| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 11 |
| 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 logarithmic integration requiring simple substitution or recognition of the standard form. Part (b) requires knowing the identities sin²x = (1-cos2x)/2 and tan²2x = sec²2x - 1, then integrating standard forms. While it involves multiple identities and careful manipulation, these are well-practiced techniques at A-level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.06f Laws of logarithms: addition, subtraction, power rules1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain form \(k\ln(3x+2)\) | \*M1 | Condone poor use of brackets if recovered later |
| Obtain correct \(4\ln(3x+2)\) | A1 | |
| Substitute limits correctly | M1 | Dependent \*M, must see \(k\ln 20-k\ln 5\) oe |
| Apply relevant logarithm properties correctly | M1 | Dependent \*M, do not allow \(\frac{4\ln 20}{4\ln 5}\) oe, must be using both the subtraction and power laws correctly |
| Obtain \(\ln 256\) nfww | A1 | AG; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use identity to obtain \(4(1-\cos 2x)\) oe | B1 | |
| Use identity to obtain \(\sec^2 2x-1\) | B1 | |
| Integrate to obtain form \(k_1x+k_2\sin 2x+k_3\tan 2x\) | \*M1 | Allow M1 if integrand contains \(p\cos 2x+q\sec^2 2x\) and no other trig terms |
| Obtain correct \(3x-2\sin 2x+\frac{1}{2}\tan 2x\) | A1 | |
| Apply limits correctly retaining exactness | M1 | Dependent \*M, allow \(\sin\frac{\pi}{3}\), \(\tan\frac{\pi}{3}\) |
| Obtain \(\frac{1}{2}\pi-\frac{1}{2}\sqrt{3}\) or exact equivalent | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $k\ln(3x+2)$ | \*M1 | Condone poor use of brackets if recovered later |
| Obtain correct $4\ln(3x+2)$ | A1 | |
| Substitute limits correctly | M1 | Dependent \*M, must see $k\ln 20-k\ln 5$ oe |
| Apply relevant logarithm properties correctly | M1 | Dependent \*M, do not allow $\frac{4\ln 20}{4\ln 5}$ oe, must be using both the subtraction and power laws correctly |
| Obtain $\ln 256$ nfww | A1 | AG; necessary detail needed |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use identity to obtain $4(1-\cos 2x)$ oe | B1 | |
| Use identity to obtain $\sec^2 2x-1$ | B1 | |
| Integrate to obtain form $k_1x+k_2\sin 2x+k_3\tan 2x$ | \*M1 | Allow M1 if integrand contains $p\cos 2x+q\sec^2 2x$ and no other trig terms |
| Obtain correct $3x-2\sin 2x+\frac{1}{2}\tan 2x$ | A1 | |
| Apply limits correctly retaining exactness | M1 | Dependent \*M, allow $\sin\frac{\pi}{3}$, $\tan\frac{\pi}{3}$ |
| Obtain $\frac{1}{2}\pi-\frac{1}{2}\sqrt{3}$ or exact equivalent | A1 | |6
\begin{enumerate}[label=(\alph*)]
\item Show that $\int _ { 1 } ^ { 6 } \frac { 12 } { 3 x + 2 } \mathrm {~d} x = \ln 256$.
\item Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( 8 \sin ^ { 2 } x + \tan ^ { 2 } 2 x \right) \mathrm { d } x$, showing all necessary working.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2018 Q6 [11]}}