| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Solve trigonometric equation via iteration |
| Difficulty | Standard +0.3 This is a straightforward fixed point iteration question requiring: (a) reading coordinates from a standard inverse cosine graph, (b) simple substitution to verify a root lies in an interval, and (c) mechanical application of a given iteration formula. All steps are routine with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(A(-1, \pi)\) | B1 | |
| \(B\left(0, \frac{\pi}{2}\right)\) | B1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\cos^{-1}x - 3x - 1 = 0\) | ||
| \(f(0.1) = 0.17\) allow \(0.2, 0.1\) | M1 | Or comparing 'sides' |
| \(f(0.2) = -0.23\) allow \(-0.2\) | ||
| Change of sign \(\therefore\) root | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x_1 = 0.1\) | M1 | |
| \(x_2 = 0.1569 = 0.157\) | A1 | |
| \(x_3 = 0.1378 = 0.138\) | ||
| \(x_4 = 0.144\) | A1 | |
| Total: 3 |
# Question 8:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $A(-1, \pi)$ | B1 | |
| $B\left(0, \frac{\pi}{2}\right)$ | B1 | |
| **Total: 2** | | |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\cos^{-1}x - 3x - 1 = 0$ | | |
| $f(0.1) = 0.17$ allow $0.2, 0.1$ | M1 | Or comparing 'sides' |
| $f(0.2) = -0.23$ allow $-0.2$ | | |
| Change of sign $\therefore$ root | A1 | |
| **Total: 2** | | |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $x_1 = 0.1$ | M1 | |
| $x_2 = 0.1569 = 0.157$ | A1 | |
| $x_3 = 0.1378 = 0.138$ | | |
| $x_4 = 0.144$ | A1 | |
| **Total: 3** | | |
---8 The diagram shows the curve $y = \cos ^ { - 1 } x$ for $- 1 \leqslant x \leqslant 1$.\\
\includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-4_492_698_1640_671}
\begin{enumerate}[label=(\alph*)]
\item Write down the exact coordinates of the points $A$ and $B$.
\item The equation $\cos ^ { - 1 } x = 3 x + 1$ has only one root. Given that the root of this equation is $\alpha$, show that $0.1 \leqslant \alpha \leqslant 0.2$.
\item Use the iteration $x _ { n + 1 } = \frac { 1 } { 3 } \left( \cos ^ { - 1 } x _ { n } - 1 \right)$ with $x _ { 1 } = 0.1$ to find the values of $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, giving your answers to three decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2007 Q8 [7]}}