| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | March |
| 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 (likely resulting in arctan). The techniques are standard P3 content with no novel insight required, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Commence division and reach quotient of the form \(2x \pm 1\) | M1 | Or by inspection \(8x^3 + 4x^2 + 2x + 7 = (4x^2 + 1)(2x \pm 1) + r\) |
| Obtain (quotient) \(2x + 1\) | A1 | |
| Obtain (remainder) \(6\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain terms \(x^2 + x\) | B1 | OE |
| Obtain term of the form \(a\tan^{-1} 2x\) | M1 | |
| Obtain term \(3\tan^{-1} 2x\) | A1 | OE |
| Use \(x = 0\) and \(x = \frac{1}{2}\) as limits in a solution containing a term of the form \(a\tan^{-1} 2x\) | M1 | \(\left(\frac{1}{2}\right)^2 + \frac{1}{2} + a\frac{\pi}{4}\), need \(\frac{\pi}{4}\) seen or implied |
| Obtain final answer \(\frac{3}{4}(1 + \pi)\), or exact equivalent | A1 | ISW; answers in degrees score A0 |
| 5 |
## Question 8:
### Part 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Commence division and reach quotient of the form $2x \pm 1$ | M1 | Or by inspection $8x^3 + 4x^2 + 2x + 7 = (4x^2 + 1)(2x \pm 1) + r$ |
| Obtain (quotient) $2x + 1$ | A1 | |
| Obtain (remainder) $6$ | A1 | |
| | **3** | |
### Part 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain terms $x^2 + x$ | B1 | OE |
| Obtain term of the form $a\tan^{-1} 2x$ | M1 | |
| Obtain term $3\tan^{-1} 2x$ | A1 | OE |
| Use $x = 0$ and $x = \frac{1}{2}$ as limits in a solution containing a term of the form $a\tan^{-1} 2x$ | M1 | $\left(\frac{1}{2}\right)^2 + \frac{1}{2} + a\frac{\pi}{4}$, need $\frac{\pi}{4}$ seen or implied |
| Obtain final answer $\frac{3}{4}(1 + \pi)$, or exact equivalent | A1 | ISW; answers in degrees score A0 |
| | **5** | |
---8
\begin{enumerate}[label=(\alph*)]
\item Find the quotient and remainder when $8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7$ is divided by $4 x ^ { 2 } + 1$.
\item Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7 } { 4 x ^ { 2 } + 1 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q8 [8]}}