| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive stationary point equation |
| Difficulty | Challenging +1.2 This question requires finding a stationary point by differentiation (using quotient rule and chain rule), then algebraic manipulation using the double angle formula for tan. The iterative part is straightforward application. While it involves multiple techniques, each step follows standard A-level procedures without requiring novel insight—slightly above average due to the algebraic manipulation needed in part (a). |
| Spec | 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 correct quotient rule or correct product rule | M1 | e.g. \(\frac{dy}{dx} = \frac{\sqrt{x} \cdot \frac{1}{1+x^2} - \tan^{-1}x \cdot \frac{1}{2\sqrt{x}}}{x}\) |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and remove inverse tangent | M1 | |
| Obtain \(a = \tan\left(\frac{2a}{1+a^2}\right)\) from correct working | A1 | AG. Accept with \(x\) in place of \(a\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculate the value of a relevant expression or pair of expressions at \(a = 1.3\) and \(a = 1.5\) | M1 | Must be using radians |
| Complete the argument correctly with correct calculated values | A1 | e.g. \(1.3 < 1.448\), \(1.5 > 1.322\) \((0.148, -0.178)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative process \(a_{n+1} = \tan\left(\frac{2a_n}{1+a_n^2}\right)\) correctly at least twice | M1 | |
| Obtain final answer \(1.39\) | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify \(1.39\) to 2 d.p. or show there is a sign change in the interval \((1.385, 1.395)\) | A1 | Allow recovery |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct quotient rule or correct product rule | M1 | e.g. $\frac{dy}{dx} = \frac{\sqrt{x} \cdot \frac{1}{1+x^2} - \tan^{-1}x \cdot \frac{1}{2\sqrt{x}}}{x}$ |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and remove inverse tangent | M1 | |
| Obtain $a = \tan\left(\frac{2a}{1+a^2}\right)$ from correct working | A1 | AG. Accept with $x$ in place of $a$ |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate the value of a relevant expression or pair of expressions at $a = 1.3$ and $a = 1.5$ | M1 | Must be using radians |
| Complete the argument correctly with correct calculated values | A1 | e.g. $1.3 < 1.448$, $1.5 > 1.322$ $(0.148, -0.178)$ |
---
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative process $a_{n+1} = \tan\left(\frac{2a_n}{1+a_n^2}\right)$ correctly at least twice | M1 | |
| Obtain final answer $1.39$ | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify $1.39$ to 2 d.p. or show there is a sign change in the interval $(1.385, 1.395)$ | A1 | Allow recovery |
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{a257f49d-5c8f-4c23-be78-46619b746fde-10_353_689_262_726}
The diagram shows the curve $y = \frac { \tan ^ { - 1 } x } { \sqrt { x } }$ and its maximum point $M$ where $x = a$.
\begin{enumerate}[label=(\alph*)]
\item Show that $a$ satisfies the equation
$$a = \tan \left( \frac { 2 a } { 1 + a ^ { 2 } } \right)$$
\item Verify by calculation that $a$ lies between 1.3 and 1.5.
\item Use an iterative formula based on the equation in part (a) to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q7 [9]}}