| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Standard +0.3 This is a standard A-level fixed point iteration question requiring routine sketching of y=sec(x) and y=3-x, straightforward substitution to verify the interval, algebraic rearrangement (which is given), and mechanical application of an iterative formula. All steps are procedural with no novel insight required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05i 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 |
|---|---|---|
| Make recognisable sketch of a relevant graph, e.g. \(y = \sec x\) | B1 | |
| Sketch an appropriate second graph, e.g. \(y = 3 - x\), correctly and justify the given statement | B1 | [2 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Consider sign of \(\sec x - (3 - x)\) at \(x = 1\) and \(x = 1.2\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | [2 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Show that the given equation is equivalent to \(\sec x = 3 - x\), or vice versa | B1 | [1 mark] |
| Answer | Marks | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.04 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p., or show there is a sign change in the interval (1.035, 1.045) | B1 | [3 marks] |
**(i)**
Make recognisable sketch of a relevant graph, e.g. $y = \sec x$ | B1 | |
Sketch an appropriate second graph, e.g. $y = 3 - x$, correctly and justify the given statement | B1 | [2 marks] |
**(ii)**
Consider sign of $\sec x - (3 - x)$ at $x = 1$ and $x = 1.2$, or equivalent | M1 | |
Complete the argument correctly with appropriate calculations | A1 | [2 marks] |
**(iii)**
Show that the given equation is equivalent to $\sec x = 3 - x$, or vice versa | B1 | [1 mark] |
**(iv)**
Use the iterative formula correctly at least once | M1 | |
Obtain final answer 1.04 | A1 | |
Show sufficient iterations to justify its accuracy to 2 d.p., or show there is a sign change in the interval (1.035, 1.045) | B1 | [3 marks] |5 (i) By sketching a suitable pair of graphs, show that the equation
$$\sec x = 3 - x$$
where $x$ is in radians, has only one root in the interval $0 < x < \frac { 1 } { 2 } \pi$.\\
(ii) Verify by calculation that this root lies between 1.0 and 1.2.\\
(iii) Show that this root also satisfies the equation
$$x = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x } \right)$$
(iv) Use the iterative formula
$$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } } \right)$$
with initial value $x _ { 1 } = 1.1$, to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2007 Q5 [8]}}