Discuss the convergence of the series \[ 1+z+z^2+...+z^n+..., \] where \(z\) may be real or complex. Prove that, if \(0 \le r < 1\) and \(\theta\) is real, \[ \sum_1^\infty r^n \sin n\theta = \frac{r \sin \theta}{1-2r\cos\theta+r^2}. \]
Express \[ \frac{2nx}{(1+x)^{2n}-(1-x)^{2n}} \] in real partial fractions, where \(n\) is an integer greater than 1. Deduce that \[ \sum_{r=1}^{n-1} (-1)^{r-1} \left(\cos \frac{r\pi}{2n}\right)^{2n-2} = \frac{1}{2}. \]
State, without proof, how the existence of a solution of the set of four equations \[ a_r x+b_r y+c_r z+d_r w=0, \quad (r=1, 2, 3, 4), \] for which not all of \(x, y, z, w\) are zero is related to the value of the determinant of the sixteen coefficients \(a_r, b_r, c_r, d_r\). Prove that, if \(p, q, r, s\) are all different from \(-1\) and if \[ \begin{vmatrix} -1 & q & r & s \\ p & -1 & r & s \\ p & q & -1 & s \\ p & q & r & -1 \end{vmatrix} = 0, \] then \[ \frac{p}{p+1} + \frac{q}{q+1} + \frac{r}{r+1} + \frac{s}{s+1} = 1. \]
Define envelope, centre of curvature. Prove that the centre of curvature of the envelope of the line \[ x \cos t + y \sin t = f(t), \] at the point where the line touches it, has co-ordinates \begin{align*} x &= -f'(t) \sin t - f''(t) \cos t, \\ y &= f'(t) \cos t - f''(t) \sin t. \end{align*}
Prove that \[ \int_0^\pi xf(\sin x) dx = \frac{\pi}{2} \int_0^\pi f(\sin x) dx. \] Evaluate the integral \[ \int_0^a \frac{x\sin x}{1+\cos^2 x} dx \] for \(a=\pi\) and \(a=2\pi\).
Explain what is meant by saying that pairs of points on a line are in homography (or projectivity); show that in general a homography on a line is determined if three pairs of corresponding points are given. A homography on a line \(l\) is determined by the three pairs of corresponding points \((A, B)\), \((B, C)\), \((C, A)\). If \(P\) is another point of the line, and \(Q, R\) are such that \((P, Q)\), \((Q, R)\) are pairs in the homography, prove that \((R, P)\) is a pair.
Interpret the equation \(S+\lambda u^2=0\), where \(S=0\) and \(u=0\) are equations of a conic and a straight line. Two conics \(S, S'\) meet in \(A, B, C, D\); \(AB, CD\) meet in \(O\). Two lines \(l, l'\) through \(O\) are harmonically conjugate with respect to \(OAB\) and \(OCD\). The line \(l\) meets \(S\) in \(P\) and \(Q\); the line \(l'\) meets \(S'\) in \(P'\) and \(Q'\). Prove that a conic exists touching \(S\) at \(P\) and \(Q\) and touching \(S'\) at \(P'\) and \(Q'\).
A horizontal beam \(AB\) is to be loaded uniformly along its length, and is supported at the end \(A\) and at some other point \(C\). Find the position of \(C\) in order that the beam may carry the greatest possible load without breaking, showing that \(AC/AB = 1/\sqrt{2}\).
Two masses \(m_1, m_2\) are connected by a light elastic string of modulus \(\lambda\) and natural length \(l\) and lie at rest at \(A, B\), respectively, on a smooth horizontal table, where \(AB=l\). If an impulse \(J\) is applied to \(m_1\) in the direction \(BA\), find the extension of the string when the relative velocity of the two masses first vanishes, and after what time this occurs.
Two gear-wheels are mounted on parallel axles. Their radii are \(a\) and \(2a\), and their moments of inertia about their axles are respectively \(I\) and \(16I\). The smaller wheel is at rest, and the larger is rotating freely with angular velocity \(\omega\). The spindles of the wheels are moved parallel to their lengths to make the wheels engage. Prove that their angular velocities become \(-\frac{4}{5}\omega\) and \(\frac{2}{5}\omega\).