| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Finding Constants from Integration After Division |
| Difficulty | Standard +0.3 This is a straightforward polynomial division question requiring standard algebraic manipulation and integration. Part (a) involves routine polynomial long division, while part (b) combines this with a definite integral that simplifies nicely. The question is slightly easier than average as it follows a predictable structure with clear signposting and standard techniques throughout. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Carry out division at least as far as \(3x^2 + n_1\) | M1 | Or equivalent (inspection, …) |
| Obtain quotient \(3x^2 + 4\) | A1 | |
| Confirm remainder is \(k - 8\) | A1 | Answer given – necessary detail needed. SC B1 for correct use of factor theorem to show remainder is \(k-8\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Synthetic division with \(-\frac{2}{3}\): row entries \(9, 6, 12, k\) then \(-6, 0, -8\) giving \(9, 0, 12, k-8\) | (M1) | Allow one sign error |
| Obtain quotient \(3x^2 + 4\) | (A1) | |
| Confirm remainder is \(k - 8\) | (A1) | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate to obtain at least a term in \(x^3\) and term of form \(n_2\ln(3x+2)\) | \*M1 | Need to be using *their* answer to part (a) |
| Obtain \(x^3 + 4x + \frac{1}{3}(k-8)\ln(3x+2)\) | A1 | FT on a quotient of \(9x^2 + 12\) |
| Apply limits correctly to expression with three terms | DM1 | |
| Obtain \(a = 235\) | A1 | FT on a quotient of \(9x^2 + 12\) |
| Equate logarithm term to \(\ln 64\) and apply appropriate logarithm properties | DM1 | |
| Obtain \(k = 17\) | A1 | |
| Total | 6 | SC 2 marks for use of quotient \(3x^2+4\) or \(9x^2+12\) to obtain either 235 or 705 if no other marks are available |
## Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out division at least as far as $3x^2 + n_1$ | M1 | Or equivalent (inspection, …) |
| Obtain quotient $3x^2 + 4$ | A1 | |
| Confirm remainder is $k - 8$ | A1 | Answer given – necessary detail needed. SC B1 for correct use of factor theorem to show remainder is $k-8$ |
**Alternative Method (Synthetic Division):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Synthetic division with $-\frac{2}{3}$: row entries $9, 6, 12, k$ then $-6, 0, -8$ giving $9, 0, 12, k-8$ | (M1) | Allow one sign error |
| Obtain quotient $3x^2 + 4$ | (A1) | |
| Confirm remainder is $k - 8$ | (A1) | |
| **Total** | **3** | |
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## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate to obtain at least a term in $x^3$ and term of form $n_2\ln(3x+2)$ | \*M1 | Need to be using *their* answer to part (a) |
| Obtain $x^3 + 4x + \frac{1}{3}(k-8)\ln(3x+2)$ | A1 | FT on a quotient of $9x^2 + 12$ |
| Apply limits correctly to expression with three terms | DM1 | |
| Obtain $a = 235$ | A1 | FT on a quotient of $9x^2 + 12$ |
| Equate logarithm term to $\ln 64$ and apply appropriate logarithm properties | DM1 | |
| Obtain $k = 17$ | A1 | |
| **Total** | **6** | SC 2 marks for use of quotient $3x^2+4$ or $9x^2+12$ to obtain either 235 or 705 if no other marks are available |7 The polynomial $\mathrm { p } ( x )$ is defined by
$$p ( x ) = 9 x ^ { 3 } + 6 x ^ { 2 } + 12 x + k$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find the quotient when $\mathrm { p } ( x )$ is divided by $( 3 x + 2 )$ and show that the remainder is $( k - 8 )$.
\item It is given that $\int _ { 1 } ^ { 6 } \frac { \mathrm { p } ( \mathrm { x } ) } { 3 \mathrm { x } + 2 } \mathrm { dx } = \mathrm { a } + \ln 64$, where $a$ is an integer.
Find the values of $a$ and $k$.\\
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q7 [9]}}