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?
Determine the following:
Define the radius of curvature \(\rho\) at a point \(P\) of a plane curve and interpret its sign. Shew that the circle which passes through \(P\) and has there the same \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) as the curve has radius \(\rho\). Hence or otherwise shew that the radius of curvature at the origin of \[ x^4+y^4+x^3+y^3=2a(x+y) \] is \(a\sqrt{2}\).
Define the envelope of a system of curves and shew how it may be found. Prove that the tangents to a curve envelope the curve itself. Find the envelope of a system of coaxal ellipses all of the same area.