| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Use trig identity before definite integration |
| Difficulty | Standard +0.3 Part (a) is a straightforward application of the standard integral of 1/x with a constant factor. Part (b) requires using the double angle identity cos(2θ) = 1 - 2sin²(θ) to rewrite sin²(3x/2) before integrating, then applying reverse chain rule - this is a standard technique taught in P2 but requires one extra step beyond routine integration. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(\frac{3}{2}\ln x\) or \(\frac{3}{2}\ln(2x)\) or \(\frac{3}{2}\ln(kx)\) | B1 | |
| Use subtraction law of logarithms correctly, showing sufficient detail | M1 | \(\ln 216 - \ln 8 = \ln\left(\dfrac{216}{8}\right)\) |
| Use power law of logarithms correctly | M1 | \(n\ln(kx) = \ln(kx)^n\) |
| Confirm \(\ln 27\) with sufficient working and no incorrect working | A1 | AG |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use appropriate identity to express integrand in form \(k_1 + k_2\cos 3x\) | \*M1 | \(k_1 \neq 0\). Allow \(2 \times \frac{3}{2}x\) for \(3x\) |
| Obtain correct \(2 - 2\cos 3x\) | A1 | |
| Integrate to obtain form \(k_3 x + k_4\sin 3x\) | DM1 | |
| Obtain correct \(2x - \frac{2}{3}\sin 3x\) | A1 | |
| Use limits to obtain \(\frac{1}{3}\pi - \frac{2}{3}\) or exact equivalent | A1 | |
| Total: 5 |
## Question 6:
### Part 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $\frac{3}{2}\ln x$ or $\frac{3}{2}\ln(2x)$ or $\frac{3}{2}\ln(kx)$ | B1 | |
| Use subtraction law of logarithms correctly, showing sufficient detail | M1 | $\ln 216 - \ln 8 = \ln\left(\dfrac{216}{8}\right)$ |
| Use power law of logarithms correctly | M1 | $n\ln(kx) = \ln(kx)^n$ |
| Confirm $\ln 27$ with sufficient working and no incorrect working | A1 | AG |
| **Total: 4** | | |
### Part 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use appropriate identity to express integrand in form $k_1 + k_2\cos 3x$ | \*M1 | $k_1 \neq 0$. Allow $2 \times \frac{3}{2}x$ for $3x$ |
| Obtain correct $2 - 2\cos 3x$ | A1 | |
| Integrate to obtain form $k_3 x + k_4\sin 3x$ | DM1 | |
| Obtain correct $2x - \frac{2}{3}\sin 3x$ | A1 | |
| Use limits to obtain $\frac{1}{3}\pi - \frac{2}{3}$ or exact equivalent | A1 | |
| **Total: 5** | | |
---6
\begin{enumerate}[label=(\alph*)]
\item Show that $\int _ { 2 } ^ { 18 } \frac { 3 } { 2 x } \mathrm {~d} x = \ln 27$.
\item Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \left( \frac { 3 } { 2 } x \right) \mathrm { d } x$. Show all necessary working.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2019 Q6 [9]}}