| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with trigonometric functions |
| Difficulty | Moderate -0.3 Part (a) requires standard integration of sin(2x) and using the double angle identity for cos²x, then evaluating at limits - routine A-level techniques. Part (b) is a straightforward application of the trapezium rule formula with two intervals. Both parts are standard textbook exercises requiring recall and careful arithmetic rather than problem-solving, making this slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use identity \(2\cos^2 x = 1 + \cos 2x\) | B1 | |
| Integrate to obtain form \(x + \frac{1}{2}\sin 2x\) | B1 | |
| Integrate to obtain \(-2\cos 2x\) | B1 | |
| Apply limits correctly, retaining exactness | M1 | Dependent on at least one B mark |
| Obtain \(4 + \frac{1}{2}\pi\) or similarly simplified equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(y\) values \(\sqrt{\ln 3}\), \(\sqrt{\ln 6}\), \(\sqrt{\ln 9}\) or decimal equivalents | B1 | Allow awrt 1.05, 1.34, 1.48; the correct level of accuracy may be implied by a correct answer |
| Use correct formula, or equivalent, with \(h = 3\), and three \(y\) values | M1 | |
| Obtain \(\frac{1}{2} \times 3\left(\sqrt{\ln 3} + 2\sqrt{\ln 6} + \sqrt{\ln 9}\right)\) and hence 7.81 | A1 | Allow greater accuracy |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use identity $2\cos^2 x = 1 + \cos 2x$ | B1 | |
| Integrate to obtain form $x + \frac{1}{2}\sin 2x$ | B1 | |
| Integrate to obtain $-2\cos 2x$ | B1 | |
| Apply limits correctly, retaining exactness | M1 | Dependent on at least one B mark |
| Obtain $4 + \frac{1}{2}\pi$ or similarly simplified equivalent | A1 | |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $y$ values $\sqrt{\ln 3}$, $\sqrt{\ln 6}$, $\sqrt{\ln 9}$ or decimal equivalents | B1 | Allow awrt 1.05, 1.34, 1.48; the correct level of accuracy may be implied by a correct answer |
| Use correct formula, or equivalent, with $h = 3$, and three $y$ values | M1 | |
| Obtain $\frac{1}{2} \times 3\left(\sqrt{\ln 3} + 2\sqrt{\ln 6} + \sqrt{\ln 9}\right)$ and hence 7.81 | A1 | Allow greater accuracy |
---4
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 4 \sin 2 x + 2 \cos ^ { 2 } x \right) \mathrm { d } x$. Show all necessary working.
\item Use the trapezium rule with two intervals to find an approximation to $\int _ { 2 } ^ { 8 } \sqrt { } ( \ln ( 1 + x ) ) \mathrm { d } x$
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2019 Q4 [8]}}