Problems

Resolve \(x^{2n}-2x^n\cos n\theta+1\) into \(n\) real quadratic factors. Express \((x+iy)^{a+ib}\) in the form \(X+iY\), and show that one of the values is real if \(\frac{1}{2}b\log(x^2+y^2)+a\tan^{-1}\frac{y}{x}\) is a multiple of \(\pi\).

By means of De Moivre's theorem, or otherwise, express \(\tan n\theta\) in terms of \(\tan\theta\). Prove that \[ \tan 10^\circ . \tan 50^\circ . \tan 70^\circ = \frac{1}{\sqrt{3}}. \] Prove that \[ \tan\theta + \frac{1}{2}\tan\frac{\theta}{2} + \frac{1}{4}\tan\frac{\theta}{4} + \dots \text{to infinity} = \frac{1}{\theta} - 2\cot 2\theta. \]

Solve the equations

1. [(i)] \(x+y=(1+xy)\sin\alpha\), \(x-y=(1-xy)\sin\beta\),
2. [(ii)] \(\frac{ax+by+cz}{x} = \frac{bx+cy+az}{y} = \frac{cx+ay+bz}{z}\).

Find the condition that the equations \(ax^2+bx+c=0\) and \(a'x^2+b'x+c'=0\) should have a common root. If \(\alpha\) is a root of the first and \(\alpha'\) of the second, find the equation whose roots are the four values of \(\alpha-\alpha'\).

Prove that the arithmetical mean of any number of positive quantities is greater than their geometrical mean. Prove that \[ 1.2^2.3^3 \dots n^n > \left(\frac{n+1}{2}\right)^{\frac{n(n+1)}{2}}. \]

If \((1+x)^n = c_0+c_1x+c_2x^2+\dots\) when \(n\) is a positive integer, find

1. [(i)] \(c_0^2+c_1^2+\dots+c_n^2\),
2. [(ii)] \(\frac{c_0}{2} + \frac{c_1}{3} + \dots + \frac{c_n}{n+2}\).

Define a differential coefficient, and find from first principles the differential coefficients of \(\log x\) and \(\cos^{-1}x\). If \(x^2+2xy+3y^2=1\), prove that \((x+3y)^3 \frac{d^2y}{dx^2} + 2 = 0\).

Find the equation of the tangent at a point on the curve given by \[ x/t = y/t^2 = 3a/(1+t^3). \] If the tangents at the points whose parameters are \(t_1, t_2, t_3, t_4\) are concurrent, prove that \[ \sum_{r=1}^4 t_r = 2 \prod_{r=1}^4 t_r. \]

If \(\phi\) is the angle between the radius vector and the tangent to the curve \(f(r,\theta)=0\), prove that \(\tan\phi=r\frac{d\theta}{dr}\). Prove that, if the tangents at \(P, Q\), two points on the curve \(r=a(1-\cos\theta)\), are parallel, the chord \(PQ\) subtends an angle \(2\pi/3\) at the pole.

Prove that if \(\rho\) is the radius of curvature at any point of a curve \[ \frac{1}{\rho^2} = \left(\frac{d^2x}{ds^2}\right)^2 + \left(\frac{d^2y}{ds^2}\right)^2. \] Prove also that \[ \frac{1}{\rho^4}\left\{1+\left(\frac{d\rho}{ds}\right)^2\right\} = \left(\frac{d^3x}{ds^3}\right)^2 + \left(\frac{d^3y}{ds^3}\right)^2. \]