Problems

By inspection, or otherwise, find all the real roots of each of the equations

1. \((x-1)^3 + (x-2)^3 = 0\),
2. \((x-1)^4 + (x-2)^4 = 1\),
3. \((x-1)^4 + (x-2)^4 = 0\).

If \begin{align*} \frac{x}{a+\lambda} + \frac{y}{b+\lambda} + \frac{z}{c+\lambda} &= 1, \\ \frac{x}{a+\mu} + \frac{y}{b+\mu} + \frac{z}{c+\mu} &= 1, \\ \frac{x}{a+\nu} + \frac{y}{b+\nu} + \frac{z}{c+\nu} &= 1, \end{align*} prove that, for all values of \(\xi\) (except \(-a, -b\) and \(-c\)), \[ \frac{x}{a+\xi} + \frac{y}{b+\xi} + \frac{z}{c+\xi} = 1 + \frac{(\lambda-\xi)(\mu-\xi)(\nu-\xi)}{(a+\xi)(b+\xi)(c+\xi)}, \] and that \[ x = \frac{(a+\lambda)(a+\mu)(a+\nu)}{(a-b)(a-c)}. \]

Prove that, if \((1+x)^n = c_0 + c_1x + \dots + c_nx^n\), then

1. \(c_0^2-c_1^2+c_2^2-\dots\pm c_n^2\) is equal to \((-1)^m (2m)!/(m!)^2\) if \(n\) is an even integer \(2m\), and is zero if \(n\) is an odd integer.
2. \(\frac{c_0}{y} - \frac{c_1}{y+1} + \frac{c_2}{y+2} - \dots \pm \frac{c_n}{y+n} = \frac{n!}{y(y+1)\dots(y+n)}\), where \(y\) is not zero or a negative integer.

Prove that, if \(k\) is real and \(|k|<1\), the function \(\cot x + k \csc x\) takes all values as \(x\) varies through real values. Prove that, if \(|k|>1\), the function takes all values except those included in an interval of length \(2\sqrt{k^2-1}\). Give rough sketches of the graph of \[ y = \cot x + k\csc x \] for \(-\pi < x < \pi\), in the cases (i) \(01\).

Prove that, if \(X+Y+Z\) is equal to \(2n\) right angles, where \(n\) is an integer, then \[ \sin 2X + \sin 2Y + \sin 2Z \] is a numerical multiple of \(\sin X \sin Y \sin Z\). Prove that, if \(A, B, C\) are the angles of a triangle, then \begin{align*} \sin^3 A \cos A + \sin^3 B \cos B + \sin^3 C \cos C \\ = \frac{1}{4}(\sin 2A + \sin 2B + \sin 2C)(\cos 2A + \cos 2B + \cos 2C). \end{align*}

The coordinates of any point on a curve are given by \(x=\phi(t)/f(t)\), \(y=\psi(t)/f(t)\), where \(t\) is a parameter; prove that the equation of the tangent is \[ \begin{vmatrix} x & \phi(t) & \phi'(t) \\ y & \psi(t) & \psi'(t) \\ 1 & f(t) & f'(t) \end{vmatrix} = 0. \] Prove that the condition that the tangents at the points of the curve \[ x=at/(t^3+bt^2+ct+d), \quad y=a/(t^3+bt^2+ct+d), \] whose parameters are \(t_1, t_2, t_3\) may be concurrent is \[ 3(t_2t_3+t_3t_1+t_1t_2)+2b(t_1+t_2+t_3)+b^2=0. \]

Prove that, if \(f\) is a homogeneous polynomial in \(x\) and \(y\) of degree \(n\) and suffixes denote partial differentiations, then

1. \(xf_x+y f_y = nf\),
2. \(xf_{xx}+y f_{xy} = (n-1)f_x\),
3. \((n-1)\begin{vmatrix} f_{xx} & f_{xy} & f_x \\ f_{xy} & f_{yy} & f_y \\ f_x & f_y & 0 \end{vmatrix} = n f (f_{xy}^2-f_{xx}f_{yy})\).

Criticize the following arguments:

1. If \(y=(2x^2+3)/(x^2+4)\), then \(dy/dx=0\) if \(x=0\) or \(\pm 1\), and so the only maximum and minimum values are given by \(x=0\), \(y=\frac{3}{4}\) (minimum) and \(x=\pm 1\), \(y=1\) (maxima). Hence \(y\) lies between \(\frac{3}{4}\) and \(1\) for all values of \(x\).
2. If \(y^2=x^3-3x+1\), then \(dy/dx=0\) if \(x=\pm 1\). Also \(d^2y/dx^2\) does not vanish for these values of \(x\). Hence \(x=1\) and \(x=-1\) give points on the curve at which the tangents are parallel to the line \(y=0\).
3. \(\int_{-1}^3 \frac{dx}{1-x} = \left[-\log(1-x)\right]_{-1}^3 = \dots = -\log 2\).

Find the equation of the straight line which is asymptotic to the curve \[ x^2(x-y)+y^2=0. \] Prove also the following facts and give a sketch of the curve:

1. the origin is a cusp;
2. no part of the curve lies between \(x=0\) and \(x=4\);
3. the curve consists of two infinite branches, one lying in the first quadrant and the other in the second and third quadrants.

(i) Find \(\int \frac{1-\tan x}{1+\tan x}dx\). (ii) Prove that, if \(a>b>0\), \[ \int_0^\pi \frac{\sin^2 x dx}{a^2 - 2ab\cos x + b^2} = \frac{\pi}{2a^2}. \] What is the value of the integral, if \(b>a>0\)?