| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Iterative formula with graphical justification |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring graph sketching of modulus and logarithmic functions, algebraic manipulation to derive an iterative formula, and systematic application of iteration. While it involves several techniques, each step is guided and follows standard A-level procedures without requiring novel insight or complex problem-solving. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw correct sketch of \(y = \ | 5 - 2x\ | \) |
| Draw correct sketch of \(y = 3\ln x\) | \*B1 | |
| Indicate the two roots either on the diagram or by a statement | DB1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(2x - 5 = 3\ln x\) and rearrange to confirm \(x = 2.5 + 1.5\ln x\) | B1 | AG – necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Consider sign of \(x - 2.5 - 1.5\ln x\), or equivalent, for \(4.5\) and \(5.0\) | M1 | |
| Obtain \(-0.25\ldots\) and \(0.08\ldots\) or equivalents and justify conclusion | A1 | AG – necessary detail needed |
| Alternative: Consider sign of \(\ | 5-2x\ | - 3\ln x\), or equivalent, for \(4.5\) and \(5.0\) |
| Obtain \(-0.51\ldots\) and \(0.17\ldots\) or equivalents and justify conclusion | A1 | AG – necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use iteration process correctly at least once | M1 | |
| Obtain final answer \(4.88\) | A1 | Answer required to exactly 3 s.f. |
| Show sufficient iterations to 5 s.f. to justify answer or show sign change in interval \([4.875,\ 4.885]\) | A1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw correct sketch of $y = \|5 - 2x\|$ | \*B1 | with vertex on positive $x$-axis |
| Draw correct sketch of $y = 3\ln x$ | \*B1 | |
| Indicate the two roots either on the diagram or by a statement | DB1 | |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $2x - 5 = 3\ln x$ and rearrange to confirm $x = 2.5 + 1.5\ln x$ | B1 | AG – necessary detail needed |
---
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Consider sign of $x - 2.5 - 1.5\ln x$, or equivalent, for $4.5$ and $5.0$ | M1 | |
| Obtain $-0.25\ldots$ and $0.08\ldots$ or equivalents and justify conclusion | A1 | AG – necessary detail needed |
| **Alternative:** Consider sign of $\|5-2x\| - 3\ln x$, or equivalent, for $4.5$ and $5.0$ | M1 | |
| Obtain $-0.51\ldots$ and $0.17\ldots$ or equivalents and justify conclusion | A1 | AG – necessary detail needed |
---
## Question 5(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use iteration process correctly at least once | M1 | |
| Obtain final answer $4.88$ | A1 | Answer required to exactly 3 s.f. |
| Show sufficient iterations to 5 s.f. to justify answer or show sign change in interval $[4.875,\ 4.885]$ | A1 | |
---5
\begin{enumerate}[label=(\alph*)]
\item By sketching the graphs of
$$y = | 5 - 2 x | \quad \text { and } \quad y = 3 \ln x$$
on the same diagram, show that the equation $| 5 - 2 x | = 3 \ln x$ has exactly two roots.
\item Show that the value of the larger root satisfies the equation $x = 2.5 + 1.5 \ln x$.
\item Show by calculation that the value of the larger root lies between 4.5 and 5.0.
\item Use an iterative formula, based on the equation in part (b), to find the value of the larger root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q5 [9]}}