Problems

If \(B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1}\,dx\) for \(p>0, q>0\), show that \[ B(p,q) = B(p+1, q) + B(p, q+1), \] and \[ B(p,q) = B(q,p). \] Hence show by induction that \[ B(p,q) = \frac{(p-1)!(q-1)!}{(p+q-1)!}, \] if \(p\) and \(q\) are integers.

By considering \((1-x)f(x)\), where \[ f(x)=c_0+c_1x+\dots+c_nx^n, \] where \(x\) is a complex number and the \(c\)'s are real numbers such that \[ c_0 > c_1 > \dots > c_n > 0, \] show that \(f(x)\) is not zero for \(|x|\le 1\).

Show that the form of a uniform heavy flexible chain hanging under gravity is given by \[ y = c\cosh x/c. \] Two long smooth straight rods \(AB, AC\) lie in a vertical plane and are each inclined at an acute angle \(\alpha\) to the downward vertical through \(A\). A uniform heavy flexible chain of length \(l\) hangs from two small weightless rings which are free to move, one on each rod. Prove that the distance between the rings is \[ l\cot\alpha\sinh^{-1}\tan\alpha. \]

A smooth wire bent into the form of a circle of radius \(a\) rotates with uniform angular velocity \(\omega\) about a vertical diameter. A bead which is free to move on the wire is released from relative rest from the point \(A\) at one end of the horizontal diameter of the circle. Show that if \(a\omega^2 > 2g\) the bead will return to \(A\) after descending a depth \(2g/\omega^2\). Prove that the time taken is given by \[ \frac{2}{\omega}\int_0^\alpha \frac{d\theta}{\sqrt{(\sin\alpha-\sin^2\theta)}}, \] where \(\sin\alpha=2g/a\omega^2\). Discuss the cases \(a\omega^2<2g\) and \(a\omega^2=2g\).

A small insect of mass \(m\) stands on a thin flat plate of mass \(M\) which rests on a horizontal table. The insect jumps off the plate so that when it lands on the table it has travelled a horizontal distance \(a\). Show that, immediately after the insect jumps, the minimum value of the total energy of the motion is \[ \frac{1}{4}ga\left[\frac{m}{M}\left\{(M+m)(M+\mu^2m)\right\}^{\frac{1}{2}} - \mu m\right], \quad (\mu<1) \] where \(\mu\) is the coefficient of impulsive friction between the plate and the table. [It is to be assumed that the plate slides on the table without rotating.]

Two particles \(A\) and \(B\) each of mass \(m\) are connected by a light inextensible string of length \(l\) which passes through a small hole at a point \(O\) in a smooth horizontal table on which the particle \(A\) can move while \(B\) hangs vertically. The particle \(B\) is attached by a light elastic spring to a fixed point which is at a distance \(3l/2\) vertically below \(O\). The elastic spring has a natural length \(l\) and modulus of elasticity \(2mg\). Initially the string \(AB\) is tight and \(B\) is released from rest while simultaneously \(A\) is projected horizontally with a velocity \(\sqrt{(2gl)}\) at a distance \(l/2\) from \(O\) and at right angles to \(OA\). Show that the mass \(B\) is next instantaneously at rest when it has moved through a distance \(l/4\).

Shew that there is a unique value of \(\lambda\) for which \(ax^4+6cx^2+4dx+e\) is expressible in the form \(A(x-\alpha)^4 + B(x-\beta)^4\), where \(\lambda, A, B, \alpha, \beta\) are independent of \(x\). If \(a, c, d, e\) are real, find the condition that \(A, B, \alpha, \beta\) shall be real.

Shew that if \(n\) be a positive integer:

1. \(n - \dfrac{n^2(n-1)}{1!2!} + \dfrac{n^2(n^2-1^2)(n-2)}{2!3!} - \dots + (-1)^{n-1} \dfrac{n^2(n^2-1^2)\dots(n^2-(n-2)^2)\cdot 1}{(n-1)!n!} = 0\).
2. \(n - \dfrac{n(n^2-1^2)}{1!2!} + \dfrac{n(n^2-1^2)(n^2-2^2)}{2!3!} - \dots + (-1)^{n-1} \dfrac{n(n^2-1^2)\dots(n^2-(n-1)^2)}{(n-1)!n!} = (-1)^{n-1}\).

1. Find the sum to \(n\) terms of the series \[ \frac{1}{1.3} + \frac{1}{2.4} + \frac{1}{3.5} + \dots. \]
2. Find the general term of the series \[ 3+4+6+10+\dots, \] given that any three successive terms satisfy a given relation of the form \[ u_r+au_{r-1}+bu_{r-2}=0. \]

State a rule for the multiplication of two determinants of the same order. By considering the determinant \[ \begin{vmatrix} x & y & z \\ z & x & y \\ y & z & x \end{vmatrix} \] or otherwise, shew that the product \(x^3+y^3+z^3-3xyz\) and \(a^3+b^3+c^3-3abc\) can be expressed in the form \(X^3+Y^3+Z^3-3XYZ\), and find the values of \(X, Y, Z\) in terms of \(x, y, z, a, b, c\). Prove that the expression \[ \begin{vmatrix} a_1 & a_2 & a_3 & \dots & a_n \\ a_n & a_1 & a_2 & \dots & a_{n-1} \\ a_{n-1} & a_n & a_1 & \dots & a_{n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_2 & a_3 & a_4 & \dots & a_1 \end{vmatrix} \] has \(n\) factors of the form \((a_1+a_2\omega+a_3\omega^2+\dots+a_n\omega^{n-1})\), where \(\omega\) is one of the \(n\)th roots of unity.