| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| 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 standard A-level fixed point iteration question requiring sketching two graphs to show root existence, verifying bounds by substitution, and applying a given iterative formula. All techniques are routine for P3/C3 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch a relevant graph, e.g. \(y = \cot\frac{1}{2}x\) | B1 | |
| Sketch a second relevant graph, e.g. \(y = 1 + e^{-x}\), and justify the given statement | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Calculate values of a relevant expression or pair of expressions at \(x = 1\) and \(x = 1.5\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.34\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify \(1.34\) to 2 d.p. or show there is a sign change in the interval \((1.335, 1.345)\) | A1 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch a relevant graph, e.g. $y = \cot\frac{1}{2}x$ | B1 | |
| Sketch a second relevant graph, e.g. $y = 1 + e^{-x}$, and justify the given statement | B1 | |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Calculate values of a relevant expression or pair of expressions at $x = 1$ and $x = 1.5$ | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $1.34$ | A1 | |
| Show sufficient iterations to 4 d.p. to justify $1.34$ to 2 d.p. or show there is a sign change in the interval $(1.335, 1.345)$ | A1 | |6
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation $\cot \frac { 1 } { 2 } x = 1 + \mathrm { e } ^ { - x }$ has exactly one root in the interval $0 < x \leqslant \pi$.
\item Verify by calculation that this root lies between 1 and 1.5.
\item Use the iterative formula $x _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 1 } { 1 + \mathrm { e } ^ { - x _ { n } } } \right)$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q6 [7]}}