Problems

An aeroplane is travelling in a straight line with constant velocity \(v\) feet per second at a constant height. At one moment a man observes it due north of him, at an angle of elevation \(\alpha\); \(t\) seconds later he sees it in a direction \(\beta\) east of north, and the angle of elevation is \(\gamma\). Prove that the aeroplane's course is \(\delta\) south of east, where \[ \tan\delta = \cot\alpha . \text{cosec}\beta . \tan\gamma - \cot\beta, \] and its height is \[ vt \text{ cosec}\beta . \tan\gamma . \cos\delta \text{ feet}. \]

Prove that in any triangle, with the usual notation,

1. [(1)] \(4R\Delta = abc\),
2. [(2)] \(a^2\cos^2 A = b^2\cos^2 B + c^2\cos^2 C + 2bc\cos B \cos C \cos 2A\).

If \(D\) is the middle point of \(BC\), prove that \[ \cot ADB = \frac{b^2-c^2}{4\Delta}. \]

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 \[ \sum_{r=1}^4 t_r = 2 \prod_{r=1}^4 t_r. \]