Shew that the condition that \(lx+my-1=0\) shall be normal to the ellipse \(x^2/a^2+y^2/b^2=1\) is \[ a^2m^2+b^2l^2 = (a^2-b^2)^2 l^2 m^2. \] Shew that two coaxal and concentric conics have four common normals in addition to the axes and prove that they are real lines for the conics \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0 \quad \text{and} \quad \frac{x^2}{a^2-\mu}+\frac{y^2}{b^2+\mu}-1=0, \] provided \[ \frac{(a^2-b^2)(a^2+3b^2)}{b^2} > 4\mu > \frac{(a^2-b^2)(3a^2+b^2)}{a^2}. \]
A family of conics have their centres at the origin and the lines \(x=\pm d\) as directrices: prove that two members of the family, which are both ellipses or both hyperbolas, pass through a chosen point in the plane provided the point lie in the regions surrounding the axis of \(x\) and bounded by the two parabolas \[ x^2-d^2-2dy=0, \quad x^2-d^2+2dy=0. \] Prove also that any member of the family touches both parabolas but that, when the member is an ellipse, the points of contact are not real if the eccentricity is less than \(1/\sqrt{2}\).
Prove that the circle of curvature at the point \((am^2, 2am)\) on the parabola \(y^2-4ax=0\) is given by the equation
\[
x^2+y^2-ax(4+6m^2)+4aym^3-3a^2m^4=0
\]
and that there are four circles of curvature real or imaginary through a chosen point.
In the case of a point \((x, 0)\), shew that no real circle of curvature passes through the point if \(0
State and prove the theorem which gives the remainder when a polynomial \(f(x)\) is divided by a linear factor \((x-a)\). Deduce that an equation of the \(n\)th degree cannot have more than \(n\) distinct roots. Determine the condition that \((x^{n+1}-ax^n+1)\) shall be divisible by \((x^2-x+1)\).
A series is such that the sum of the \(r\)th term and the \((r+1)\)th is always \(r^4\). Prove that
Determine the number of combinations of \(n\) things \(r\) at a time, and shew that \[ {}_{n+1}C_{r+1} = {}_rC_r + {}_{r+1}C_r + \dots + {}_nC_r. \] A set of \(n\) points in a plane is such that \(p\) of them lie on one straight line but not more than two on any other straight line. How many different triangles can be formed having these points for vertices?
Shew how to find the equation whose roots are the squares of the roots of a given algebraic equation. If \(\alpha, \beta, \gamma \dots\) are the roots of \[ x^n+a_1x^{n-1}+a_2x^{n-2}+\dots+a_n=0, \] shew that for all values of \(k\) \[ (k^2+\alpha^2)(k^2+\beta^2)(k^2+\gamma^2)\dots = (k^n-a_2k^{n-2}+a_4k^{n-4}-\dots)^2 + (a_1k^{n-1}-a_3k^{n-3}+\dots)^2. \]
If \(Z(=X+iY), z(=x+iy)\) are points of an Argand diagram, what is the geometrical meaning of the transformations (i) \(Z=Az\), (ii) \(Z=z+A\), (iii) \(Z=\frac{1}{z}\), where \(A\) is a complex number? If \(Z=\frac{z-i}{z+i}\), shew that when \(z\) lies above the real axis \(Z\) will lie within the unit circle which has centre at the origin. How will \(Z\) move as \(z\) travels along the real axis from \(-\infty\) to \(+\infty\)?
Prove, by integrating the inequality \(\cos\theta \le 1\), that \(\cos\theta\) lies between \[ \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \dots + \frac{(-1)^n\theta^{2n}}{(2n)!}\right) \quad \text{and} \quad \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \dots + \frac{(-1)^{n+1}\theta^{2n+2}}{(2n+2)!}\right). \] Deduce the infinite series for \(\cos\theta\). If an infinite series \(u_1+u_2+\dots+u_n+\dots\) of positive terms is convergent, shew that so also is \[ u_1^2+u_2^2+\dots+u_n^2+\dots. \]
Define a "maximum" of a function of \(x\). \(y\) is determined by the equations: \begin{align*} y &= \cos x - \log\left(\frac{\cos x}{\cos 1}\right) + 1 - \cos 1 \quad \text{for } 0 \le x < 1, \\ y &= \frac{1}{4}\left(x+\frac{3}{x}\right) \quad \text{for } x \ge 1. \end{align*} Find the greatest value of \(y\) for values of \(x\) in the interval \((0, 3)\) and shew that this occurs for two values of \(x\). Is \(y\) a maximum at the points in question?