| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| 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 multi-part question on numerical methods requiring standard techniques: sketching y = 3ln(x) and y = 15 - x³ to show intersection, substituting bounds to verify sign change, 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 | 1.02q 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) Sketch, showing the correct shape of each, \(y = 3\ln x\) and \(y = 15 - x^3\) | B1 | |
| Indicate the correct intercepts \((1,0)\) and \((0,15)\) | B1 | |
| Indicate one real root from two correct sketches | B1 | [3] |
| (ii) Consider sign of \(3\ln x + x^3 - 15\) for \(2.0\) and \(2.5\) or equivalent | M1 | |
| Justify conclusion with correct calculations (\(-4.9\) and \(3.4\) or equivalents) | A1 | [2] |
| (iii) Use the iteration process correctly at least once | M1 | |
| Obtain final answer \(2.319\) | A1 | |
| Show sufficient iterations to \(5\) decimal places to justify answer or show a sign change in the interval \((2.3185, 2.3195)\) | A1 | [3] |
**(i)** Sketch, showing the correct shape of each, $y = 3\ln x$ and $y = 15 - x^3$ | B1 |
Indicate the correct intercepts $(1,0)$ and $(0,15)$ | B1 |
Indicate one real root from two correct sketches | B1 | [3]
**(ii)** Consider sign of $3\ln x + x^3 - 15$ for $2.0$ and $2.5$ or equivalent | M1 |
Justify conclusion with correct calculations ($-4.9$ and $3.4$ or equivalents) | A1 | [2]
**(iii)** Use the iteration process correctly at least once | M1 |
Obtain final answer $2.319$ | A1 |
Show sufficient iterations to $5$ decimal places to justify answer or show a sign change in the interval $(2.3185, 2.3195)$ | A1 | [3]4 (i) By sketching a suitable pair of graphs, show that the equation
$$3 \ln x = 15 - x ^ { 3 }$$
has exactly one real root.\\
(ii) Show by calculation that the root lies between 2.0 and 2.5.\\
(iii) Use the iterative formula $x _ { n + 1 } = \sqrt [ 3 ] { } \left( 15 - 3 \ln x _ { n } \right)$ to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P2 2014 Q4 [8]}}