| Exam Board | CAIE |
|---|---|
| Module | Further Paper 3 (Further Paper 3) |
| Year | 2020 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Centre of mass of composite shapes |
| Difficulty | Challenging +1.2 This is a standard Further Maths mechanics problem involving center of mass of composite shapes and equilibrium conditions. Part (a) requires routine application of the composite body formula with straightforward algebra to verify the given expression. Part (b) involves setting up an equilibrium condition (center of mass must be above the pivot edge) and solving a quadratic equation, with the final answer requiring surd form. While it requires multiple techniques and careful algebraic manipulation, the methods are standard for FM mechanics and the question provides significant scaffolding by giving one expression to verify. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
\includegraphics{figure_4}
A uniform square lamina $ABCD$ has sides of length $10\text{cm}$. The point $E$ is on $BC$ with $EC = 7.5\text{cm}$, and the point $F$ is on $DC$ with $CF = x\text{cm}$. The triangle $EFC$ is removed from $ABCD$ (see diagram). The centre of mass of the resulting shape $ABEFD$ is a distance $\bar{x}\text{cm}$ from $CB$ and a distance $\bar{y}\text{cm}$ from $CD$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\bar{x} = \frac{400 - x^2}{80 - 3x}$ and find a corresponding expression for $\bar{y}$. [4]
\end{enumerate}
The shape $ABEFD$ is in equilibrium in a vertical plane with the edge $DF$ resting on a smooth horizontal surface.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the greatest possible value of $x$, giving your answer in the form $a + b\sqrt{2}$, where $a$ and $b$ are constants to be determined. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 3 2020 Q4 [7]}}