| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| 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 standard fixed-point iteration question requiring routine graph sketching (parabola and secant curve), sign-change verification, and iterative calculation. While it involves multiple parts and the secant function, each step follows textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02q Use intersection points: of graphs to solve equations1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch relevant graph, e.g. \(y = 4 - x^2\) | B1 | Needs \((0, 4)\) or marks on axis and \((2, 0)\) or \((\pi, 0)\) |
| Sketch second relevant graph, e.g. \(y = \sec\frac{1}{2}x\), and justify the given statement | B1 | Needs \((0, 1)\) or mark on axis and \((\pi, 0)\); asymptote NOT required, but must NOT reach \(x = \pi\); sec graph must exist over at least interval \(\left[0, \frac{3\pi}{4}\right]\) and quadratic graph over \([0, 2.5]\) |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Calculate value of relevant expression or values of pair of relevant expressions at \(x = 1\) and \(x = 2\) | M1 | Need all 4 values or the 2 values correct for M1; angles in degrees score M0 |
| Complete the argument with correct calculated values | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the iterative process correctly at least twice | M1 | |
| Obtain final answer \(1.60\) | A1 | Must be 2 d.p. |
| Show sufficient iterations to 4 d.p. to justify \(1.60\) to 2 d.p. or show there is a sign change in the interval \((1.595, 1.605)\) | A1 | |
| 3 |
## Question 7:
### Part 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch relevant graph, e.g. $y = 4 - x^2$ | B1 | Needs $(0, 4)$ or marks on axis and $(2, 0)$ or $(\pi, 0)$ |
| Sketch second relevant graph, e.g. $y = \sec\frac{1}{2}x$, and justify the given statement | B1 | Needs $(0, 1)$ or mark on axis and $(\pi, 0)$; asymptote NOT required, but must NOT reach $x = \pi$; sec graph must exist over at least interval $\left[0, \frac{3\pi}{4}\right]$ and quadratic graph over $[0, 2.5]$ |
| | **2** | |
### Part 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Calculate value of relevant expression or values of pair of relevant expressions at $x = 1$ and $x = 2$ | M1 | Need all 4 values or the 2 values correct for M1; angles in degrees score M0 |
| Complete the argument with correct calculated values | A1 | |
| | **2** | |
### Part 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the iterative process correctly at least twice | M1 | |
| Obtain final answer $1.60$ | A1 | Must be 2 d.p. |
| Show sufficient iterations to 4 d.p. to justify $1.60$ to 2 d.p. or show there is a sign change in the interval $(1.595, 1.605)$ | A1 | |
| | **3** | |
---7
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation $4 - x ^ { 2 } = \sec \frac { 1 } { 2 } x$ has exactly one root in the interval $0 \leqslant x < \pi$.
\item Verify by calculation that this root lies between 1 and 2 .
\item Use the iterative formula $x _ { n + 1 } = \sqrt { 4 - \sec \frac { 1 } { 2 } x _ { n } }$ 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 2022 Q7 [7]}}