| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find intersection point coordinates |
| Difficulty | Moderate -0.3 This is a straightforward application of fixed point iteration with clear scaffolding: part (i) is simple substitution to verify a bracket, part (ii) is algebraic rearrangement (equating the two expressions and manipulating), and part (iii) is mechanical iteration with a given formula and starting value. While it requires multiple steps, each step is routine and the question guides students through the entire process without requiring problem-solving insight or novel approaches. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Consider sign of \(\frac{6}{x^2} - x - 1\) at \(x = 1.4\) and \(x = 1.6\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | [2] |
| (ii) State \(\frac{6}{x^2} = x + 1\) | B1 | |
| Rearrange equation to given equation or vice versa | B1 | [2] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.54\) | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval \((1.535, 1.545)\) | B1 | [3] |
**(i)** Consider sign of $\frac{6}{x^2} - x - 1$ at $x = 1.4$ and $x = 1.6$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations | A1 | [2]
**(ii)** State $\frac{6}{x^2} = x + 1$ | B1 |
Rearrange equation to given equation or vice versa | B1 | [2]
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer $1.54$ | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval $(1.535, 1.545)$ | B1 | [3]6 The curve with equation $y = \frac { 6 } { x ^ { 2 } }$ intersects the line $y = x + 1$ at the point $P$.\\
(i) Verify by calculation that the $x$-coordinate of $P$ lies between 1.4 and 1.6.\\
(ii) Show that the $x$-coordinate of $P$ satisfies the equation
$$x = \sqrt { } \left( \frac { 6 } { x + 1 } \right)$$
(iii) Use the iterative formula
$$x _ { n + 1 } = \sqrt { } \left( \frac { 6 } { x _ { n } + 1 } \right)$$
with initial value $x _ { 1 } = 1.5$, to determine the $x$-coordinate of $P$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2010 Q6 [7]}}