| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.3 This is a standard C3 fixed point iteration question with routine tasks: showing a root exists in an interval (sign change), rearranging an equation algebraically, applying an iterative formula with a calculator, and drawing a cobweb diagram. All parts follow textbook procedures with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(0.5) = -0.875\) | M1 | |
| \(f(1) = 2\) | ||
| Change of sign \(\therefore\) root | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x^3 + 4x - 3 = 0 \Rightarrow 4x = 3 - x^3\) | B1 | AG |
| \(x = \frac{3-x^3}{4}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x_1 = 0.5\) | M1 | |
| \(x_2 = 0.71875 \approx 0.72\) AWRT | A1 | |
| \(x_3 = 0.66\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Cobweb diagram shown | M1 | For cobweb, \(x_1\) to curve |
| A1 | For \(x_2\) | |
| A1 | All correct |
## Question 6:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(0.5) = -0.875$ | M1 | |
| $f(1) = 2$ | | |
| Change of sign $\therefore$ root | A1 | |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x^3 + 4x - 3 = 0 \Rightarrow 4x = 3 - x^3$ | B1 | AG |
| $x = \frac{3-x^3}{4}$ | | |
### Part (c)(i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x_1 = 0.5$ | M1 | |
| $x_2 = 0.71875 \approx 0.72$ AWRT | A1 | |
| $x_3 = 0.66$ | A1 | |
### Part (c)(ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Cobweb diagram shown | M1 | For cobweb, $x_1$ to curve |
| | A1 | For $x_2$ |
| | A1 | All correct |
---6 [Figure 1, printed on the insert, is provided for use in this question.]\\
The curve $y = x ^ { 3 } + 4 x - 3$ intersects the $x$-axis at the point $A$ where $x = \alpha$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\alpha$ lies between 0.5 and 1.0.
\item Show that the equation $x ^ { 3 } + 4 x - 3 = 0$ can be rearranged into the form $x = \frac { 3 - x ^ { 3 } } { 4 }$.\\
(1 mark)
\item \begin{enumerate}[label=(\roman*)]
\item Use the iteration $x _ { n + 1 } = \frac { 3 - x _ { n } { } ^ { 3 } } { 4 }$ with $x _ { 1 } = 0.5$ to find $x _ { 3 }$, giving your answer to two decimal places.\\
(3 marks)
\item The sketch on Figure 1 shows parts of the graphs of $y = \frac { 3 - x ^ { 3 } } { 4 }$ and $y = x$, and the position of $x _ { 1 }$.
On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of $x _ { 2 }$ and $x _ { 3 }$ on the $x$-axis.\\
(3 marks)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2006 Q6 [12]}}