| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.8 This is a straightforward textbook exercise on fixed-point iteration requiring only routine application of a given formula. Part (a) involves simple substitution to verify a sign change, part (b) is trivial algebraic rearrangement, and part (c) is mechanical calculator work with no problem-solving required. Significantly easier than average A-level questions. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(2) = -1\) and \(f(2.1) = +0.161\) with change of sign: \(-2 < \alpha < 2.1\) | M1 A1 | both attempted; 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^3 - x - 7 = 0 \Rightarrow x^3 = x + 7 \Rightarrow x = \sqrt[3]{x+7}\) | B1 | AG; 1 mark total |
| Answer | Marks | Guidance |
|---|---|---|
| \(x_1 = 2\); \(x_2 = 2.0801...\); \(x_3 = 2.0862...\); \(x_4 = 2.09\) | M1 A1 A1 | AWRT 2.08 and AWRT 2.09; 3 marks total |
### 1(a)
$f(2) = -1$ and $f(2.1) = +0.161$ with change of sign: $-2 < \alpha < 2.1$ | M1 A1 | both attempted; 2 marks total
### 1(b)
$x^3 - x - 7 = 0 \Rightarrow x^3 = x + 7 \Rightarrow x = \sqrt[3]{x+7}$ | B1 | AG; 1 mark total
### 1(c)
$x_1 = 2$; $x_2 = 2.0801...$; $x_3 = 2.0862...$; $x_4 = 2.09$ | M1 A1 A1 | AWRT 2.08 and AWRT 2.09; 3 marks total1 The curve $y = x ^ { 3 } - x - 7$ intersects the $x$-axis at the point where $x = \alpha$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\alpha$ lies between 2.0 and 2.1.
\item Show that the equation $x ^ { 3 } - x - 7 = 0$ can be rearranged in the form $x = \sqrt [ 3 ] { x + 7 }$.
\item Use the iteration $x _ { n + 1 } = \sqrt [ 3 ] { x _ { n } + 7 }$ with $x _ { 1 } = 2$ to find the values of $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, giving your answers to three significant figures.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2006 Q1 [6]}}