| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply iteration to find root (pure fixed point) |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring factor theorem verification (routine), polynomial division (standard technique), algebraic rearrangement to obtain an iterative formula, and applying fixed-point iteration. All steps are guided and use standard A-level techniques with no novel insight required. Slightly easier than average due to the scaffolding provided. |
| 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 |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(x = 3\) and attempt evaluation | M1 | |
| Obtain \(0\) and confirm factor \(x - 3\) | A1 | AG |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Divide quartic expression by \(x - 3\) at least as far as \(x^3 + kx^2\) | M1 | |
| Obtain \(x^3 - 2x^2\) | A1 | |
| Obtain \(x^3 - 2x^2 + 5\) | A1 | With no errors seen |
| Attempt rearrangement of their cubic expression to \(x = \ldots\) | M1 | Or \(a = \ldots\) |
| Confirm \(a = -\sqrt{\dfrac{5}{2-a}}\) | A1 | AG |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use iteration process correctly at least once | M1 | Need to see 3 values including *their* starting value |
| Obtain final answer \(-1.24\) | A1 | Answer required to exactly 3 significant figures |
| Show sufficient iterations to 5 sf to justify answer or show a sign change in the interval \([-1.245,\ -1.235]\) | A1 | |
| Total | 3 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $x = 3$ and attempt evaluation | M1 | |
| Obtain $0$ and confirm factor $x - 3$ | A1 | AG |
| **Total** | **3** | |
---
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Divide quartic expression by $x - 3$ at least as far as $x^3 + kx^2$ | M1 | |
| Obtain $x^3 - 2x^2$ | A1 | |
| Obtain $x^3 - 2x^2 + 5$ | A1 | With no errors seen |
| Attempt rearrangement of their cubic expression to $x = \ldots$ | M1 | Or $a = \ldots$ |
| Confirm $a = -\sqrt{\dfrac{5}{2-a}}$ | A1 | AG |
| **Total** | **5** | |
---
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use iteration process correctly at least once | M1 | Need to see 3 values including *their* starting value |
| Obtain final answer $-1.24$ | A1 | Answer required to exactly 3 significant figures |
| Show sufficient iterations to 5 sf to justify answer or show a sign change in the interval $[-1.245,\ -1.235]$ | A1 | |
| **Total** | **3** | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{c473f577-1e96-4d11-a0d5-cdfa4873c295-12_650_720_260_708}
A curve has equation $y = \mathrm { f } ( x )$ where $\mathrm { f } ( x ) = x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } + 5 x - 15$. As shown in the diagram, the curve crosses the $x$-axis at the points $A$ and $B$ with coordinates $( a , 0 )$ and $( b , 0 )$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Use the factor theorem to show that $( x - 3 )$ is a factor of $\mathrm { f } ( x )$.
\item By first finding the quotient when $\mathrm { f } ( x )$ is divided by $( x - 3 )$, show that
$$a = - \sqrt { \frac { 5 } { 2 - a } } .$$
\item Use an iterative formula, based on the equation in part (b), to find the value of $a$ correct to 3 significant figures. Give the result of each iteration to 5 significant figures.\\
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 2020 Q7 [10]}}