| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | March |
| Marks | 12 |
| 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 (a) requires routine implicit differentiation and algebraic manipulation, part (b) is simple substitution to verify a bracket, and part (c) applies a given iterative formula mechanically. All steps are standard textbook 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 diagrams1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| 5(a) | Differentiate using quotient rule | *M1 |
| Answer | Marks | Guidance |
|---|---|---|
| (x+2)2 | A1 | OE |
| Equate first derivative to 6 and simplify at least as far as x3 =... | DM1 | |
| Confirm x=312x+12 | A1 | Answer given – necessary detail needed. |
| Answer | Marks | Guidance |
|---|---|---|
| 5(b) | Consider sign of x−312x+12 , or equivalent, for 3.8 and 4.0 | M1 |
| Obtain −0.06... and 0.08..., or equivalents, and justify conclusion | A1 | Answer given – necessary detail needed. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 5(c) | Use iterative process correctly at least once | M1 |
| Obtain final answer 3.88 | A1 | Answer required to exactly 3 sf. |
| Answer | Marks |
|---|---|
| interval [3.875, 3.885] | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 5:
--- 5(a) ---
5(a) | Differentiate using quotient rule | *M1 | OE
(x+2)3x2−x3
Obtain
(x+2)2 | A1 | OE
Equate first derivative to 6 and simplify at least as far as x3 =... | DM1
Confirm x=312x+12 | A1 | Answer given – necessary detail needed.
4
--- 5(b) ---
5(b) | Consider sign of x−312x+12 , or equivalent, for 3.8 and 4.0 | M1
Obtain −0.06... and 0.08..., or equivalents, and justify conclusion | A1 | Answer given – necessary detail needed.
2
Question | Answer | Marks | Guidance
--- 5(c) ---
5(c) | Use iterative process correctly at least once | M1
Obtain final answer 3.88 | A1 | Answer required to exactly 3 sf.
Show sufficient iterations to 5 sf to justify answer or show sign change in the
interval [3.875, 3.885] | A1
3
Question | Answer | Marks | Guidance\includegraphics{figure_5}
The diagram shows part of the curve with equation $y = \frac{x^3}{x + 2}$. At the point $P$, the gradient of the curve is 6.
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$-coordinate of $P$ satisfies the equation $x = \sqrt[3]{12x + 12}$. [4]
\item Show by calculation that the $x$-coordinate of $P$ lies between 3.8 and 4.0. [2]
\item Use an iterative formula, based on the equation in part (a), to find the $x$-coordinate of $P$ correct to 3 significant figures. Show the result of each iteration to 5 significant figures. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q5 [12]}}