| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show convergence to specific root |
| Difficulty | Challenging +1.2 This is a structured multi-part question on fixed point iteration requiring differentiation to find a stationary point equation, numerical verification of an interval, showing convergence to a specific root (substitution-based proof), and performing iterations. While it involves several A-level techniques, each part is guided and follows standard procedures without requiring novel insight or complex proof techniques. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07r 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 |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct product rule | M1 | Condone incorrect/missing chain rule |
| Obtain correct derivative in any form | A1 | e.g. \(\frac{dy}{dx} = \sqrt{\sin x} + \frac{x\cos x}{2\sqrt{\sin x}}\) or \(2y\frac{dy}{dx} = 2x\sin x + x^2\cos x\) |
| Equate derivative to zero and obtain an equation in \(\tan x\) or \(\tan a\) | M1 | |
| Obtain \(\tan a = -\frac{1}{2}a\) correctly | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculate the value of a relevant expression or pair of expressions at \(a=2\) and \(a=2.5\) | M1 | Must be working in radians. At least one correct |
| Complete the argument correctly with correct calculated values | A1 | e.g. \(-1 > -2.18\) and \(-1.25 < -0.747\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State a suitable equation, e.g. \(x = \pi - \tan^{-1}\!\left(\frac{1}{2}x\right)\) | B1 | A correct equation without subscripts or quote \(\tan\theta = -\tan(\pi-\theta)\) |
| Using \(\tan(A\pm B)\) formula, or otherwise, rearrange this as \(\tan x = -\frac{1}{2}x\) | B1 | Complete argument correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the iterative process correctly at least once | M1 | Must be working in radians |
| Obtain answer \(a = 2.29\) | A1 | |
| Show sufficient iterations to 4 dp to justify \(2.29\) to 2 dp or show there is a sign change in the interval \((2.285, 2.295)\) | A1 | e.g. \(2.25, 2.2974, 2.2871, 2.2893, 2.2888, \ldots\) |
| Total | 3 |
## Question 10:
### Part 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct product rule | M1 | Condone incorrect/missing chain rule |
| Obtain correct derivative in any form | A1 | e.g. $\frac{dy}{dx} = \sqrt{\sin x} + \frac{x\cos x}{2\sqrt{\sin x}}$ or $2y\frac{dy}{dx} = 2x\sin x + x^2\cos x$ |
| Equate derivative to zero and obtain an equation in $\tan x$ or $\tan a$ | M1 | |
| Obtain $\tan a = -\frac{1}{2}a$ correctly | A1 | AG |
### Part 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate the value of a relevant expression or pair of expressions at $a=2$ and $a=2.5$ | M1 | Must be working in radians. At least one correct |
| Complete the argument correctly with correct calculated values | A1 | e.g. $-1 > -2.18$ and $-1.25 < -0.747$ |
### Part 10(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| State a suitable equation, e.g. $x = \pi - \tan^{-1}\!\left(\frac{1}{2}x\right)$ | B1 | A correct equation without subscripts or quote $\tan\theta = -\tan(\pi-\theta)$ |
| Using $\tan(A\pm B)$ formula, or otherwise, rearrange this as $\tan x = -\frac{1}{2}x$ | B1 | Complete argument correctly |
## Question 10(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the iterative process correctly at least once | **M1** | Must be working in radians |
| Obtain answer $a = 2.29$ | **A1** | |
| Show sufficient iterations to 4 dp to justify $2.29$ to 2 dp or show there is a sign change in the interval $(2.285, 2.295)$ | **A1** | e.g. $2.25, 2.2974, 2.2871, 2.2893, 2.2888, \ldots$ |
| **Total** | **3** | |10\\
\includegraphics[max width=\textwidth, alt={}, center]{c1fbc9ef-2dc6-43c3-bc58-179f683c9acf-18_471_686_276_717}
The curve $y = x \sqrt { \sin x }$ has one stationary point in the interval $0 < x < \pi$, where $x = a$ (see diagram).
\begin{enumerate}[label=(\alph*)]
\item Show that $\tan a = - \frac { 1 } { 2 } a$.
\item Verify by calculation that $a$ lies between 2 and 2.5.
\item Show that if a sequence of values in the interval $0 < x < \pi$ given by the iterative formula $x _ { n + 1 } = \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } \right)$ converges, then it converges to $a$, the root of the equation in part (a). [2]
\item Use the iterative formula given in part (c) to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q10 [11]}}