| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find coordinate from gradient condition |
| Difficulty | Standard +0.3 This is a straightforward multi-part question involving standard techniques: differentiating to find the touching condition (part a), applying a given iterative formula (part b), and substituting to find k (part c). The 'touching' condition is a common A-level concept, and the iteration is mechanical with the formula provided. Slightly above average due to the multi-step nature and requiring understanding of what 'touching' means, but all techniques are standard P3 material. |
| Spec | 1.05o Trigonometric equations: solve in given intervals1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\cos p = \dfrac{k}{1+p}\) | B1 | |
| Differentiate both equations and equate derivatives at \(x = p\) | M1 | |
| Obtain a correct equation in any form, e.g. \(-\sin p = -\dfrac{k}{(1+p)^2}\) | A1 | |
| Eliminate \(k\) | M1 | |
| Obtain the given answer showing sufficient working | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(p = 0.568\) | A1 | |
| Show sufficient iterations to justify \(0.568\) to 3 d.p., or show there is a sign change in the interval \((0.5675, 0.5685)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to find \(k\) | M1 | |
| Obtain answer \(k = 1.32\) | A1 |
## Question 9:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\cos p = \dfrac{k}{1+p}$ | B1 | |
| Differentiate both equations and equate derivatives at $x = p$ | M1 | |
| Obtain a correct equation in any form, e.g. $-\sin p = -\dfrac{k}{(1+p)^2}$ | A1 | |
| Eliminate $k$ | M1 | |
| Obtain the **given answer** showing sufficient working | A1 | |
**Total: 5 marks**
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $p = 0.568$ | A1 | |
| Show sufficient iterations to justify $0.568$ to 3 d.p., or show there is a sign change in the interval $(0.5675, 0.5685)$ | A1 | |
**Total: 3 marks**
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to find $k$ | M1 | |
| Obtain answer $k = 1.32$ | A1 | |
**Total: 2 marks**
---9\\
\includegraphics[max width=\textwidth, alt={}, center]{3149080d-ad1a-4d2e-8e20-eb9977ced619-14_558_686_260_726}
The diagram shows the curves $y = \cos x$ and $y = \frac { k } { 1 + x }$, where $k$ is a constant, for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$. The curves touch at the point where $x = p$.
\begin{enumerate}[label=(\alph*)]
\item Show that $p$ satisfies the equation $\tan p = \frac { 1 } { 1 + p }$.
\item Use the iterative formula $p _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + p _ { n } } \right)$ to determine the value of $p$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
\item Hence find the value of $k$ correct to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q9 [10]}}