| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | March |
| 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 fixed point iteration requiring standard techniques: sketching y=sec(x) and y=2-x/2 to show intersection, substituting boundary values to verify the root location, and applying a given iterative formula. All steps are routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks |
|---|---|
| Sketch the graph \(y = \sec x\) | M1 |
| Sketch the graph \(y = 2-\frac{1}{2}x\), and justify the given statement | A1 |
| Answer | Marks |
|---|---|
| Calculate the values of a relevant expression or pair of expressions at \(x=0.8\) and \(x=1\) | M1 |
| Complete the argument correctly with correct calculated values | A1 |
| Answer | Marks |
|---|---|
| Use the iterative formula correctly at least once | M1 |
| Obtain final answer \(0.88\) | A1 |
| Show sufficient iterations to 4 d.p. to justify \(0.88\) to 2 d.p., or show there is a sign change in the interval \((0.875, 0.885)\) | A1 |
## Question 3(a):
| Sketch the graph $y = \sec x$ | M1 | |
| Sketch the graph $y = 2-\frac{1}{2}x$, and justify the given statement | A1 | |
## Question 3(b):
| Calculate the values of a relevant expression or pair of expressions at $x=0.8$ and $x=1$ | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
## Question 3(c):
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $0.88$ | A1 | |
| Show sufficient iterations to 4 d.p. to justify $0.88$ to 2 d.p., or show there is a sign change in the interval $(0.875, 0.885)$ | A1 | |
---3
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation $\sec x = 2 - \frac { 1 } { 2 } x$ has exactly one root in the interval $0 \leqslant x < \frac { 1 } { 2 } \pi$.
\item Verify by calculation that this root lies between 0.8 and 1 .
\item Use the iterative formula $x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 2 } { 4 - x _ { n } } \right)$ 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 2020 Q3 [7]}}