| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing routine A-level techniques: factor theorem verification (simple substitution), algebraic rearrangement to isolate x, and applying a given iterative formula. All steps are standard procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| 4(i) | Substitute –2 and simplify | M1 |
| Obtain 16−16+8+24−32 and hence zero and conclude | A1 | AG; necessary detail needed |
| Answer | Marks |
|---|---|
| 4(ii) | Attempt division by x+2 to reach at least partial quotient x3+kx or use of |
| identity or inspection | M1 |
| Obtain x3 +2x−16 | A1 |
| Equate to zero and obtain x= 316−2x | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 4(iii) | Use iteration process correctly at least once | M1 |
| Obtain final answer 2.256 | A1 |
| Answer | Marks |
|---|---|
| interval (2.2555,2.2565) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(i) ---
4(i) | Substitute –2 and simplify | M1
Obtain 16−16+8+24−32 and hence zero and conclude | A1 | AG; necessary detail needed
2
--- 4(ii) ---
4(ii) | Attempt division by x+2 to reach at least partial quotient x3+kx or use of
identity or inspection | M1
Obtain x3 +2x−16 | A1
Equate to zero and obtain x= 316−2x | A1
3
Question | Answer | Marks | Guidance
--- 4(iii) ---
4(iii) | Use iteration process correctly at least once | M1
Obtain final answer 2.256 | A1
Show sufficient iterations to 6 sf to justify answer or show a sign change in the
interval (2.2555,2.2565) | A1
3
Question | Answer | Marks | Guidance\includegraphics{figure_4}
The diagram shows the curve with equation
$$y = x^4 + 2x^3 + 2x^2 - 12x - 32.$$
The curve crosses the $x$-axis at points with coordinates $(\alpha, 0)$ and $(\beta, 0)$.
\begin{enumerate}[label=(\roman*)]
\item Use the factor theorem to show that $(x + 2)$ is a factor of
$$x^4 + 2x^3 + 2x^2 - 12x - 32.$$ [2]
\item Show that $\beta$ satisfies an equation of the form $x = \sqrt[3]{p + qx}$, and state the values of $p$ and $q$. [3]
\item Use an iterative formula based on the equation in part (ii) to find the value of $\beta$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2018 Q4 [11]}}