| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.3 This is a straightforward fixed point iteration question requiring routine algebraic manipulation to rearrange x⁴ + 2x³ + 2x² - 12x - 32 = 0 into the form x = ∛(p + qx). The factor theorem part is basic substitution, and the iteration itself is mechanical application of a formula. While it involves multiple parts, each step follows standard A-level procedures without requiring novel insight or complex problem-solving. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(-2\) and simplify | M1 | |
| Obtain \(16 - 16 + 8 + 24 - 32\) and hence zero and conclude | A1 | AG; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt division by \(x+2\) to reach at least partial quotient \(x^3 + kx\) or use of identity or inspection | M1 | |
| Obtain \(x^3 + 2x - 16\) | A1 | |
| Equate to zero and obtain \(x = \sqrt[3]{16-2x}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 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 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $-2$ and simplify | M1 | |
| Obtain $16 - 16 + 8 + 24 - 32$ and hence zero and conclude | A1 | AG; necessary detail needed |
---
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt division by $x+2$ to reach at least partial quotient $x^3 + kx$ or use of identity or inspection | M1 | |
| Obtain $x^3 + 2x - 16$ | A1 | |
| Equate to zero and obtain $x = \sqrt[3]{16-2x}$ | A1 | |
---
## Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| 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 | |
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-06_652_789_260_676}
The diagram shows the curve with equation
$$y = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 12 x - 32$$
The curve crosses the $x$-axis at points with coordinates $( \alpha , 0 )$ and $( \beta , 0 )$.\\
(i) Use the factor theorem to show that $( x + 2 )$ is a factor of
$$x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 12 x - 32$$
(ii) Show that $\beta$ satisfies an equation of the form $x = \sqrt [ 3 ] { } ( p + q x )$, and state the values of $p$ and $q$. [3]\\
(iii) 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.\\
\hfill \mbox{\textit{CAIE P2 2018 Q4 [8]}}