Let \(m\) be a positive integer and \(y \ne \pm 1\). Put \[ (m,0)=1; \quad (m,j) = \frac{(1-y^{2m})(1-y^{2m-2})\dots(1-y^{2m+2-2j})}{(1-y^2)(1-y^4)\dots(1-y^{2j})} \quad (0< j\le m). \] Show that, for \(0 \le j < m\), \[ (m+1, j+1)=(m,j+1)+y^{2m-2j}(m,j), \] and hence, or otherwise, that \[ (m,0)+(m,1)y+(m,2)y^2+\dots+(m,j)y^j+\dots+(m,m)y^m = (1+y)(1+y^3)\dots(1+y^{2m-1}). \] What corresponds to this identity for \(y = \pm 1\)?
The ends of a uniform heavy chain of length \(l\) are attached to light rings threaded on two smooth rods \(OA, OB\). which are fixed in the same vertical plane and make equal angles \(\alpha\) (\(<\frac{1}{2}\pi\)) with the downward vertical through \(O\). Show that the chain can rest in equilibrium under gravity with the rings at the same level and at a distance \(kl\) apart, where \(k\) is such that \[ \sinh(k \tan\alpha) = \tan\alpha. \] Show that, given \(\alpha\), \(k\) is determined uniquely by this equation.
A parallelogram \(ABCD\) of freely jointed rods is in equilibrium on a smooth horizontal table. If \(T_1, T_2\) are the tensions in two strings, of which the first joins a point \(P\) of \(AB\) to a point \(Q\) of \(CD\), while the second joins a point \(R\) of \(BC\) to a point \(S\) of \(DA\), prove that \[ \frac{(AP-DQ)}{AB} \frac{T_1}{PQ} = \frac{(BR-AS)}{DA} \frac{T_2}{RS}, \] and explain the significance of this equation if \((AP-DQ)/(BR-AS)\) is zero or negative.
A particle of unit mass is projected vertically upwards in a medium whose resistance is \(k\) times the square of the velocity of the particle. If the initial velocity is \(V\), prove that the velocity \(u\) after rising through a distance \(s\) is given by \[ u^2 = V^2 e^{-2ks} + \frac{g}{k}(e^{-2ks}-1). \] Deduce an expression for the maximum height of the particle above its point of projection.
The two ends \(A\) and \(B\) of a uniform rod of length \(2a\) and mass \(m\) are attached by light rings to a smooth vertical wire and a smooth horizontal wire respectively. The wires are fixed in space so that the shortest distance between them is equal to \(a\). The rod is released from rest with the end \(A\) at a vertical height \(a\) above the level of the horizontal wire. Find the speed of \(A\) when the rod \(AB\) becomes horizontal.
A particle \(P\) of mass \(m\) is attached by a light elastic string, of unstretched length \(l\) and modulus of elasticity \(\lambda m\), to a point \(O\) on a smooth horizontal plane. Initially the particle is at rest on the plane and \(OP\) is of length \(l\). The particle is then given an initial velocity \(V\) on the plane in a direction perpendicular to \(OP\). Prove that, if \(3V^2 < 4\lambda l\), the length of \(OP\) in the ensuing motion never exceeds \(2l\).
Show that if \(a, b, c\) are real numbers different from \(\pm 1\) and such that \[ a^2+b^2+c^2+2abc=1, \] then all three or none of them lie between \(-1\) and \(+1\).
Show that in any algebraic equation \[ x^n - p_1x^{n-1} + p_2x^{n-2} - \dots + (-1)^n p_n = 0 \] the coefficient \(p_r\) is the sum of all the products formed by taking the roots together \(r\) at a time. If the roots are \(x_1, x_2, \dots, x_n\), prove that \[ (1-p_2+p_4-\dots)^2 + (p_1-p_3+p_5-\dots)^2 = (1+x_1^2)(1+x_2^2)\dots(1+x_n^2). \]
If a sequence of quantities \(x_0, x_1, x_2, \dots\) satisfy the recurrence relation \[ x_{n+2} - 2x_{n+1} + 2x_n = 0 \] and \(x_0=1, x_1=2\), show that \(x_{4n}=(-4)^n\), and find \(x_{4n-1}, x_{4n-2}\) and \(x_{4n-3}\). Prove also that \[ \sum_{r=1}^{4n} x_r^2 = \frac{8}{5}(2^{4n}-1). \]
A pack of 52 playing cards is shuffled and dealt to four players. One person finds he has 5 cards of a particular suit. Prove that the chance that his partner holds all the remaining 8 cards of the same suit is rather greater than 1 in 50,000.