| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.3 This is a standard A-level fixed point iteration question requiring graph sketching, interval verification, algebraic rearrangement, and iterative calculation. All steps are routine applications of taught techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Sketch the other relevant graph and justify the given statement | B1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Complete the argument with correct calculated values | A1 | [2] |
| Answer | Marks |
|---|---|
| B1 | [1] |
| Answer | Marks |
|---|---|
| Obtain final answer 1.13 | A1 |
| Show sufficient iterations to 4 d.p. to justify 1.13 to 2 d.p., or show there is a sign change in the interval (1.125, 1.135) | A1 |
| [SR: Successive evaluation of the iterative function with \(x = 1, 2, \ldots\) scores M0.] | [3] |
**(i) Make recognisable sketch of a relevant graph over the given interval**
Sketch the other relevant graph and justify the given statement | B1 | [2]
**(ii) Consider the sign of $\sec x - (3 - \frac{1}{2}x^2)$ at $x = 1$ and $x = 1.4$, or equivalent**
Complete the argument with correct calculated values | A1 | [2]
**(iii) Convert the given equation to $\sec x = 3 - \frac{1}{2}x^2$ or work vice versa**
| B1 | [1]
**(iv) Use a correct iterative formula correctly at least once**
Obtain final answer 1.13 | A1 |
Show sufficient iterations to 4 d.p. to justify 1.13 to 2 d.p., or show there is a sign change in the interval (1.125, 1.135) | A1 |
[SR: Successive evaluation of the iterative function with $x = 1, 2, \ldots$ scores M0.] | [3]5 (i) By sketching a suitable pair of graphs, show that the equation
$$\sec x = 3 - \frac { 1 } { 2 } x ^ { 2 }$$
where $x$ is in radians, has a root in the interval $0 < x < \frac { 1 } { 2 } \pi$.\\
(ii) Verify by calculation that this root lies between 1 and 1.4.\\
(iii) Show that this root also satisfies the equation
$$x = \cos ^ { - 1 } \left( \frac { 2 } { 6 - x ^ { 2 } } \right)$$
(iv) Use an iterative formula based on the equation in part (iii) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2011 Q5 [8]}}