A pencil of conics \(S\) passes through the four fixed points \(A_1, A_2, A_3, A_4\). Shew that the locus of the poles of a fixed line \(l\) with respect to the conics \(S\) is a conic \(C\) passing through the diagonal points of the quadrangle \(A_1A_2A_3A_4\). If \(A_1A_2\) meets \(l\) in \(A_{12}\), and \(A_{12}'\) is the harmonic conjugate of \(A_{12}\) with respect to \(A_1, A_2\), prove that \(C\) passes through the six points such as \(A_{12}'\). Shew also that \(C\) passes through the double points of the involution determined on \(l\) by the conics \(S\). What does \(C\) become when \(A_1, A_2, A_3, A_4\) are the vertices and orthocentre of a triangle and \(l\) is the line at infinity?
Shew that by suitable choice of homogeneous coordinates any conic can be represented by the parametric equations \[ x:y:z=t^2:t:1. \] Two conics \(S\) and \(S'\) have the two points \(A\) and \(B\) in common. \(P\) is a variable point on \(S'\) and the lines \(PA, PB\) meet \(S\) again in \(Q, R\). Prove that the line \(QR\) envelopes a conic.
Prove that the polar lines of a fixed line with respect to a system of confocal quadrics generate a paraboloid, which touches each of the three principal planes of the confocal system.
Give definitions of the tangent, principal normal, binormal, curvature (\(1/\rho\)), torsion (\(1/\sigma\)), and centre of curvature of a twisted curve, explaining carefully any conventions of sign involved in your definitions. Shew that, if from a fixed point \(O\) lines \(OT, ON, OB\) are drawn parallel to the positive directions of the tangent, principal normal, and binormal, respectively, at a point \(P\) of a curve \(C\), then if \(P\) moves along \(C\) with unit velocity the triad \(OTNB\) has at any instant an angular velocity whose components about \(OT, ON, OB\) are respectively \(1/\sigma, 0, 1/\rho\). A curve \(C\) drawn on the surface of a right circular cone of semi-vertical angle \(\alpha\) cuts the generators at a constant angle \(\beta\). Shew that the curvature and torsion of \(C\) at a point \(P\) at distance \(r\) from the axis of the cone are given by \[ \frac{r}{\rho} = |\sin\beta|\sqrt{1-\cos^2\alpha\cos^2\beta}, \quad \frac{r}{\sigma} = \pm \cos\alpha\sin\beta\cos\beta, \] explaining the ambiguity in the second formula. Shew that as \(P\) describes \(C\), the centre of curvature describes a curve lying on the surface of a right circular cone and cutting the generators at a constant angle.
Establish Newton's formulae for expressing the sums of powers of the roots of an equation \[ x^n+a_1x^{n-1}+\dots+a_n=0 \] in terms of the coefficients. Shew that, if \(1 \le m \le n\), then the sum of the \(m\)th powers may be expressed in the form \[ s_m = (-1)^m \begin{vmatrix} a_1 & 1 & 0 & \dots & 0 & 0 \\ 2a_2 & a_1 & 1 & \dots & 0 & 0 \\ 3a_3 & a_2 & a_1 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ (m-1)a_{m-1} & a_{m-2} & a_{m-3} & \dots & a_1 & 1 \\ ma_m & a_{m-1} & a_{m-2} & \dots & a_2 & a_1 \end{vmatrix}. \]
A function \(f(x)\) satisfies, in the interval \((\alpha,\beta)\) (\(\alpha<\beta\)), the conditions \[ f(\alpha)<0, \quad f(\beta)>0, \quad f''(x)>0 \quad (\alpha \le x \le \beta). \] Shew that, if the equation \(f(x)=0\) has a single root \(\xi\) in \((\alpha,\beta)\), and that if \(\alpha', \beta'\) are defined by \[ \alpha' = \frac{\alpha f(\beta) - \beta f(\alpha)}{f(\beta)-f(\alpha)}, \quad \beta' = \beta - \frac{f(\beta)}{f'(\beta)}, \] then \(\alpha < \alpha' < \xi < \beta' < \beta\). Explain the geometrical significance of these results.
The function \(y=f(x)\) is continuous in the interval \(a \le x \le b\) (\(a < b\)), and increases (in the strict sense) from \(A\) to \(B\) as \(x\) increases from \(a\) to \(b\). Shew that there is a unique (inverse) function \(x=F(y)\), defined in \(A \le y \le B\), and satisfying the equation \[ f[F(y)]=y \] identically throughout this interval. Shew further that \(F(y)\) is a continuous strictly increasing function in \(A \le y \le B\). Shew that, for every \(a >1\), the equation \[ \tan x = ax \] has a unique root in the interval \(0 < x < \frac{1}{2}\pi\), and that this root is a continuous function of \(a\).
State and prove Taylor's Theorem with Lagrange's form of remainder. Shew that, if \(s\) is any positive number and \(n\) any positive integer, then \[ \int_0^\infty \frac{e^{-sx}dx}{\sqrt{1+x^2}} = \sum_{v=0}^{n-1} \frac{(-1)^v c_v}{s^{2v+1}} + \theta_n(s) \frac{(-1)^n c_n}{s^{2n+1}}, \] where \[ c_0=1, \quad c_v = 1^2 . 3^2 \dots (2v-1)^2 \quad (v \ge 1), \] and \(\theta_n(s)\) satisfies the inequality \(0 < \theta_n(s) < 1\). Give reasons for the existence of any integrals which you use.
A sequence of numbers \(a_1, a_2, \dots\), all different from \(-1\), is such that \[ a_n = \frac{\gamma}{n} + b_n, \] where \(\gamma\) is a constant and \(\Sigma|b_n|\) is convergent. Shew, by considering the infinite product \[ \prod \left\{(1+a_n)(1-\frac{\gamma}{n})\right\} \] or otherwise, that \[ P_n = \prod_{v=1}^n (1+a_v) \sim An^\gamma \] as \(n\to\infty\), \(A\) being a constant different from zero. If further \(b_n=O(n^{-1-\delta})\), where \(0<\delta<1\), prove that \[ P_n = An^\gamma + O(n^{\gamma-\delta}) \quad (n\to\infty). \] Discuss the convergence, for all real values of \(\gamma, \alpha, \beta\), and all values (real or complex) of \(x\), of the series \[ \Sigma \frac{x^n}{n^\gamma}, \quad \Sigma \frac{\alpha(\alpha+1)(\alpha+2)\dots(\alpha+n-1)}{\beta(\beta+1)(\beta+2)\dots(\beta+n-1)}x^n, \] distinguishing between absolute and conditional convergence.
Shew that, if \(\Sigma u_n(x)\) is uniformly convergent over the infinite range \(x \ge a\), and if, for each \(n\), \(u_n(x)\) tends to a limit \(u_n\) as \(x\to\infty\), then \(\Sigma u_n\) is convergent, and \[ \Sigma u_n(x) \to \Sigma u_n, \quad \text{as } x\to\infty. \] Prove that \(e^y(1-y)\) is a decreasing function of \(y\) in the interval \(0 \le y \le 1\), and deduce that, as \(x\to\infty\), \[ \sum_{n=1}^{[x]} a_n e^n \left(1-\frac{n}{x}\right)^x \to \sum_{n=1}^\infty a_n, \] provided that the series on the right is convergent. Here \([x]\) denotes the integral part of \(x\).