\includegraphics{figure_3}

A particle $P$ of mass 4 kg is attached to one end of a light inextensible string. A particle $Q$ of mass $m$ kg is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac{3}{4}$ and the second plane is inclined to the horizontal at an angle $\beta$, where $\tan \beta = \frac{4}{3}$. Particle $P$ is on the first plane and particle $Q$ is on the second plane with the string taut, as shown in Figure 3.

The first plane is rough and the coefficient of friction between $P$ and the plane is $\frac{1}{4}$. The second plane is smooth. The system is in limiting equilibrium.

Given that $P$ is on the point of slipping down the first plane,

\begin{enumerate}[label=(\alph*)]
\item find the value of $m$,
[10]

\item find the magnitude of the force exerted on the pulley by the string,
[4]

\item find the direction of the force exerted on the pulley by the string.
[1]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M1 2016 Q7 [15]}}