A plane is inclined at an angle \(\arctan \frac34\) to the horizontal and a small, smooth, light pulley~\(P\) is fixed to the top of the plane. A string, \(APB\), passes over the pulley. A particle of mass~\(m_1\) is attached to the string at \(A\) and rests on the inclined plane with \(AP\) parallel to a line of greatest slope in the plane. A particle of mass \(m_2\), where \(m_2>m_1\), is attached to the string at \(B\) and hangs freely with \(BP\) vertical. The coefficient of friction between the particle at \(A\) and the plane is \( \frac{1}{2}\). The system is released from rest with the string taut. Show that the acceleration of the particles is \(\ds \frac{m_2-m_1}{m_2+m_1}g\). At a time \(T\) after release, the string breaks. Given that the particle at \(A\) does not reach the pulley at any point in its motion, find an expression in terms of \(T\) for the time after release at which the particle at \(A\) reaches its maximum height. It is found that, regardless of when the string broke, this time is equal to the time taken by the particle at \(A\) to descend from its point of maximum height to the point at which it was released. Find the ratio \(m_1 : m_2\). \noindent [Note that \(\arctan \frac34\) is another notation for \(\tan^{-1} \frac34\,\).]
The twins Anna and Bella share a computer and never sign their e-mails. When I e-mail them, only the twin currently online responds. The probability that it is Anna who is online is \(p\) and she answers each question I ask her truthfully with probability \(a\), independently of all her other answers, even if a question is repeated. The probability that it is Bella who is online is~\(q\), where \(q=1-p\), and she answers each question truthfully with probability \(b\), independently of all her other answers, even if a question is repeated.
The number of printing errors on any page of a large book of \(N\) pages is modelled by a Poisson variate with parameter \(\lambda\) and is statistically independent of the number of printing errors on any other page. The number of pages in a random sample of \(n\) pages (where \(n\) is much smaller than \(N\) and \(n\ge2\)) which contain fewer than two errors is denoted by \(Y\). Show that \(\P(Y=k) = \binom n k p^kq^{n-k}\) where \(p=(1+\lambda)e^{-\lambda}\) and \(q=1-p\,\). Show also that, if \(\lambda\) is sufficiently small,
The probability density function \(\f(x)\) of the random variable \(X\) is given by $$ \f(x) = k\left[{\phi}(x) + {\lambda}\g(x)\right],\,\,\,\, $$ where \({\phi}(x)\) is the probability density function of a normal variate with mean~0 and variance~1, \(\lambda \) is a positive constant, and \(\g(x)\) is a probability density function defined by \[ \g(x)= \begin{cases} 1/\lambda & \mbox{for \(0 \le x \le {\lambda}\)}\,;\\ 0& \mbox{otherwise} . \end{cases} \] Find \(\mu\), the mean of \(X\), in terms of \(\lambda\), and prove that \(\sigma\), the standard deviation of \(X\), satisfies. $$ \sigma^2 = \frac{\lambda^4 +4{\lambda}^3+12{\lambda}+12} {12(1 + \lambda )^2}\;. $$ In the case \(\lambda=2\):