| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Difficulty | Moderate -0.8 Part (a) requires the standard double-angle identity cos²(θ/2) = (1 + cos θ)/2, then straightforward integration. Part (b) is a routine reverse chain rule application with logarithms. Both are textbook exercises testing direct recall of standard techniques with minimal problem-solving, making this easier than average but not trivial. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.06d Natural logarithm: ln(x) function and properties1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Express integrand in the form \(p\cos\theta + 2\) | M1 | |
| State correct \(2\cos\theta + 2\) | A1 | |
| Integrate to obtain \(2\sin\theta + 2\theta\ (+c)\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate to obtain form \(k\ln(2x+3)\) | M1 | |
| Obtain correct \(\frac{1}{2}\ln(2x+3)\) | A1 | |
| Apply limits correctly | DM1 | |
| Obtain \(\frac{1}{2}\ln 15\) | A1 | [4] |
## Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Express integrand in the form $p\cos\theta + 2$ | M1 | |
| State correct $2\cos\theta + 2$ | A1 | |
| Integrate to obtain $2\sin\theta + 2\theta\ (+c)$ | A1 | [3] |
## Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate to obtain form $k\ln(2x+3)$ | M1 | |
| Obtain correct $\frac{1}{2}\ln(2x+3)$ | A1 | |
| Apply limits correctly | DM1 | |
| Obtain $\frac{1}{2}\ln 15$ | A1 | [4] |
---3
\begin{enumerate}[label=(\alph*)]
\item Find $\int 4 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta$.
\item Find the exact value of $\int _ { - 1 } ^ { 6 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2014 Q3 [7]}}