| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | November |
| Marks | 9 |
| 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 combining polynomial division (a standard algebraic technique) with integration. Part (a) is routine division with verification of the remainder. Part (b) requires recognizing that the quotient from part (a) simplifies the integrand, leaving a logarithmic term from the remainder and polynomial terms to integrate—all standard P2 techniques with no novel insight required. Slightly easier than average due to clear scaffolding. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out division at least as far as the \(3x^2 + kx\) stage | M1 | Where \(k\) should be zero |
| Obtain quotient \(3x^2 + 2\) | A1 | |
| Confirm remainder is \(2\) | A1 | AG – necessary detail needed. SC If M0A0 obtained then allow B1 if remainder theorem is used to obtain 2. Must see sufficient detail. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identify integrand as \(2 + \frac{2}{4x-3}\) | B1 FT | FT *their* quotient |
| Integrate to obtain at least \(k\ln(4x-3)\) term | \*M1 | |
| Obtain correct \(2x + \frac{1}{2}\ln(4x-3)\) | A1 | |
| Apply limits correctly | DM1 | |
| Use relevant logarithm properties | M1 | Must be in the form \(k\ln(4x-3)\) |
| Obtain \(20 + \ln 3\) | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out division at least as far as the $3x^2 + kx$ stage | M1 | Where $k$ should be zero |
| Obtain quotient $3x^2 + 2$ | A1 | |
| Confirm remainder is $2$ | A1 | AG – necessary detail needed. SC If M0A0 obtained then allow B1 if remainder theorem is used to obtain 2. Must see sufficient detail. |
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## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Identify integrand as $2 + \frac{2}{4x-3}$ | B1 FT | FT *their* quotient |
| Integrate to obtain at least $k\ln(4x-3)$ term | \*M1 | |
| Obtain correct $2x + \frac{1}{2}\ln(4x-3)$ | A1 | |
| Apply limits correctly | DM1 | |
| Use relevant logarithm properties | M1 | Must be in the form $k\ln(4x-3)$ |
| Obtain $20 + \ln 3$ | A1 | |
---6 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = 12 x ^ { 3 } - 9 x ^ { 2 } + 8 x - 4$$
\begin{enumerate}[label=(\alph*)]
\item Find the quotient when $\mathrm { p } ( x )$ is divided by $( 4 x - 3 )$ and show that the remainder is 2 .
\item Hence find $\int _ { 2 } ^ { 12 } \left( \frac { \mathrm { p } ( x ) } { 4 x - 3 } - 3 x ^ { 2 } \right) \mathrm { d } x$, giving your answer in the form $a + \ln b$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q6 [9]}}