairs of concurrent lines should belong to an involution pencil and deduce (or prove otherwise) that if two of the pairs are at right angles the third pair are also at right angles. \par (ii) \(A, B\) are the ends of a fixed diameter of a conic and \(P\) is a variable point on a fixed line \(\lambda\) through \(B\); if the tangents from \(P\) to the conic meet the tangent at \(A\) at \(Q, Q'\), prove that (a) \(Q, Q'\) are a pair in an involution, (b) \(AQ+AQ'\) is constant, (c) if the line \(\lambda\) cuts the conic again at \(C\) and the tangent at \(A\) at \(D\), then the tangent to the conic at \(C\) bisects \(AD\).

Prove that

1. [(i)] the coordinates \((x,y)\) of any point on any given conic can be written in the form \[ x:y:1 = \alpha t^2 + 2\alpha' t + \alpha'' : \beta t^2 + 2\beta' t + \beta'' : \gamma t^2 + 2\gamma' t + \gamma'', \] where \(t\) is a parameter;
2. [(ii)] the condition that the line \(lx+my+n=0\) should touch the conic is \[ (\alpha l + \beta m + \gamma n)(\alpha'' l + \beta'' m + \gamma'' n) - (\alpha' l + \beta' m + \gamma' n)^2 = 0. \]

If the coordinate axes are rectangular and \(x=(t-1)(t-2)\), \(y=t^2+1\), prove that the conic is a parabola and find the equation of its directrix and the coordinates of its focus.

Shew that, if \(a_n \to a\) and \(b_n \to b\) as \(n \to \infty\), then

1. [(i)] \(\text{Max}\,(a_n, b_n) \to \text{Max}\,(a,b)\), where \(\text{Max}\,(a,b)\) is equal to \(a\) if \(a \geq b\), and is equal to \(b\) if \(b > a\),
2. [(ii)] if \(b \neq 0\), then \(\frac{a_n}{b_n} \to \frac{a}{b}\),
3. [(iii)] if \(P(x)\) and \(Q(x)\) are polynomials and if \(Q(a) \neq 0\), then \(\frac{P(a_n)}{Q(a_n)} \to \frac{P(a)}{Q(a)}\).

Define \(\log x\) for positive values of \(x\), and prove from your definition that

1. [(i)] \(\log(xy) = \log x + \log y\),
2. [(ii)] \(\log x\) increases strictly from \(-\infty\) to \(\infty\) as \(x\) increases from \(0\) to \(\infty\),
3. [(iii)] \(\frac{d}{dx} \log x = \frac{1}{x}\),
4. [(iv)] if \(n\) is an integer, then \(\frac{(\log x)^n}{x} \to 0\) as \(x \to \infty\).

State, without proof, any general theorems that you use.

If \(x=r\cos\theta\), \(y=r\sin\theta\), find the values of \(A, B, C, D\) such that \begin{align*} \delta x &= A\delta r + B\delta\theta + \epsilon (|\delta x| + |\delta\theta|), \\ \delta y &= C\delta r + D\delta\theta + \epsilon' (|\delta x| + |\delta\theta|), \end{align*} where \(\epsilon\) and \(\epsilon'\) both tend to zero as \(|\delta x| + |\delta\theta| \to 0\). \par If \(\phi(x,y)\) is a function of \(x\) and \(y\), then it may be regarded as a function of any two of the variables \(x, y, r\) and \(\theta\); find the values of \(\frac{\partial\phi}{\partial x}\) and \(\frac{\partial^2\phi}{\partial x^2}\) (i) when the independent variables are \(x\) and \(r\), (ii) when they are \(x\) and \(\theta\). These values are to be expressed in terms of the partial derivatives of \(\phi\) when the independent variables are \(x\) and \(y\).

Prove that a real rational function of \(x\) may be expressed as the sum of a polynomial and real partial fractions. \par Express in this way \(\frac{1}{x(x^2+1)^2}\).

A uniform thin rigid plank of weight \(W\) has one end on rough horizontal ground and rests, at an inclination \(2\alpha\) to the horizontal, against a rough heavy uniform cylinder which is in contact with the ground along a generator parallel to, and at a distance \(d\) from, the lower end of the plank. The vertical plane through the middle line of the plank contains the centre of gravity of the cylinder. The coefficient of friction between the plank and the ground, the cylinder and the ground, and the plank and the cylinder is \(2 \tan\alpha\) in each case. The length of the horizontal projection of the plank is \(c\). Prove that the only condition necessary for equilibrium is \(3c<4d\). \par Prove that a body of weight \(\frac{1}{2}W\) can be placed in any position on the middle line of the plank without equilibrium being broken if \(c

A uniform elliptic cylinder of weight \(W\) is loaded with a particle of weight \(kW\) at an end of the major axis of the normal cross-section through its centre of gravity, and is placed with its axis horizontal on a smooth horizontal table. Determine the possible positions of equilibrium and consider the stability of the symmetrical positions.

A particle is projected with velocity \(V_0\) at an angle \(\alpha\) with the horizontal and moves under the action of gravity and of an air resistance, equal to \(f\) per unit mass, which is directly opposed to the direction of motion and whose magnitude is given as a function of the velocity of the particle. After time \(t\) the length of the path described is denoted by \(s\), the resultant velocity by \(V\), its horizontal component by \(u\), and the tangent of the angle between the direction of motion and the horizontal by \(p\). Prove that \[ \frac{du}{ds} = -\frac{u}{V^2}f \] and \[ \frac{dp}{dt} = u \sqrt{1+p^2} \frac{dp}{ds} = -\frac{g}{u}, \] and that in the case when \(f=kV^2\), \(k\) being constant, \[ u = V_0 e^{-ks} \cos \alpha \] and \[ F(p_0) - F(p) = \frac{g}{kV_0^2 \cos^2\alpha} (e^{2ks} - 1), \] where \(p_0 = \tan\alpha\) and \[ F(p) = p\sqrt{1+p^2} + \log(p+\sqrt{1+p^2}). \] Deduce that \[ t = \frac{1}{\sqrt(kg)} \int_p^{p_0} \frac{dp}{\sqrt{G(p)}}, \] where \[ G(p) = F(p_0) - F(p) + \frac{g}{kV_0^2 \cos^2\alpha}. \]

A light rigid rod of length \(l\), carrying a heavy particle rigidly attached at one end, is whirled with constant angular velocity \(\omega\) round the vertical through the other end, which is fixed. If \(\omega^2 > g/l\), shew that it is possible for the particle to move steadily in a circle, and that its distance from the vertical axis of rotation is then \(\sqrt{l^2 - g^2/\omega^4}\). \par If, with the angular velocity \(\omega\) maintained unaltered and constant, the distance of the particle from the axis is very slightly altered from the value \(\sqrt{l^2 - g^2/\omega^4}\), find the period of the resulting small oscillations.