| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Polynomial division before integration |
| Difficulty | Moderate -0.3 This is a straightforward two-part question requiring polynomial long division followed by integration of the resulting polynomial plus a logarithmic term. Both techniques are standard P2 procedures with no novel insight required, making it slightly easier than average but still requiring careful algebraic manipulation across multiple steps. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out division at least as far as \(3x^2+k_1x\) | M1 | OE (e.g. by inspection) |
| Obtain quotient \(3x^2-4x-10\) | A1 | |
| Confirm given result of remainder is 6 with sufficient detail | A1 | AG. SC If remainder \(=6\) shown using remainder theorem allow B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate to obtain at least \(x^3\) and term of form \(k_2\ln(2x+1)\) | *M1 | ln term must be added |
| Obtain \(x^3-2x^2-10x+3\ln(2x+1)\) | A1 | |
| Apply limits correctly to expression with four terms | DM1 | |
| Apply appropriate logarithm properties correctly to obtain the form \(k_3\ln a\) | DM1 | |
| Obtain \(195+\ln 27\) | A1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out division at least as far as $3x^2+k_1x$ | M1 | OE (e.g. by inspection) |
| Obtain quotient $3x^2-4x-10$ | A1 | |
| Confirm given result of remainder is 6 with sufficient detail | A1 | AG. SC If remainder $=6$ shown using remainder theorem allow B1 |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain at least $x^3$ and term of form $k_2\ln(2x+1)$ | *M1 | ln term must be added |
| Obtain $x^3-2x^2-10x+3\ln(2x+1)$ | A1 | |
| Apply limits correctly to expression with four terms | DM1 | |
| Apply appropriate logarithm properties correctly to obtain the form $k_3\ln a$ | DM1 | |
| Obtain $195+\ln 27$ | A1 | |5
\begin{enumerate}[label=(\alph*)]
\item Find the quotient when $6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4$ is divided by ( $2 x + 1$ ), and show that the remainder is 6 .
\item Hence find
$$\int _ { 2 } ^ { 7 } \frac { 6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4 } { 2 x + 1 } d x$$
giving your answer in the form $a + \ln b$, where $a$ and $b$ are integers.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2023 Q5 [8]}}