| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| 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 straightforward multi-part question on numerical methods requiring standard techniques: sketching y = cot x and y = 1 + x² to show intersection, substituting boundary values to verify the root location, and applying a given iterative formula. All steps are routine 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 graphs1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Make recognisable sketch of a relevant graph over the given range | B1 | |
| Sketch the other relevant graph and justify the given statement | B1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Consider the sign of \(\cot x - (1 + x^2)\) at \(x = 0.5\) and \(x = 0.8\), or equivalent | M1 | |
| Complete the argument with correct calculated values | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once with \(0.5 \leq x_n \leq 0.8\) | M1 | |
| Obtain final answer 0.62 | A1 | |
| Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval (0.615, 0.625) | A1 | [3] |
**(i)**
Make recognisable sketch of a relevant graph over the given range | B1 |
Sketch the other relevant graph and justify the given statement | B1 | [2]
**(ii)**
Consider the sign of $\cot x - (1 + x^2)$ at $x = 0.5$ and $x = 0.8$, or equivalent | M1 |
Complete the argument with correct calculated values | A1 | [2]
**(iii)**
Use the iterative formula correctly at least once with $0.5 \leq x_n \leq 0.8$ | M1 |
Obtain final answer 0.62 | A1 |
Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval (0.615, 0.625) | A1 | [3]6 (i) By sketching a suitable pair of graphs, show that the equation
$$\cot x = 1 + x ^ { 2 }$$
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 0.5 and 0.8.\\
(iii) Use the iterative formula
$$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + x _ { n } ^ { 2 } } \right)$$
to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2011 Q6 [7]}}