| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find gradient at a point - direct evaluation |
| Difficulty | Moderate -0.3 Part (a) requires chain rule application to differentiate sin²(3x) and direct substitution - a standard technique. Part (b) requires using the double angle identity cos(6x) = 1 - 2sin²(3x) to integrate, which is a common textbook exercise. Both parts are routine applications of standard methods with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Differentiate to obtain form \(k\sin 3x\cos 3x\) or \(k\sin 6x\) | M1 | |
| Obtain correct \(24\sin 3x\cos 3x\) or \(12\sin 6x\) | A1 | |
| Substitute \(x = \tfrac{1}{4}\pi\) to obtain \(-12\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Express integrand in form \(k_1 + k_2\cos 6x\) | M1 | Where \(k_1 k_2 \neq 0\) |
| Obtain \(2 - 2\cos 6x\) | A1 | |
| Integrate to obtain \(2x - \tfrac{1}{3}\sin 6x\) | A1 | OE |
| Total | 3 |
**Question 2(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differentiate to obtain form $k\sin 3x\cos 3x$ or $k\sin 6x$ | M1 | |
| Obtain correct $24\sin 3x\cos 3x$ or $12\sin 6x$ | A1 | |
| Substitute $x = \tfrac{1}{4}\pi$ to obtain $-12$ | A1 | |
| **Total** | **3** | |
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**Question 2(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Express integrand in form $k_1 + k_2\cos 6x$ | M1 | Where $k_1 k_2 \neq 0$ |
| Obtain $2 - 2\cos 6x$ | A1 | |
| Integrate to obtain $2x - \tfrac{1}{3}\sin 6x$ | A1 | OE |
| **Total** | **3** | |2 Let $\mathrm { f } ( x ) = 4 \sin ^ { 2 } 3 x$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\mathrm { f } ^ { \prime } \left( \frac { 1 } { 4 } \pi \right)$.
\item Find $\int \mathrm { f } ( x ) \mathrm { d } x$.\\
\includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-05_2723_35_101_20}
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q2 [6]}}