CAIE P2 2024 June — Question 6 9 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2024
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicNumerical integration
TypeTrapezium rule with stated number of strips
DifficultyModerate -0.3 Part (a) is a straightforward application of the trapezium rule formula with two intervals—pure recall and substitution. Part (b) requires applying the volume of revolution formula π∫y²dx, which is standard P2 content. Both parts are routine textbook exercises with no problem-solving insight required, making this slightly easier than average.
Spec1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes
  1. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.
  2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_65_1548_379_349} \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_67_1566_466_328} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_646_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_735_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_826_324} \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_74_1570_916_324} ........................................................................................................................................ . \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1572_1096_322} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_1187_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_67_1570_1279_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_1367_324} \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_62_1570_1462_324} .......................................................................................................................................... ......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_1724_324} .......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_71_1570_1905_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_74_1570_1994_324} \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_76_1570_2083_324} ......................................................................................................................................... . ........................................................................................................................................ ......................................................................................................................................... ........................................................................................................................................ . ......................................................................................................................................... . ........................................................................................................................................
Question 6(a):
AnswerMarks Guidance
AnswerMarks Guidance
Use \(y\)-values \((0)\), \(\sqrt{\sin\frac{1}{6}\pi + \sin^2\frac{1}{6}\pi}\), \(\sqrt{\sin\frac{1}{3}\pi + \sin^2\frac{1}{3}\pi}\) or decimal equivalentsB1 \((0),\ \sqrt{0.75}\) or \(0.866,\ \sqrt{1.616}\) or \(1.271\)
Use correct formula, or equivalent, with \(h = \frac{1}{12}\pi\)M1 Must be using '\(y\)' values. May do as 2 separate trapezia \((0.113339 + 0.27976)\)
Obtain \(0.39\)A1 Allow \(0.393\) but not greater accuracy
Question 6(b):
AnswerMarks Guidance
AnswerMarks Guidance
Use \((\pi)\int(\sin 2x + \sin^2 2x)\,dx\)M1 OE
Express integrand in the form \(k_1\sin 2x + k_2 + k_3\cos 4x\)*M1 Where \(k_1 k_2 k_3 \neq 0\)
Obtain correct \(\sin 2x + \frac{1}{2} - \frac{1}{2}\cos 4x\)A1 Or \(\pi\) times this
Integrate to obtain \(-\frac{1}{2}\cos 2x + \frac{1}{2}x - \frac{1}{8}\sin 4x\)A1 FT Following *their* integrand only if of correct form
Apply limits correctly to obtain exact valueDM1
Obtain volume \(= \frac{1}{4}\pi + \frac{1}{12}\pi^2 - \frac{1}{16}\pi\sqrt{3}\)A1 Or exact equivalent 
## Question 6(a):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $y$-values $(0)$, $\sqrt{\sin\frac{1}{6}\pi + \sin^2\frac{1}{6}\pi}$, $\sqrt{\sin\frac{1}{3}\pi + \sin^2\frac{1}{3}\pi}$ or decimal equivalents | B1 | $(0),\ \sqrt{0.75}$ or $0.866,\ \sqrt{1.616}$ or $1.271$ |
| Use correct formula, or equivalent, with $h = \frac{1}{12}\pi$ | M1 | Must be using '$y$' values. May do as 2 separate trapezia $(0.113339 + 0.27976)$ |
| Obtain $0.39$ | A1 | Allow $0.393$ but not greater accuracy |

## Question 6(b):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $(\pi)\int(\sin 2x + \sin^2 2x)\,dx$ | M1 | OE |
| Express integrand in the form $k_1\sin 2x + k_2 + k_3\cos 4x$ | *M1 | Where $k_1 k_2 k_3 \neq 0$ |
| Obtain correct $\sin 2x + \frac{1}{2} - \frac{1}{2}\cos 4x$ | A1 | Or $\pi$ times this |
| Integrate to obtain $-\frac{1}{2}\cos 2x + \frac{1}{2}x - \frac{1}{8}\sin 4x$ | A1 FT | Following *their* integrand only if of correct form |
| Apply limits correctly to obtain exact value | DM1 | |
| Obtain volume $= \frac{1}{4}\pi + \frac{1}{12}\pi^2 - \frac{1}{16}\pi\sqrt{3}$ | A1 | Or exact equivalent |
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.
\item The shaded region is rotated completely about the $x$-axis.

Find the exact volume of the solid produced.\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_65_1548_379_349}\\
\includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_67_1566_466_328} ........................................................................................................................................\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_646_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_735_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_826_324}\\
\includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_74_1570_916_324} ........................................................................................................................................ .\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1572_1096_322}\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_1187_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_67_1570_1279_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_1367_324}\\
\includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_62_1570_1462_324} .......................................................................................................................................... ......................................................................................................................................... .\\
\includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_1724_324} .......................................................................................................................................... .\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_71_1570_1905_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_74_1570_1994_324}\\
\includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_76_1570_2083_324} ......................................................................................................................................... . ........................................................................................................................................ ......................................................................................................................................... ........................................................................................................................................ . ......................................................................................................................................... . ........................................................................................................................................
\end{enumerate}

\hfill \mbox{\textit{CAIE P2 2024 Q6 [9]}}
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