| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive stationary point equation |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard calculus techniques: finding dy/dx using quotient rule, setting equal to zero, and algebraic manipulation to reach the given form. The iteration in part (c) is routine application of a provided rearrangement. While it involves several steps, each is a standard A-level procedure with no novel insight required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use quotient or product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative at \(x = p\) to zero and obtain the given equation | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate a relevant expression or pair of relevant expressions at \(p = 2.5\) and \(p = 3\) | M1 | |
| Complete the argument with correct calculated values | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the iterative formula \(p_{n+1} = 3(1 - e^{-p_n})\) correctly at least once | M1 | |
| Obtain final answer \(p = 2.82\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify 2.82 to 2 d.p., or show there is a sign change in the interval \((2.815, 2.825)\) | A1 | |
| Total | 3 |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use quotient or product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative at $x = p$ to zero and obtain the given equation | A1 | |
## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate a relevant expression or pair of relevant expressions at $p = 2.5$ and $p = 3$ | M1 | |
| Complete the argument with correct calculated values | A1 | |
## Question 8(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the iterative formula $p_{n+1} = 3(1 - e^{-p_n})$ correctly at least once | M1 | |
| Obtain final answer $p = 2.82$ | A1 | |
| Show sufficient iterations to 4 d.p. to justify 2.82 to 2 d.p., or show there is a sign change in the interval $(2.815, 2.825)$ | A1 | |
| **Total** | **3** | |
---8 The curve with equation $y = \frac { x ^ { 3 } } { \mathrm { e } ^ { x } - 1 }$ has a stationary point at $x = p$, where $p > 0$.
\begin{enumerate}[label=(\alph*)]
\item Show that $p = 3 \left( 1 - \mathrm { e } ^ { - p } \right)$.
\item Verify by calculation that $p$ lies between 2.5 and 3 .
\item Use an iterative formula based on the equation in part (a) to determine $p$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q8 [8]}}