Problems

The end \(O\) of a smooth light rod \(OA\) of length \(2a\) is a fixed point. The rod \(OA\) makes a fixed angle \(\sin^{-1}\frac{3}{5}\) with the downward vertical \(ON,\) but is free to rotate about \(ON.\) A particle of mass \(m\) is attached to the rod at \(A\) and a small ring \(B\) of mass \(m\) is free to slide on the rod but is joined to a spring of natural length \(a\) and modulus of elasticity \(kmg\). The vertical plane containing the rod \(OA\) rotates about \(ON\) with constant angular velocity \(\sqrt{5g/2a}\) and \(B\) is at rest relative to the rod. Show that the length of \(OB\) is \[ \frac{(10k+8)a}{10k-9}. \] Given that the reaction of the rod on the particle at \(A\) makes an angle \(\tan^{-1}\frac{13}{21}\) with the horizontal, find the value of \(k\). Find also the magnitude of the reaction between the rod and the ring \(B\).

A pack of \(2n\) (where \(n\geqslant4\)) cards consists of two each of \(n\) different sorts. If four cards are drawn from the pack without replacement show that the probability that no pairs of identical cards have been drawn is \[ \frac{4(n-2)(n-3)}{(2n-1)(2n-3)}. \] Find the probability that exactly one pair of identical cards is included in the four. If \(k\) cards are drawn without replacement and \(2 < k < 2n,\) find an expression for the probability that there are exactly \(r\) pairs of identical cards included when \(r < \frac{1}{2}k.\) For even values of \(k\) show that the probability that the drawn cards consist of \(\frac{1}{2}k\) pairs is \[ \frac{1\times3\times5\times\cdots\times(k-1)}{(2n-1)(2n-3)\cdots(2n-k+1)}. \]

The random variables \(X\) and \(Y\) take integer values \(x\) and \(y\) respectively which are restricted by \(x\geqslant1,\) \(y\geqslant1\) and \(2x+y\leqslant2a\) where \(a\) is an integer greater than 1. The joint probability is given by \[ \mathrm{P}(X=x,Y=y)=c(2x+y), \] where \(c\) is a positive constant, within this region and zero elsewhere. Obtain, in terms of \(x,c\) and \(a,\) the marginal probability \(\mathrm{P}(X=x)\) and show that \[ c=\frac{6}{a(a-1)(8a+5)}. \] Show that when \(y\) is an even number the marginal probability \(\mathrm{P}(Y=y)\) is \[ \frac{3(2a-y)(2a+2+y)}{2a(a-1)(8a+5)} \] and find the corresponding expression when \(y\) is off. Evaluate \(\mathrm{E}(Y)\) in terms of \(a\).