| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Centre of mass of lamina by integration |
| Difficulty | Standard +0.8 This is a standard M3 centre of mass problem requiring integration to find ȳ = (∫y·dA)/(∫dA), but involves multiple steps: setting up the correct integral for a vertical strip, integrating (x+1)^(1/2) and (x+1), evaluating at limits, and simplifying fractions. The integration itself is routine, but the setup and algebraic manipulation across several stages makes it moderately challenging for M3 level. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_0^3 \sqrt{(x+1)}\, dx\) | M1 | Use of \(\int_0^3 \sqrt{(x+1)}\, dx\); limits not needed; accept \(k \times \int_0^3 \sqrt{(x+1)}\, dx\); attempt at integration seen when powers increase by 1 |
| \(= \frac{2}{3}\left[(x+1)^{\frac{3}{2}}\right]_0^3\) | A1 | Correct integrated expression with correct limits |
| \(\dfrac{\int_0^3 \frac{1}{2}(\sqrt{(x+1)})^2\, dx}{\int_0^3 \sqrt{(x+1)}\, dx}\) or \(\dfrac{\int_0^3 \frac{1}{2}(x+1)\, dx}{\int_0^3 \sqrt{(x+1)}\, dx}\) | M1 | Formula must be correct; allow constant multiple on both numerator and denominator; must see correct formula and attempt at integrating numerator |
| \(= \dfrac{\frac{1}{2}\left[\frac{1}{2}x^2+x\right]_0^3}{\frac{2}{3}\left[(x+1)^{\frac{3}{2}}\right]_0^3}\) or \(\dfrac{\frac{1}{2}\left[\frac{1}{2}(x+1)^2\right]_0^3}{\frac{2}{3}\left[(x+1)^{\frac{3}{2}}\right]_0^3}\) | A1 | Correct integrated expression for numerator in correct formula with correct limits |
| \(= \dfrac{45}{56}\) (0.80 or better) | A1 (5) | Correct answer; both previous A marks must have been awarded; numerical substitution does not need to be seen |
# Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_0^3 \sqrt{(x+1)}\, dx$ | M1 | Use of $\int_0^3 \sqrt{(x+1)}\, dx$; limits not needed; accept $k \times \int_0^3 \sqrt{(x+1)}\, dx$; attempt at integration seen when powers increase by 1 |
| $= \frac{2}{3}\left[(x+1)^{\frac{3}{2}}\right]_0^3$ | A1 | Correct integrated expression with correct limits |
| $\dfrac{\int_0^3 \frac{1}{2}(\sqrt{(x+1)})^2\, dx}{\int_0^3 \sqrt{(x+1)}\, dx}$ or $\dfrac{\int_0^3 \frac{1}{2}(x+1)\, dx}{\int_0^3 \sqrt{(x+1)}\, dx}$ | M1 | Formula must be correct; allow constant multiple on both numerator and denominator; must see correct formula and attempt at integrating numerator |
| $= \dfrac{\frac{1}{2}\left[\frac{1}{2}x^2+x\right]_0^3}{\frac{2}{3}\left[(x+1)^{\frac{3}{2}}\right]_0^3}$ or $\dfrac{\frac{1}{2}\left[\frac{1}{2}(x+1)^2\right]_0^3}{\frac{2}{3}\left[(x+1)^{\frac{3}{2}}\right]_0^3}$ | A1 | Correct integrated expression for numerator in correct formula with correct limits |
| $= \dfrac{45}{56}$ (0.80 or better) | A1 (5) | Correct answer; both previous A marks must have been awarded; numerical substitution does not need to be seen |
---\begin{enumerate}
\item In this question you must show all stages in your working.
\end{enumerate}
Solutions relying on calculator technology are not acceptable.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-02_579_1059_386_502}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The finite region $R$, shown shaded in Figure 1, is bounded by the $x$-axis, the line with equation $x = 3$, the curve with equation $y = \sqrt { ( x + 1 ) }$ and the $y$-axis.\\
Find the $\boldsymbol { y }$ coordinate of the centre of mass of a uniform lamina in the shape of $R$.
\hfill \mbox{\textit{Edexcel M3 2023 Q1 [5]}}