| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Use trig identity before indefinite integration |
| Difficulty | Moderate -0.3 Part (a) is a straightforward reverse chain rule application with exponentials. Part (b) requires knowing the standard double-angle identity cos(2θ) = 1 - 2sin²(θ), rearranging it, then integrating—this is a common textbook exercise testing identity manipulation and basic integration, slightly easier than average due to being routine once the identity is recalled. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain integral of the form \(kc^{1-2x}\) with any non-zero \(k\) | M1 | |
| Correct integral | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to use double angle formula to expand \(\cos(3x + 3y)\) | M1 | |
| State correct expression \(\frac{1}{2} - \frac{1}{2}\cos 6x\) or equivalent | A1 | |
| Integrate an expression of the form \(a + b\cos 6x\), where \(ab \neq 0\), correctly | M1 | |
| State correct integral \(\frac{1}{2}x - \frac{1}{12}\sin 6x\), or equivalent | A1 | [4] |
**(a)**
| Obtain integral of the form $kc^{1-2x}$ with any non-zero $k$ | M1 |
| Correct integral | A1 | [2] |
**(b)**
| Attempt to use double angle formula to expand $\cos(3x + 3y)$ | M1 |
| State correct expression $\frac{1}{2} - \frac{1}{2}\cos 6x$ or equivalent | A1 |
| Integrate an expression of the form $a + b\cos 6x$, where $ab \neq 0$, correctly | M1 |
| State correct integral $\frac{1}{2}x - \frac{1}{12}\sin 6x$, or equivalent | A1 | [4] |4
\begin{enumerate}[label=(\alph*)]
\item Find $\int \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x$.
\item Express $\sin ^ { 2 } 3 x$ in terms of $\cos 6 x$ and hence find $\int \sin ^ { 2 } 3 x \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2010 Q4 [6]}}