Problems

A sequence of terms \(u_1, u_2, \dots, u_n, \dots\) is such that any four consecutive terms are connected by the relation \[ u_n - 4u_{n-1} + 5u_{n-2} - 2u_{n-3} = 0. \] If \(u_1=1, u_2=0, u_3=-5\), find \(u_n\).

Find the relation between \(p\) and \(q\) necessary in order that the equation \(x^3-px+q=0\) may be put into the form \[ (x^2+mx+n)^2 = x^4. \] Hence or otherwise solve the equation \[ 8x^3 - 36x + 27 = 0. \]

Prove that

1. [(i)] \(\displaystyle\frac{1}{4} + \frac{1.3}{4.6} + \frac{1.3.5}{4.6.8} + \frac{1.3.5.7}{4.6.8.10} + \dots \text{ to infinity} = 1\);
2. [(ii)] \(\displaystyle\frac{1}{1.2.3} + \frac{1}{3.4.5} + \frac{1}{5.6.7} + \frac{1}{7.8.9} + \dots \text{ to infinity} = \log_e 2 - \frac{1}{2}\).

\item[(i)] If there are \(n\) straight lines, produced indefinitely, lying in one plane, and no three of them meet in a point, prove that the number of \(n\)-sided polygons they form is \(\frac{1}{2}(n-1)!\). \item[(ii)] Prove that the arithmetic mean of all numbers less than \(n\) and prime to it is \(\frac{1}{2}n\). \item[(i)] Solve the equation \[ \cos x + \cos(x-\alpha) = \cos(x-\beta) + \cos(x-\alpha+\beta). \] \item[(ii)] Prove that \[ \left(1-\tan^2\frac{x}{2}\right)\left(1-\tan^2\frac{x}{2^2}\right)\left(1-\tan^2\frac{x}{2^3}\right)\dots \text{ to infinity} = \frac{x}{\tan x}. \]

A man standing at a distance \(c\) from a straight line of railway sees a train standing on the line, having its nearer end at a distance \(a\) from the point in the railway nearest him. He observes the angle \(\alpha\), which the train subtends, and thence calculates its length. If in observing \(\alpha\) he makes a small error \(\theta\), prove that the percentage error in the calculated length of the train is \[ \frac{100c\theta}{\sin\alpha(c\cos\alpha-a\sin\alpha)}. \]

Prove that the radius \(R\) of the circle that touches externally each of three circles of radii \(a, b, c\), that touch one another externally, is given by \[ \{Rbc(b+c+R)\}^{\frac{1}{2}} + \{Rca(c+a+R)\}^{\frac{1}{2}} + \{Rab(a+b+R)\}^{\frac{1}{2}} = \{abc(a+b+c)\}^{\frac{1}{2}}. \]

If \(m<1\), and \(\theta\) and \(\phi\) are acute angles, and if \[ \theta = \phi - m\sin2\phi + \frac{1}{2}m^2\sin4\phi - \frac{1}{3}m^3\sin6\phi + \dots \text{ to infinity}, \] prove that \[ (1+m)\tan\theta = (1-m)\tan\phi. \]

If \(i=\sqrt{-1}\), if \(x, y, u\) and \(v\) are real quantities, and if \[ \tan(x+iy) = \sin(u+iv), \] prove that \[ \tan u \cdot \sinh 2y = \tanh v \cdot \sin 2x. \]

Two circles are given. Show how to construct a rhombus \(ABCD\) with \(A, C\) on one circle and \(B, D\) on the other. Show that all such rhombuses have equal sides.

\(OM, ON\) are fixed lines through \(O\), a point on a hyperbola. Through \(P\), a variable point on the hyperbola, a line \(PM\) is drawn parallel to one of the asymptotes to meet \(OM\) in \(M\), and \(PN\) is drawn parallel to the other asymptote to meet \(ON\) in \(N\). Prove that \(MN\) passes through a fixed point.

\(A, B\) are conjugate points with respect to a conic. \(R\) is a variable point on the conic and \(RA, RB\) meet the conic again in \(P, Q\). Show that \(PQ\) passes through a fixed point \(C\). Show that the triangles \(ABC, PQR\) are in perspective and that as \(R\) varies the centre of perspective describes a conic.