| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| 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 = 4x² - 1 and y = cot x to show intersection, substituting boundary values to verify the root location, and applying a given iterative formula. All steps are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| (i) Make recognisable sketch of a relevant graph over the given range | B1 |
| Sketch the other relevant graph on the same diagram and justify the given statement | B1 |
| Answer | Marks |
|---|---|
| (ii) Consider sign of \(4x^2 - 1 - \cot x\) at \(x = 0.6\) and \(x = 1\), or equivalent | M1 |
| Complete the argument correctly with correct calculated values | A1 |
| Answer | Marks |
|---|---|
| (iii) Use the iterative formula correctly at least once | M1 |
| Obtain final answer 0.73 | A1 |
| Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval (0.725, 0.735) | A1 |
**(i)** Make recognisable sketch of a relevant graph over the given range | B1 |
Sketch the other relevant graph on the same diagram and justify the given statement | B1 |
[2 marks total]
**(ii)** Consider sign of $4x^2 - 1 - \cot x$ at $x = 0.6$ and $x = 1$, or equivalent | M1 |
Complete the argument correctly with correct calculated values | A1 |
[2 marks total]
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 0.73 | A1 |
Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval (0.725, 0.735) | A1 |
[3 marks total]4 (i) By sketching suitable graphs, show that the equation
$$4 x ^ { 2 } - 1 = \cot x$$
has only one root in the interval $0 < x < \frac { 1 } { 2 } \pi$.\\
(ii) Verify by calculation that this root lies between 0.6 and 1 .\\
(iii) Use the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \sqrt { } \left( 1 + \cot x _ { n } \right)$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2010 Q4 [7]}}