Problems

(i) Solve the equations \[ x^2 + 2xy^2 + 2y^4 = 1, \quad \frac{1}{x^2} - \frac{2}{xy^2} + \frac{2}{y^4} = 1. \] (ii) If \[ \frac{a}{x-md} + \frac{b}{x-mc} + \frac{c}{x+mb} + \frac{d}{x+ma} = 0, \] and \(a+b+c+d=0\), prove that the only finite value of \(x\) is \(\displaystyle\frac{m(ac+bd)}{a+b}\).

Sum to \(n\) terms the series

1. [(i)] \(\displaystyle\frac{x}{(1+x)(1+ax)} + \frac{ax}{(1+ax)(1+a^2x)} + \frac{a^2x}{(1+a^2x)(1+a^3x)} + \dots\)
2. [(ii)] \(\displaystyle\frac{1}{4!} + \frac{2\cdot 3}{5!} + \frac{3\cdot 3^2}{6!} + \frac{4\cdot 3^3}{7!} + \dots\)

If \(a_r\) is the coefficient of \(x^r\) in the expansion of \((1+x+x^2)^n\) in a series of ascending powers of \(x\), prove that

1. [(i)] \(a_0+a_1+a_2+\dots+a_{2n}=3^n\),
2. [(ii)] \(a_1+2a_2+3a_3+\dots+2na_{2n}=n\cdot3^{n-1}\),
3. [(iii)] \(a_0^2-a_1^2+a_2^2-\dots+a_{2n}^2=a_n\).

If £\(P\) is the present value of an annuity of £\(A\), to continue for \(n\) years, at \(100r\) per cent. per annum compound interest, prove that \[ \frac{Pr}{A} = 1-(1+r)^{-n}. \] If £\(Q\) is the present value of an annuity of £1 on the life of a man, shew that in order to receive £\(R\) at his death the payment to be made immediately and repeated annually is \[ £\frac{R(1-Qr)}{Q(1+r)}. \]

Find the number of combinations of \(m\) unlike things \(r\) at a time. Prove that the number of combinations \(n\) at a time of \(2n\) things, of which \(n\) are alike and the rest all different, is \(2^n\).

If \[ \frac{\cos(\alpha-3\theta)}{\cos^3\theta} = \frac{\sin(\alpha-3\theta)}{\sin^3\theta} = m, \] prove that \[ m^2+m\cos\alpha=2. \]

A gun is fired from a fort \(A\), and the intervals between seeing the flash and hearing the report at two stations \(B,C\) are \(t,t'\) respectively. \(D\) is a point in \(BC\) produced at a known distance \(a\) from \(A\). Prove that if \(BD=b\) and \(CD=c\), the speed of sound is \[ \left\{\frac{(b-c)(a^2-bc)}{bt'^2-ct^2}\right\}^{\frac{1}{2}}. \] Examine the case when \(a^2=bc\).

In a triangle \(ABC\), with the usual notation, prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] A circle is drawn touching the circumscribed circle of the triangle internally, and also touching \(AB, AC\). Prove that its radius is \[ r\sec^2\frac{A}{2}. \]

If \[ \tan\alpha = \cos2\omega\cdot\tan\lambda, \] prove that \[ \lambda-\alpha = \tan^2\omega\cdot\sin2\alpha + \frac{1}{2}\tan^4\omega\cdot\sin4\alpha + \frac{1}{3}\tan^6\omega\cdot\sin6\alpha + \dots \]

Express \(\tan n\theta\) in terms of \(\tan\theta\). Prove that the values of \(x\) which satisfy the equation \[ 1-nx - \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots + (-1)^{\frac{1}{2}n(n-1)}x^n=0 \] are given by \[ x=\tan\frac{(4r+1)\pi}{4n}, \] where \(r\) is any integer.