| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Integration Using Polynomial Division |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring polynomial long division (routine A-level technique), integration of the resulting polynomial plus logarithmic term (standard P2 content), and a substitution to solve an exponential equation. All parts follow predictable patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08d Evaluate definite integrals: between limits1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out division at least as far as \(3x^2 + kx\) | M1 | |
| Obtain quotient \(3x^2 - 4x - 4\) | A1 | |
| Confirm remainder is 9 | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain at least \(k_1x^3\) and \(k_2\ln(3x+2)\) terms | \*M1 | |
| Obtain \(x^3 - 2x^2 - 4x + 3\ln(3x+2)\) | A1FT | FT from quotient in part (a) |
| Apply limits correctly | DM1 | |
| Apply appropriate logarithm properties correctly | M1 | |
| Obtain \(125 + \ln 64\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(9x^3 - 6x^2 - 20x - 8 = (3x+2)(3x^2-4x-4)\) | B1FT | FT from quotient in part (a) |
| Attempt to solve cubic equation to find positive value of \(x\) (or of \(e^{3y}\)) | M1 | |
| Use logarithms to solve equation of form \(e^{3y} = k\) where \(k > 0\) | M1 | |
| Obtain \(\frac{1}{3}\ln 2\) or exact equivalent | A1 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out division at least as far as $3x^2 + kx$ | M1 | |
| Obtain quotient $3x^2 - 4x - 4$ | A1 | |
| Confirm remainder is 9 | A1 | AG |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain at least $k_1x^3$ and $k_2\ln(3x+2)$ terms | \*M1 | |
| Obtain $x^3 - 2x^2 - 4x + 3\ln(3x+2)$ | A1FT | FT from quotient in part (a) |
| Apply limits correctly | DM1 | |
| Apply appropriate logarithm properties correctly | M1 | |
| Obtain $125 + \ln 64$ | A1 | |
---
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $9x^3 - 6x^2 - 20x - 8 = (3x+2)(3x^2-4x-4)$ | B1FT | FT from quotient in part (a) |
| Attempt to solve cubic equation to find positive value of $x$ (or of $e^{3y}$) | M1 | |
| Use logarithms to solve equation of form $e^{3y} = k$ where $k > 0$ | M1 | |
| Obtain $\frac{1}{3}\ln 2$ or exact equivalent | A1 | |7
\begin{enumerate}[label=(\alph*)]
\item Find the quotient when $9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1$ is divided by ( $3 x + 2$ ), and show that the remainder is 9 .
\item Hence find $\int _ { 1 } ^ { 6 } \frac { 9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1 } { 3 x + 2 } \mathrm {~d} x$, giving the answer in the form $a + \ln b$ where $a$ and $b$ are integers.
\item Find the exact root of the equation $9 e ^ { 9 y } - 6 e ^ { 6 y } - 20 e ^ { 3 y } - 8 = 0$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q7 [12]}}