Prove that the three (distinct) complex numbers \(z_1, z_2, z_3\) represent the vertices of an equilateral triangle in the Argand diagram if and only if \[ z_1^2+z_2^2+z_3^2-z_2z_3-z_3z_1-z_1z_2 = 0. \]
If \(\epsilon\) is small in magnitude compared with unity, show that the perimeter of the curve \[ r = 1 + \epsilon \cos\theta \] is approximately \(\frac{1}{2}\pi(4+\epsilon^2)\).
Obtain the equation and perimeter of the evolute (locus of centres of curvature) of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]
Sketch the curve \[ (x+y)(x^2+y^2) = 2xy, \] and obtain the area of its loop.
Solution: Let \(X = x+y\) and \(Y = x-y\), notice that \(4xy = X^2-Y^2\) so our equation becomes \begin{align*} && \frac12 (X^2-Y^2) &= X(X^2-\frac12(X^2-Y^2)) \\ &&&= \frac12X(X^2+Y^2) \\ \Rightarrow && Y^2(X+1) &= X^2 - X^3 \\ \Rightarrow && Y^2 &= \frac{X^2-X^3}{X+1} \end{align*}
Obtain a recurrence relation between integrals of the type \[ \int x \sec^n x \,dx. \] Evaluate \[ \int_0^{\pi/4} x \sec^4 x \,dx. \]
Prove that \[ \int_1^x \frac{dt}{t+\alpha} \le \log x \le \int_1^x \frac{dt}{t-\alpha}, \] where \(x>0\) and \(\alpha>0\). Hence show that \[ \log x = \lim_{n\to\infty} n(\sqrt[n]{x}-1). \] Use the above expression for \(\log x\) to prove that \[ \log(x^m) = m \log x \] for positive integral values of \(m\).
It is given that \(u=f(x,y)\) satisfies the relation \[ x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = nu, \] where \(n\) is a constant. Prove that \[ k\frac{\partial}{\partial k}f(kx, ky) = n f(kx, ky), \] and deduce that \(f(x,y)\) is a homogeneous function of degree \(n\) (i.e. \(f(kx, ky)=k^n f(x,y)\) for all positive \(k\)).
Solve the differential equation \[ \frac{d^2y}{dx^2} + 6\frac{dy}{dx} + 9y = 0 \] with the conditions \(y=2\) and \(\dfrac{dy}{dx}=-5\) at \(x=0\). Hence, or otherwise, find \(u_n\), given that \[ u_{n+2}+6u_{n+1}+9u_n = 0 \] for \(n\ge 0\), and \(u_0=2, u_1=-5\).
Prove that \[ \int f \frac{d^n g}{dx^n} dx = \sum_{r=1}^n (-1)^{r-1} \frac{d^{r-1}f}{dx^{r-1}}\frac{d^{n-r}g}{dx^{n-r}} + (-1)^n \int g \frac{d^n f}{dx^n} dx. \] Evaluate \[ \int_a^\infty x^m e^{-x} dx \quad \text{and} \quad \int_{-\infty}^\infty x^m \frac{d^n}{dx^n}(e^{-x^2}) dx, \] where \(m\) and \(n\) are positive integers. [The result \(\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}\) may be quoted without proof.]
(i) Prove that \[ \begin{vmatrix} a_1+x & a_1 & \dots & a_1 \\ a_2 & a_2+x & \dots & a_2 \\ \vdots & \vdots & \ddots & \vdots \\ a_n & a_n & \dots & a_n+x \end{vmatrix} = x^{n-1}(x+a_1+a_2+\dots+a_n). \] (ii) Prove that \[ \begin{vmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \\ a_4 & b_4 & c_4 & d_4 \end{vmatrix} = \begin{vmatrix} d_4 & d_3 & d_2 & d_1 \\ c_4 & c_3 & c_2 & c_1 \\ b_4 & b_3 & b_2 & b_1 \\ a_4 & a_3 & a_2 & a_1 \end{vmatrix}. \] If \(a, b, c, d\) are real numbers, and \(p, q, r, s, t, u\) are complex numbers with respective conjugate complexes \(\bar{p}, \bar{q}, \bar{r}, \bar{s}, \bar{t}, \bar{u}\), show that all the coefficients of the polynomial in \(x\) \[ \begin{vmatrix} r-x & q & p & a \\ t & s-x & \bar{p} & b \\ u & c & \bar{s}-x & \bar{q} \\ d & \bar{u} & \bar{t} & \bar{r}-x \end{vmatrix} \] are real.