| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Integration Using Polynomial Division |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring polynomial long division followed by integration of the resulting quotient plus remainder term. Part (i) is routine algebraic manipulation, and part (ii) involves standard integration techniques (polynomial terms plus a logarithmic term from the remainder). The 'hence' structure guides students clearly, and expressing the final answer in the given form requires only basic logarithm laws. Slightly easier than average due to its predictable structure and standard techniques. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out division to obtain quotient of form \(x^2 + k\) | M1 | |
| Obtain quotient \(x^2 - 4\) | A1 | Allow use of an identity |
| Obtain remainder 4 | A1 | SC: If only the remainder theorem is used to obtain 4 then B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain at least \(k_1x^3\) and \(k_2\ln(2x+1)\) terms using the result from (i) | *M1 | |
| Obtain correct \(\frac{1}{3}x^3 - 4x + 2\ln(2x+1)\) | A1 | |
| Apply limits and note or imply that constant \(k_3\) can be written \(\ln e^{k_3}\) | DM1 | |
| Apply appropriate logarithm properties correctly | M1 | |
| Obtain \(\ln(49e^{-3})\) | A1 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out division to obtain quotient of form $x^2 + k$ | M1 | |
| Obtain quotient $x^2 - 4$ | A1 | Allow use of an identity |
| Obtain remainder 4 | A1 | **SC**: If only the remainder theorem is used to obtain 4 then B1 |
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## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain at least $k_1x^3$ and $k_2\ln(2x+1)$ terms using the result from (i) | *M1 | |
| Obtain correct $\frac{1}{3}x^3 - 4x + 2\ln(2x+1)$ | A1 | |
| Apply limits and note or imply that constant $k_3$ can be written $\ln e^{k_3}$ | DM1 | |
| Apply appropriate logarithm properties correctly | M1 | |
| Obtain $\ln(49e^{-3})$ | A1 | |
---5 (i) Find the quotient and remainder when $2 x ^ { 3 } + x ^ { 2 } - 8 x$ is divided by ( $2 x + 1$ ).\\
(ii) Hence find the exact value of $\int _ { 0 } ^ { 3 } \frac { 2 x ^ { 3 } + x ^ { 2 } - 8 x } { 2 x + 1 } \mathrm {~d} x$, giving the answer in the form $\ln \left( k \mathrm { e } ^ { a } \right)$ where $k$ and $a$ are constants.\\
\hfill \mbox{\textit{CAIE P2 2019 Q5 [8]}}