| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive stationary point equation |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: differentiation to find the stationary point equation (routine chain/product rule), simple fixed-point iteration with a given formula (mechanical calculation), and integration by parts. All steps are procedural with no novel insight required, making it slightly easier than average. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07m Tangents and normals: gradient and equations1.07q Product and quotient rules: differentiation1.08i Integration by parts1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| 10(i) | Use correct product rule | M1 |
| Answer | Marks |
|---|---|
| Obtain correct derivative in any form y′=2xcos2x−2x 2 sin2x | A1 |
| Equate to zero and derive the given equation | A1 |
| Total: | 3 |
| Answer | Marks |
|---|---|
| 10(ii) | Use the iterative formula correctly at least once e.g. |
| 0.5→0.55357→0.53261→0.54070→0.53755 | M1 |
| Obtain final answer 0.54 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| the interval (0.535, 0.545) | A1 | |
| Total: | 3 | |
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 10(iii) | Integrate by parts and reach ax 2 sin2x+b∫xsin2x dx | *M1 |
| Answer | Marks |
|---|---|
| 2 2 | A1 |
| Answer | Marks |
|---|---|
| 2 2 4 | A1 |
| Answer | Marks |
|---|---|
| 4 | DM1 |
| Answer | Marks |
|---|---|
| 32 | A1 |
| Total: | 5 |
Question 10:
--- 10(i) ---
10(i) | Use correct product rule | M1
( )
Obtain correct derivative in any form y′=2xcos2x−2x 2 sin2x | A1
Equate to zero and derive the given equation | A1
Total: | 3
--- 10(ii) ---
10(ii) | Use the iterative formula correctly at least once e.g.
0.5→0.55357→0.53261→0.54070→0.53755 | M1
Obtain final answer 0.54 | A1
Show sufficient iterations to 4 d.p. to justify 0.54 to 2 d.p., or show there is a sign change in
the interval (0.535, 0.545) | A1
Total: | 3
Question | Answer | Marks
--- 10(iii) ---
10(iii) | Integrate by parts and reach ax 2 sin2x+b∫xsin2x dx | *M1
1 2 1
Obtain x sin2x−∫2x. sin2x dx
2 2 | A1
1 2 sin2x+1 1
Complete integration and obtain x xcos2x− sin2x, or equivalent
2 2 4 | A1
Substitute limits x = 0, x= 1 π, having integrated twice
4 | DM1
1 2
Obtain answer (π −8), or exact equivalent
32 | A1
Total: | 5\includegraphics{figure_10}
The diagram shows the curve $y = x^2 \cos 2x$ for $0 \leq x \leq \frac{1}{4}\pi$. The curve has a maximum point at $M$ where $x = p$.
\begin{enumerate}[label=(\roman*)]
\item Show that $p$ satisfies the equation $p = \frac{1}{2} \tan^{-1} \left(\frac{1}{p}\right)$. [3]
\item Use the iterative formula $p_{n+1} = \frac{1}{2} \tan^{-1} \left(\frac{1}{p_n}\right)$ to determine the value of $p$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
\item Find, showing all necessary working, the exact area of the region bounded by the curve and the $x$-axis. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q10 [11]}}