The angles of elevation of the top of a mountain from three points \(A, B, C\) in a base line are observed to be \(\alpha, \beta, \gamma\) respectively. Prove that the height of the mountain is \[ (-AB \cdot BC \cdot CA)^{\frac{1}{2}}(BC\cot^2\alpha+CA\cot^2\beta+AB\cot^2\gamma)^{-\frac{1}{2}}, \] where regard is paid to the sense of the lines.
Find the sum \(s_n\) of \(n\) terms of the series \[ \sin x + \sin 2x + \sin 3x + \dots, \] and prove that \[ \text{Limit}_{n\to\infty} \frac{s_1+s_2+\dots+s_n}{n} = \tfrac{1}{2}\cot\tfrac{1}{2}x. \]
If \(A=a^2(a+b+c)+3abc\), \(B=b^2(a+b+c)+3abc\) and \(C=c^2(a+b+c)+3abc\), where \(ab+bc+ca=0\), then \((AB+BC+CA)abc = ABC\).
Solve the equations \begin{align*} a_1x+b_1y+c_1z &= d_1, \\ a_2x+b_2y+c_2z &= d_2, \\ a_3x+b_3y+c_3z &= d_3. \end{align*} And eliminate \(a, b, c\) from the equations \[ x = \frac{a}{b+c}, \quad y=\frac{b}{c+a}, \quad z=\frac{c}{a+b}. \]
(i) Prove with the usual notation that \({}^nC_r = \frac{n}{r}{}^{n-1}C_{r-1}\) and derive the number of combinations of \(n\) letters \(r\) at a time. (ii) Shew that the number of parts into which a plane is divided by \(n\) straight lines no two of which are parallel and no three concurrent is \(\frac{1}{2}(n^2+n+2)\).
Prove that, if \(p\) and \(q\) are positive integers, \(e^{p/q} = 1 + \dfrac{p}{q} + \dfrac{p^2}{2q^2} + \dfrac{p^3}{3!q^3} + \dots\). If \(u = 1 + \dfrac{1}{1.3} + \dfrac{1}{1.3.5} + \dfrac{1}{1.3.5.7} + \dots\), and \(v = 1 - \dfrac{1}{1}\cdot\dfrac{1}{3} + \dfrac{1}{4}\cdot\dfrac{1}{2\cdot 5} - \dfrac{1}{8}\cdot\dfrac{1}{3\cdot 7} + \dfrac{1}{16}\cdot\dfrac{1}{4\cdot 9} - \dots\), prove that, to four places of decimals, \(u^2/v^2=e\).
Shew that, if \(a_1, a_2 \dots a_n\) are unequal positive numbers, then \[ \frac{a_1+a_2+\dots+a_n}{n} > (a_1 a_2 \dots a_n)^{1/n} \] and, if also, \(a_1+a_2+\dots+a_n < 1\) then \((1+a_1)(1+a_2)\dots(1+a_n) < [1-(a_1+a_2+\dots+a_n)]^{-1}\).
(i) Sum to \(m\) terms the series whose \(n\)th term is \[ (a+\overline{n-1}b)(a+nb)\dots(a+\overline{n+r-2}b). \] (ii) Prove that \[ \frac{1^2}{n-1}\frac{1}{n+1} + \frac{2^2}{n-2}\frac{1}{n+2} + \dots + \frac{(n-1)^2}{1}\frac{1}{2n-1} + \frac{n^2}{2n} = \frac{2^{2n-3}}{2n-1}. \]
Find from first principles the differential coefficients of (i) \(\sin x\), (ii) \(\log_e(1+x^2)\). Find the value of \(\dfrac{du}{dx}\) when \(u = \log\dfrac{x^2+\sqrt{2}x+1}{x^2-\sqrt{2}x+1} + 2\tan^{-1}\dfrac{\sqrt{2}x}{1-x^2}\).
State Maclaurin's Theorem for the expansion of \(f(x)\). Apply this method to the expansion of \(\sin\left(x+\dfrac{\pi}{4}\right)\) in ascending powers of \(x\).