6.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_245_435_356_817}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

The shaded region, shown in Figure 4, is bounded by the $x$-axis, the line with equation $x = 6$, the line with equation $y = 2$ and the $y$-axis.

This region is rotated through $360 ^ { \circ }$ about the $\boldsymbol { x }$-axis to form a solid of revolution. This solid is used to model a non-uniform cylinder of height 6 cm and radius 2 cm .

The mass per unit volume of the cylinder at the point $( x , y , z )$ is $\lambda ( x + 2 ) \mathrm { kg } \mathrm { cm } ^ { - 3 }$, where $0 \leqslant x \leqslant 6$ and $\lambda$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that the mass of the cylinder is $120 \lambda \pi \mathrm {~kg}$.
\item Show that the centre of mass of the cylinder is 3.6 cm from $O$.

The point $O$ is the centre of one end of the cylinder. The point $A$ is the centre of the other end of the cylinder.

A uniform solid hemisphere of radius 3 cm has density $\lambda \mathrm { kg } \mathrm { cm } ^ { - 3 }$. The hemisphere is attached to the cylinder with the centre of its circular face in contact with the point $A$ on the cylinder to form the model shown in Figure 5.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_309_673_1713_696}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}

The model is placed with the end containing $O$ on a rough inclined plane which is inclined at angle $\alpha ^ { \circ }$ to the horizontal. The plane is sufficiently rough to prevent the model from sliding. The model is on the point of toppling.
\item Find the value of $\alpha$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel FM2 2024 Q6 [13]}}