| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Solve trigonometric equation via iteration |
| Difficulty | Standard +0.3 This is a straightforward iterative methods question requiring graph sketching, interval verification by substitution, and applying a given formula repeatedly. All techniques are standard P3 fare with no novel problem-solving required—slightly easier than average due to the formula being provided and the mechanical nature of the iterations. |
| 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. \(y = \cot x\): \(x\) intercept should be correct. Not touching the \(y\)-axis. No incorrect curvature. Ignore anything outside \(0 < x \leq \dfrac{1}{2}\pi\) | B1 | |
| Sketch a second relevant graph and justify the given statement, e.g. \(y = 2 - \cos x\). Condone if looks almost straight, but not if drawn with a ruler. Correct \(y\) intercept. Needs to be drawn for \(0 < x \leq \dfrac{1}{2}\pi\) | B1 | \(2^{\text{nd}}\) B1 requires a mark at the point of intersection or a suitable comment for the justification |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculate value of relevant expression or values of a pair of expressions at \(x = 0.6\) and \(x = 0.8\). Must be working in radians. Values correct to at least 2 significant figures. Need all relevant values but only one pair needs to be correct to award M1. Complete set of values for their expression required | M1 | e.g. \(1.17 < 1.46\), \(1.30 > 0.971\), \(-0.29 < 0\), \(0.33 > 0\); \(-0.20 < 0\), \(0.342 > 0\) from \(\tan x(2-\cos x)-1=0\); \(0.80 < 1\), \(1.34 > 1\) from \(\tan x(2-\cos x)=1\); \(0.146 > 0\), \(-0.105 < 0\) from \(x - \tan^{-1}\left(\dfrac{1}{2-\cos x}\right)\) |
| Complete the argument correctly with correct calculated values (awrt 2 s.f.). Clear comparison for their expression. Allow work on a smaller interval | A1 | Accept truncated values. If comparing with 0 can either indicate different signs or a negative product |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative process correctly at least once. Must be working in radians | M1 | |
| Obtain final answer 0.68 | A1 | Must be a clear conclusion |
| Show sufficient iterations to at least 4 d.p. to justify 0.68 to 2 d.p. or show there is a sign change in the interval \((0.675, 0.685)\). Allow recovery. Allow truncation. Allow small differences in the \(4^{\text{th}}\) s.f. | A1 | e.g. 0.7, 0.6806, 0.6855, 0.6843, 0.6846; 0.6, 0.7053, 0.6792, 0.6858, 0.6842, 0.6846; 0.8, 0.6545, 0.6920, 0.6826, 0.6850, 0.6844, 0.6845 |
| Total | 3 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch a relevant graph, e.g. $y = \cot x$: $x$ intercept should be correct. Not touching the $y$-axis. No incorrect curvature. Ignore anything outside $0 < x \leq \dfrac{1}{2}\pi$ | B1 | |
| Sketch a second relevant graph **and justify the given statement**, e.g. $y = 2 - \cos x$. Condone if looks almost straight, but not if drawn with a ruler. Correct $y$ intercept. Needs to be drawn for $0 < x \leq \dfrac{1}{2}\pi$ | B1 | $2^{\text{nd}}$ B1 requires a mark at the point of intersection or a suitable comment for the justification |
| **Total** | **2** | |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate value of relevant expression or values of a pair of expressions at $x = 0.6$ and $x = 0.8$. Must be working in radians. Values correct to at least 2 significant figures. Need all relevant values but only one pair needs to be correct to award M1. Complete set of values for their expression required | M1 | e.g. $1.17 < 1.46$, $1.30 > 0.971$, $-0.29 < 0$, $0.33 > 0$; $-0.20 < 0$, $0.342 > 0$ from $\tan x(2-\cos x)-1=0$; $0.80 < 1$, $1.34 > 1$ from $\tan x(2-\cos x)=1$; $0.146 > 0$, $-0.105 < 0$ from $x - \tan^{-1}\left(\dfrac{1}{2-\cos x}\right)$ |
| Complete the argument correctly with correct calculated values (awrt 2 s.f.). Clear comparison for their expression. Allow work on a smaller interval | A1 | Accept truncated values. If comparing with 0 can either indicate different signs or a negative product |
| **Total** | **2** | |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative process correctly at least once. Must be working in radians | M1 | |
| Obtain final answer 0.68 | A1 | Must be a clear conclusion |
| Show sufficient iterations to at least 4 d.p. to justify 0.68 to 2 d.p. or show there is a sign change in the interval $(0.675, 0.685)$. Allow recovery. Allow truncation. Allow small differences in the $4^{\text{th}}$ s.f. | A1 | e.g. 0.7, 0.6806, 0.6855, 0.6843, 0.6846; 0.6, 0.7053, 0.6792, 0.6858, 0.6842, 0.6846; 0.8, 0.6545, 0.6920, 0.6826, 0.6850, 0.6844, 0.6845 |
| **Total** | **3** | |
---6
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation
$$\cot x = 2 - \cos x$$
has one root in the interval $0 < x \leqslant \frac { 1 } { 2 } \pi$.
\item Show by calculation that this root lies between 0.6 and 0.8 .
\item Use the iterative formula $x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 2 - \cos x _ { n } } \right)$ 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 2023 Q6 [7]}}