| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | November |
| Marks | 9 |
| 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 on differentiation and fixed-point iteration. Part (i) requires applying the product rule to find dy/dx and setting it equal to 3, then rearranging—a routine algebraic manipulation. Parts (ii) and (iii) are mechanical: substituting values to verify a sign change, then iterating a given formula until convergence. All techniques are standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.07q Product and quotient rules: differentiation1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) At any stage, state the correct derivative of \(e^{\frac{1}{2}x}\) | B1 | |
| Use product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to 3 and obtain given equation correctly | A1 | [4] |
| (ii) Consider sign of \(2 + 6e^{\frac{1}{2}x} - x\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | [2] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 3.21 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (3.205, 3.215) | B1 | [3] |
**(i)** At any stage, state the correct derivative of $e^{\frac{1}{2}x}$ | B1 |
Use product rule | M1 |
Obtain correct derivative in any form | A1 |
Equate derivative to 3 and obtain given equation correctly | A1 | [4]
**(ii)** Consider sign of $2 + 6e^{\frac{1}{2}x} - x$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations | A1 | [2]
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 3.21 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (3.205, 3.215) | B1 | [3]7\\
\includegraphics[max width=\textwidth, alt={}, center]{e82fee05-0c55-4fe2-b781-e5e82186c153-2_608_999_1430_571}
The diagram shows the curve $y = ( x - 4 ) \mathrm { e } ^ { \frac { 1 } { 2 } x }$. The curve has a gradient of 3 at the point $P$.\\
(i) Show that the $x$-coordinate of $P$ satisfies the equation
$$x = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x }$$
(ii) Verify that the equation in part (i) has a root between $x = 3.1$ and $x = 3.3$.\\
(iii) Use the iterative formula $x _ { n + 1 } = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } }$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2011 Q7 [9]}}