| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| 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 requiring polynomial long division followed by term-by-term integration. Part (a) is mechanical division, and part (b) involves integrating a polynomial plus a simple rational function (resulting in ln and arctan terms). While it requires multiple techniques, each step follows standard procedures with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Commence division and reach quotient of the form \(2x + k\) | M1 | |
| Obtain quotient \(2x - 1\) | A1 | |
| Obtain remainder \(6\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain terms \(x^2 - x\) | B1FT | FT on quotient of the form \(2x+k\) |
| Obtain term of the form \(a\tan^{-1}\left(\dfrac{x}{\sqrt{3}}\right)\) | M1 | |
| Obtain term \(\dfrac{6}{\sqrt{3}}\tan^{-1}\left(\dfrac{x}{\sqrt{3}}\right)\) | A1FT | FT on a constant remainder |
| Use \(x=1\) and \(x=3\) as limits in a solution containing a term of the form \(a\tan^{-1}(bx)\) | M1 | |
| Obtain final answer \(\dfrac{1}{\sqrt{3}}\pi + 6\), or exact equivalent | A1 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Commence division and reach quotient of the form $2x + k$ | M1 | |
| Obtain quotient $2x - 1$ | A1 | |
| Obtain remainder $6$ | A1 | |
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## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain terms $x^2 - x$ | B1FT | FT on quotient of the form $2x+k$ |
| Obtain term of the form $a\tan^{-1}\left(\dfrac{x}{\sqrt{3}}\right)$ | M1 | |
| Obtain term $\dfrac{6}{\sqrt{3}}\tan^{-1}\left(\dfrac{x}{\sqrt{3}}\right)$ | A1FT | FT on a constant remainder |
| Use $x=1$ and $x=3$ as limits in a solution containing a term of the form $a\tan^{-1}(bx)$ | M1 | |
| Obtain final answer $\dfrac{1}{\sqrt{3}}\pi + 6$, or exact equivalent | A1 | |
---5
\begin{enumerate}[label=(\alph*)]
\item Find the quotient and remainder when $2 x ^ { 3 } - x ^ { 2 } + 6 x + 3$ is divided by $x ^ { 2 } + 3$.
\item Using your answer to part (a), find the exact value of $\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } - x ^ { 2 } + 6 x + 3 } { x ^ { 2 } + 3 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q5 [8]}}