| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Difficulty | Standard +0.3 Part (a) requires recognizing the Pythagorean identity sin²θ + cos²θ = 1 to simplify the integrand, then splitting into standard integrals. Part (b) is a straightforward logarithmic integration with algebraic manipulation. Both parts are routine applications of standard techniques with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain integrand of form \(a\sec^2\theta + b\) | M1 | |
| Obtain correct \(5\sec^2\theta - 1\) | A1 | |
| Integrate to obtain form \(a\tan\theta + b\theta\) | M1 | |
| Obtain \(5\tan\theta - \theta + c\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain integral of form \(k\ln(3x+1)\) | \*M1 | |
| Apply limits and obtain \(\frac{2}{3}\ln(3a+1) = \ln 16\) | A1 | |
| Obtain equation with no presence of \(\ln\) | DM1 | |
| Obtain \(21\) | A1 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain integrand of form $a\sec^2\theta + b$ | M1 | |
| Obtain correct $5\sec^2\theta - 1$ | A1 | |
| Integrate to obtain form $a\tan\theta + b\theta$ | M1 | |
| Obtain $5\tan\theta - \theta + c$ | A1 | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain integral of form $k\ln(3x+1)$ | \*M1 | |
| Apply limits and obtain $\frac{2}{3}\ln(3a+1) = \ln 16$ | A1 | |
| Obtain equation with no presence of $\ln$ | DM1 | |
| Obtain $21$ | A1 | |
---4
\begin{enumerate}[label=(\alph*)]
\item Find $\int \frac { 4 + \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } \mathrm {~d} \theta$.
\item Given that $\int _ { 0 } ^ { a } \frac { 2 } { 3 x + 1 } \mathrm {~d} x = \ln 16$, find the value of the positive constant $a$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2017 Q4 [8]}}