| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2009 |
| Session | November |
| Marks | 9 |
| 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 calculus question requiring product rule differentiation, setting derivative to zero, numerical verification of a root, and applying a given iterative formula. All techniques are standard A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| Use product rule | M1 |
| Obtain correct derivative in any form | A1 |
| Equate derivative to zero and express \(\tan x\) in terms of \(x\) | M1 |
| Obtain given answer | A1 |
| Answer | Marks |
|---|---|
| Consider sign of \(\tan x - \frac{2}{x}\) at \(x = 1\) and \(x = 1.2\), or equivalent | M1 |
| Complete the argument with correct calculations | A1 |
| Answer | Marks |
|---|---|
| Use the iterative formula correctly at least once | M1 |
| Obtain final answer 1.08 | A1 |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval \((1.075, 1.085)\) | A1 |
**(i)**
| Use product rule | M1 |
| Obtain correct derivative in any form | A1 |
| Equate derivative to zero and express $\tan x$ in terms of $x$ | M1 |
| Obtain given answer | A1 |
**Total: [4]**
**(ii)**
| Consider sign of $\tan x - \frac{2}{x}$ at $x = 1$ and $x = 1.2$, or equivalent | M1 |
| Complete the argument with correct calculations | A1 |
**Total: [2]**
**(iii)**
| Use the iterative formula correctly at least once | M1 |
| Obtain final answer 1.08 | A1 |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval $(1.075, 1.085)$ | A1 |
**Total: [3]**
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{729aa2f6-2b62-445f-a2aa-a63b45cb6b64-3_604_971_262_587}
The diagram shows the curve $y = x ^ { 2 } \cos x$, for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$, and its maximum point $M$.\\
(i) Show by differentiation that the $x$-coordinate of $M$ satisfies the equation
$$\tan x = \frac { 2 } { x }$$
(ii) Verify by calculation that this equation has a root (in radians) between 1 and 1.2.\\
(iii) Use the iterative formula $x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { x _ { n } } \right)$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2009 Q7 [9]}}