| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Standard +0.8 This question combines graph sketching of transcendental functions (cosec and exponential), understanding of root existence, and iterative numerical methods with careful attention to domain restrictions and precision. While the iteration itself is straightforward to execute, the rearrangement to the given form and verification of convergence requires solid understanding of both the graphical and analytical aspects of fixed-point iteration, placing it moderately above average difficulty. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.06a Exponential function: a^x and e^x graphs and properties1.09c 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. \(y = \cosec x\) | B1 | \(\cosec x\), U shaped, roughly symmetrical about \(x = \frac{\pi}{2}\), \(y\!\left(\frac{\pi}{2}\right) = 1\) and domain at least \(\left(\frac{\pi}{6}, \frac{5\pi}{6}\right)\) |
| Sketch a second relevant graph, e.g. \(y = 1 + e^{-\frac{1}{2}x}\), and justify the given statement | B1 | Exponential graph needs \(y(0) = 2\), negative gradient, always increasing, and \(y(\pi) > 1\). Needs to mark intersections with dots, crosses, or say roots at points of intersection, or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the iterative formula correctly at least twice | M1 | 2, 2.3217, 2.2760, 2.2824… Need to see 2 iterations and following value inserted correctly |
| Obtain final answer 2.28 | A1 | Must be supported by iterations |
| Show sufficient iterations to at least 4 d.p. to justify 2.28 to 2 d.p., or show there is a sign change in the interval (2.275, 2.285) | A1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch a relevant graph, e.g. $y = \cosec x$ | B1 | $\cosec x$, U shaped, roughly symmetrical about $x = \frac{\pi}{2}$, $y\!\left(\frac{\pi}{2}\right) = 1$ and domain at least $\left(\frac{\pi}{6}, \frac{5\pi}{6}\right)$ |
| Sketch a second relevant graph, e.g. $y = 1 + e^{-\frac{1}{2}x}$, and justify the given statement | B1 | Exponential graph needs $y(0) = 2$, negative gradient, always increasing, and $y(\pi) > 1$. Needs to mark intersections with dots, crosses, or say roots at points of intersection, or equivalent |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the iterative formula correctly at least twice | M1 | 2, 2.3217, 2.2760, 2.2824… Need to see 2 iterations and following value inserted correctly |
| Obtain final answer 2.28 | A1 | Must be supported by iterations |
| Show sufficient iterations to at least 4 d.p. to justify 2.28 to 2 d.p., or show there is a sign change in the interval (2.275, 2.285) | A1 | |
---5
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation $\operatorname { cosec } x = 1 + \mathrm { e } ^ { - \frac { 1 } { 2 } x }$ has exactly two roots in the interval $0 < x < \pi$.
\item The sequence of values given by the iterative formula
$$x _ { n + 1 } = \pi - \sin ^ { - 1 } \left( \frac { 1 } { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } + 1 } \right)$$
with initial value $x _ { 1 } = 2$, converges to one of these roots.\\
Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q5 [5]}}