| 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 | Find coordinate from gradient condition |
| Difficulty | Standard +0.8 This question requires implicit differentiation of a quotient involving trigonometric functions, algebraic manipulation to derive an iterative formula, and systematic application of fixed-point iteration. While the calculus is standard A-level, the multi-step derivation and iterative numerical method elevate it above typical exercises, though it remains accessible with careful working. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07m Tangents and normals: gradient and equations1.07q Product and quotient rules: differentiation1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct product or quotient rule | M1 | |
| Obtain correct derivative in any form, e.g. \(\frac{dy}{dx} = \frac{\cos^2 x + 2x\sin x\cos x}{\cos^4 x}\) or \(\frac{dy}{dx} = \sec^2 x + 2x\sec^2 x\tan x\) | A1 | |
| Equate derivative at \(x=a\) to 12 and obtain \(a = \cos^{-1}\left(\sqrt[3]{\frac{\cos a + 2a\sin a}{12}}\right)\) | A1 | AG |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate a relevant expression or pair of expressions at \(a=0.9\) and \(a=1\) | M1 | Must be calculated in radians |
| Complete the argument correctly with correct calculated values | A1 | e.g. \(\cos 0.9 = 0.622 > 0.553\), \(0.9 < 0.985\), \(0.0846 > 0\); or \(\cos 1 = 0.540 < 0.570\), \(1 > 0.964\), \(-0.0358 < 0\); or could be looking at values of the gradient \(8.46\) & \(14.1\) |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the process \(a_{n+1} = \cos^{-1}\left(\sqrt[3]{\frac{\cos a_n + 2a_n\sin a_n}{12}}\right)\) correctly at least once | M1 | Must be working in radians |
| Obtain final answer \(0.97\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify \(0.97\) to 2 d.p., or show there is a sign change in the interval \((0.965, 0.975)\) | A1 | e.g. \(0.95, 0.9743, 0.9694, 0.9704\) |
| Total | 3 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct product or quotient rule | M1 | |
| Obtain correct derivative in any form, e.g. $\frac{dy}{dx} = \frac{\cos^2 x + 2x\sin x\cos x}{\cos^4 x}$ or $\frac{dy}{dx} = \sec^2 x + 2x\sec^2 x\tan x$ | A1 | |
| Equate derivative at $x=a$ to 12 and obtain $a = \cos^{-1}\left(\sqrt[3]{\frac{\cos a + 2a\sin a}{12}}\right)$ | A1 | AG |
| **Total** | **3** | |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate a relevant expression or pair of expressions at $a=0.9$ and $a=1$ | M1 | Must be calculated in radians |
| Complete the argument correctly with correct calculated values | A1 | e.g. $\cos 0.9 = 0.622 > 0.553$, $0.9 < 0.985$, $0.0846 > 0$; or $\cos 1 = 0.540 < 0.570$, $1 > 0.964$, $-0.0358 < 0$; or could be looking at values of the gradient $8.46$ & $14.1$ |
| **Total** | **2** | |
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the process $a_{n+1} = \cos^{-1}\left(\sqrt[3]{\frac{\cos a_n + 2a_n\sin a_n}{12}}\right)$ correctly at least once | M1 | Must be working in radians |
| Obtain final answer $0.97$ | A1 | |
| Show sufficient iterations to 4 d.p. to justify $0.97$ to 2 d.p., or show there is a sign change in the interval $(0.965, 0.975)$ | A1 | e.g. $0.95, 0.9743, 0.9694, 0.9704$ |
| **Total** | **3** | |7 The equation of a curve is $y = \frac { x } { \cos ^ { 2 } x }$, for $0 \leqslant x < \frac { 1 } { 2 } \pi$. At the point where $x = a$, the tangent to the curve has gradient equal to 12 .
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \cos ^ { - 1 } \left( \sqrt [ 3 ] { \frac { \cos a + 2 a \sin a } { 12 } } \right)$.
\item Verify by calculation that $a$ lies between 0.9 and 1 .
\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 2022 Q7 [8]}}