| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on fixed point iteration requiring standard techniques: sketching two familiar curves (exponential and parabola), verifying a root by substitution, and applying a given iterative formula. All steps are routine with no novel insight required, making it slightly easier than average. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sketch a relevant graph, e.g. \(y = e^{-\frac{1}{2}x}\) | B1 | |
| Sketch a second relevant graph, e.g. \(y = 4 - x^2\), and justify the given statement | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculate the value of a relevant expression or values of a pair of expressions at \(x = -1\) and \(x = -1.5\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(-1.41\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify \(-1.41\) to 2 d.p., or show there is a sign change in the interval \((-1.415, -1.405)\) | A1 |
## Question 3(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch a relevant graph, e.g. $y = e^{-\frac{1}{2}x}$ | B1 | |
| Sketch a second relevant graph, e.g. $y = 4 - x^2$, and justify the given statement | B1 | |
**Total: 2**
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## Question 3(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate the value of a relevant expression or values of a pair of expressions at $x = -1$ and $x = -1.5$ | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
**Total: 2**
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## Question 3(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $-1.41$ | A1 | |
| Show sufficient iterations to 4 d.p. to justify $-1.41$ to 2 d.p., or show there is a sign change in the interval $(-1.415, -1.405)$ | A1 | |
**Total: 3**
---3 (i) By sketching suitable graphs, show that the equation $\mathrm { e } ^ { - \frac { 1 } { 2 } x } = 4 - x ^ { 2 }$ has one positive root and one negative root.\\
(ii) Verify by calculation that the negative root lies between - 1 and - 1.5 .\\
(iii) Use the iterative formula $x _ { n + 1 } = - \sqrt { } \left( 4 - e ^ { - \frac { 1 } { 2 } x _ { n } } \right)$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2017 Q3 [7]}}