| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| 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, algebraic manipulation, and applying a given iterative formula. All techniques are standard A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use the product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate 2-term derivative to zero and obtain the given answer correctly | A1 | [3] |
| (ii) Use calculations to consider the sign of a relevant expression at \(p = 2\) and \(p = 2.5\), or compare values of relevant expressions at \(p = 2\) and \(p = 2.5\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 | [2] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 2.15 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 2.15 to 2 d.p., or show there is a sign change in the interval (2.145, 2.155) | A1 | [3] |
**(i)** Use the product rule | M1 |
Obtain correct derivative in any form | A1 |
Equate 2-term derivative to zero and obtain the given answer correctly | A1 | [3]
**(ii)** Use calculations to consider the sign of a relevant expression at $p = 2$ and $p = 2.5$, or compare values of relevant expressions at $p = 2$ and $p = 2.5$ | M1 |
Complete the argument correctly with correct calculated values | A1 | [2]
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 2.15 | A1 |
Show sufficient iterations to 4 d.p. to justify 2.15 to 2 d.p., or show there is a sign change in the interval (2.145, 2.155) | A1 | [3]
---6 The curve with equation $y = x ^ { 2 } \cos \frac { 1 } { 2 } x$ has a stationary point at $x = p$ in the interval $0 < x < \pi$.\\
(i) Show that $p$ satisfies the equation $\tan \frac { 1 } { 2 } p = \frac { 4 } { p }$.\\
(ii) Verify by calculation that $p$ lies between 2 and 2.5.\\
(iii) Use the iterative formula $p _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 4 } { p _ { n } } \right)$ to determine the value of $p$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2016 Q6 [8]}}