| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| 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 (a) is routine algebraic manipulation, and part (b) involves standard integration techniques (polynomial terms plus logarithmic integration). The question is slightly easier than average as it guides students through the process and requires only well-practiced A-level techniques with no novel problem-solving. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out division at least as far as \(3x^2 + k_1x\) | M1 | Or equivalent (inspection, …) |
| Obtain quotient \(3x^2 + 4x - 1\) | A1 | |
| Confirm remainder is \(6\) | A1 | Answer given – necessary detail needed. SC B1 for use of factor theorem to show remainder is 6 if no other marks awarded |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Synthetic division: \(-\frac{2}{3}\) \ | \(9\) \ | \(18\) \ |
| Obtain quotient \(3x^2 + 4x - 1\) | (A1) | |
| Confirm remainder is \(6\) | (A1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Identify integrand as \(3x^2 + 4x - 1 + \frac{6}{3x+2}\) | B1 FT | Following *their* quotient |
| Integrate to obtain at least \(x^3\) and \(k_2\ln(3x+2)\) terms | *M1 | |
| Obtain \(x^3 + 2x^2 - x + 2\ln(3x+2)\) | A1 | |
| Apply limits and appropriate logarithm properties | DM1 | |
| Obtain \(14 + \ln 16\) | A1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out division at least as far as $3x^2 + k_1x$ | M1 | Or equivalent (inspection, …) |
| Obtain quotient $3x^2 + 4x - 1$ | A1 | |
| Confirm remainder is $6$ | A1 | Answer given – necessary detail needed. SC B1 for use of factor theorem to show remainder is 6 if no other marks awarded |
**Alternative Method 5(a) — Synthetic division:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Synthetic division: $-\frac{2}{3}$ \| $9$ \| $18$ \| $5$ \| $4$ → row: $-6$, $8$, $-2$ → result: $9$, $12$, $-3$, $6$ | (M1) | |
| Obtain quotient $3x^2 + 4x - 1$ | (A1) | |
| Confirm remainder is $6$ | (A1) | |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Identify integrand as $3x^2 + 4x - 1 + \frac{6}{3x+2}$ | B1 FT | Following *their* quotient |
| Integrate to obtain at least $x^3$ and $k_2\ln(3x+2)$ terms | *M1 | |
| Obtain $x^3 + 2x^2 - x + 2\ln(3x+2)$ | A1 | |
| Apply limits and appropriate logarithm properties | DM1 | |
| Obtain $14 + \ln 16$ | A1 | |5 The polynomial $\mathrm { p } ( x )$ is defined by $\mathrm { p } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } + 5 x + 4$.
\begin{enumerate}[label=(\alph*)]
\item Find the quotient when $\mathrm { p } ( x )$ is divided by $( 3 x + 2 )$, and show that the remainder is 6 .\\
\includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-08_2713_33_146_2012}\\
\includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-09_2723_33_138_20}
\item Find the value of $\int _ { 0 } ^ { 2 } \frac { \mathrm { p } ( x ) } { 3 x + 2 } \mathrm {~d} x$ ,giving your answer in the form $a + \ln b$ where $a$ and $b$ are integers.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q5 [8]}}