| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| 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 algebraic manipulation and integration. Part (a) involves routine polynomial division by a linear factor, and part (b) requires integrating a rational function after simplification and matching coefficients—all standard A-level techniques with no novel insight required. Slightly easier than average due to the guided structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out algebraic long division at least as far as \(2x^2 + kx\) | M1 | |
| Obtain quotient \(2x^2 + x + a - 2\) | A1 | |
| Confirm remainder is \(4\) | A1 | AG – necessary detail needed; SC B1 for use of remainder theorem to obtain 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Identify integrand as \(2x^2 + x + a - 2 + \frac{4}{x+2}\) | B1 FT | Following *their* quotient, may be implied |
| Integrate to obtain at least 2 terms from the form \(k_1 x^3 + k_2 x^2 + k_3\ln(x+2)\) | M1 | for non-zero \(k_1, k_2, k_3\) |
| Obtain correct \(\frac{2}{3}x^3 + \frac{1}{2}x^2 + ax - 2x + 4\ln(x+2)\) | A1 | |
| Apply limits correctly and attempt correct process to find \(a\) or \(b\) | M1 | Must have the correct form |
| Obtain \(-\frac{8}{3} + 2a = \frac{22}{3}\) or equivalent and hence \(a = 5\) | A1 | |
| Obtain \(4\ln 3\) and hence \(b = 81\) | B1 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out algebraic long division at least as far as $2x^2 + kx$ | M1 | |
| Obtain quotient $2x^2 + x + a - 2$ | A1 | |
| Confirm remainder is $4$ | A1 | AG – necessary detail needed; SC B1 for use of remainder theorem to obtain 4 |
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## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Identify integrand as $2x^2 + x + a - 2 + \frac{4}{x+2}$ | B1 FT | Following *their* quotient, may be implied |
| Integrate to obtain at least 2 terms from the form $k_1 x^3 + k_2 x^2 + k_3\ln(x+2)$ | M1 | for non-zero $k_1, k_2, k_3$ |
| Obtain correct $\frac{2}{3}x^3 + \frac{1}{2}x^2 + ax - 2x + 4\ln(x+2)$ | A1 | |
| Apply limits correctly and attempt correct process to find $a$ or $b$ | M1 | Must have the correct form |
| Obtain $-\frac{8}{3} + 2a = \frac{22}{3}$ or equivalent and hence $a = 5$ | A1 | |
| Obtain $4\ln 3$ and hence $b = 81$ | B1 | |7 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + a x + 2 a$$
where $a$ is an integer.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $x$ and $a$, the quotient when $\mathrm { p } ( x )$ is divided by ( $x + 2$ ), and show that the remainder is 4 .
\item It is given that $\int _ { - 1 } ^ { 1 } \frac { \mathrm { p } ( x ) } { x + 2 } \mathrm {~d} x = \frac { 22 } { 3 } + \ln b$, where $b$ is an integer.
Find the values of $a$ and $b$.\\
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 2022 Q7 [9]}}