| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find constant from definite integral |
| Difficulty | Standard +0.3 Part (a) is a straightforward reverse chain rule application with standard forms (ln and tan). Part (b) requires using the double angle identity cos²(x/2) = (1+cos x)/2, then integrating standard functions and solving for k from the definite integral equation. This is slightly above average due to the identity manipulation and definite integral calculation, but remains a standard textbook exercise with no novel insight required. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Express \(\dfrac{8}{\cos^2(4x+1)}\) as \(8\sec^2(4x+1)\) | B1 | SOI |
| Integrate to obtain the form \(a\ln(4x+1)\) | M1 | |
| Integrate to obtain \(b\tan(4x+1)\) | M1 | |
| Obtain \(2\ln(4x+1) + 2\tan(4x+1) + c\) | A1 | Condone use of brackets rather than modulus signs |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Express \(4\cos^2\frac{1}{2}x\) in the form \(p + q\cos x\) | M1 | For constants where \(pq \neq 0\) |
| Obtain correct \(2 + 2\cos x\) | A1 | |
| Integrate to obtain form \(px + q\sin x + r\cos 2x\) | \*M1 | For constants where \(pqr \neq 0\) |
| Obtain correct \(5x + 2\sin x - \frac{1}{2}k\cos 2x\) | A1 | Allow \(3x + 2x\) in place of \(5x\) |
| Apply limits correctly, equate to \(10\) and solve for \(k\) | DM1 | |
| Obtain \(k = 8 - \dfrac{5}{2}\pi\) | A1 | CWO |
| Total | 6 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Express $\dfrac{8}{\cos^2(4x+1)}$ as $8\sec^2(4x+1)$ | B1 | SOI |
| Integrate to obtain the form $a\ln(4x+1)$ | M1 | |
| Integrate to obtain $b\tan(4x+1)$ | M1 | |
| Obtain $2\ln(4x+1) + 2\tan(4x+1) + c$ | A1 | Condone use of brackets rather than modulus signs |
| **Total** | **4** | |
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## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Express $4\cos^2\frac{1}{2}x$ in the form $p + q\cos x$ | M1 | For constants where $pq \neq 0$ |
| Obtain correct $2 + 2\cos x$ | A1 | |
| Integrate to obtain form $px + q\sin x + r\cos 2x$ | \*M1 | For constants where $pqr \neq 0$ |
| Obtain correct $5x + 2\sin x - \frac{1}{2}k\cos 2x$ | A1 | Allow $3x + 2x$ in place of $5x$ |
| Apply limits correctly, equate to $10$ and solve for $k$ | DM1 | |
| Obtain $k = 8 - \dfrac{5}{2}\pi$ | A1 | CWO |
| **Total** | **6** | |
---6
\begin{enumerate}[label=(\alph*)]
\item Find $\int \left( \frac { 8 } { 4 x + 1 } + \frac { 8 } { \cos ^ { 2 } ( 4 x + 1 ) } \right) \mathrm { d } x$.
\item It is given that $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 3 + 4 \cos ^ { 2 } \frac { 1 } { 2 } x + k \sin 2 x \right) \mathrm { d } x = 10$.
Find the exact value of the constant $k$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q6 [10]}}