| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Moderate -0.3 This is a straightforward fixed point iteration question requiring standard techniques: sketching y=ln(x) and y=4-x/2 to show intersection, sign-change verification with calculator, and applying a given iterative formula. All steps are routine A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02p Interpret algebraic solutions: graphically1.02q Use intersection points: of graphs to solve equations1.06d Natural logarithm: ln(x) function and properties1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Make a recognisable sketch of \(y = \ln x\). Draw straight line with negative gradient crossing positive y-axis and justify one real root | B1 B1 | [2] |
| (ii) Consider sign of \(\ln x + \frac{1}{2}x - 4\) at 4.5 and 5.0 or equivalent. Complete the argument correctly with appropriate calculations | M1 A1 | [2] |
| (iii) Use the iterative formula correctly at least once. Obtain final answer \(4.84\). Show sufficient iterations to justify accuracy to 2 d.p. or show sign change in interval \((4.835, 4.845)\) | M1 A1 A1 | [3] |
(i) Make a recognisable sketch of $y = \ln x$. Draw straight line with negative gradient crossing positive y-axis and justify one real root | B1 B1 | [2]
(ii) Consider sign of $\ln x + \frac{1}{2}x - 4$ at 4.5 and 5.0 or equivalent. Complete the argument correctly with appropriate calculations | M1 A1 | [2]
(iii) Use the iterative formula correctly at least once. Obtain final answer $4.84$. Show sufficient iterations to justify accuracy to 2 d.p. or show sign change in interval $(4.835, 4.845)$ | M1 A1 A1 | [3]4 (i) By sketching a suitable pair of graphs, show that the equation
$$\ln x = 4 - \frac { 1 } { 2 } x$$
has exactly one real root, $\alpha$.\\
(ii) Verify by calculation that $4.5 < \alpha < 5.0$.\\
(iii) Use the iterative formula $x _ { n + 1 } = 8 - 2 \ln x _ { n }$ to find $\alpha$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2015 Q4 [7]}}