| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find coordinate from gradient condition |
| Difficulty | Standard +0.3 This is a multi-part question involving differentiation, perpendicular gradient condition, and fixed-point iteration. Part (i) requires finding derivatives and using the perpendicular gradient property (m₁m₂ = -1), which is standard A-level technique. Part (ii) is routine substitution. Part (iii) applies a given iterative formula mechanically. While it combines several topics, each step follows standard procedures without requiring novel insight or complex problem-solving, making it slightly easier than average. |
| Spec | 1.07b Gradient as rate of change: dy/dx notation1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State at least one correct derivative | B1 | \(-2\sin\frac{1}{2}x\), \(\frac{1}{(4-x)^2}\) |
| Equate product of derivatives to \(-1\) | M1 | or equivalent |
| Obtain a correct equation, e.g. \(2\sin\frac{1}{2}x = (4-x)^2\) | A1 | |
| Rearrange correctly to obtain \(a = 4 - \sqrt{2\sin\frac{a}{2}}\) AG | A1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Calculate values of a relevant expression or pair of expressions at \(a=2\) and \(a=3\) | M1 | e.g. \(a=2\): \(2 < 2.7027..\) \(\begin{pmatrix}0.703\end{pmatrix}\) \(\begin{pmatrix}2.317\end{pmatrix}\); \(a=3\): \(3 > 2.587..\) \(\begin{pmatrix}-0.412\end{pmatrix}\) \(\begin{pmatrix}-0.995\end{pmatrix}\). Values correct to at least 2 dp |
| Complete the argument correctly with correct calculated values | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the iterative formula \(a_{n+1} = 4 - \sqrt{(2\sin\frac{1}{2}a_n)}\) correctly at least once | M1 | |
| Obtain final answer 2.611 | A1 | |
| Show sufficient iterations to 5 d.p. to justify 2.611 to 3 d.p., or show there is a sign change in the interval \((2.6105, 2.6115)\) | A1 | 2, 2.70272, 2.60285, 2.61152, 2.61070, 2.61077; 2.5, 2.62233, 2.60969, 2.61087, 2.61076; 3, 2.58756, 2.61301, 2.61056, 2.61079. Condone truncation. Accept more than 5 dp |
| Total | 3 |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State at least one correct derivative | B1 | $-2\sin\frac{1}{2}x$, $\frac{1}{(4-x)^2}$ |
| Equate product of derivatives to $-1$ | M1 | or equivalent |
| Obtain a correct equation, e.g. $2\sin\frac{1}{2}x = (4-x)^2$ | A1 | |
| Rearrange correctly to obtain $a = 4 - \sqrt{2\sin\frac{a}{2}}$ **AG** | A1 | |
| **Total** | **4** | |
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Calculate values of a relevant expression or pair of expressions at $a=2$ and $a=3$ | M1 | e.g. $a=2$: $2 < 2.7027..$ $\begin{pmatrix}0.703\end{pmatrix}$ $\begin{pmatrix}2.317\end{pmatrix}$; $a=3$: $3 > 2.587..$ $\begin{pmatrix}-0.412\end{pmatrix}$ $\begin{pmatrix}-0.995\end{pmatrix}$. Values correct to at least 2 dp |
| Complete the argument correctly with correct calculated values | A1 | |
| **Total** | **2** | |
## Question 7(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the iterative formula $a_{n+1} = 4 - \sqrt{(2\sin\frac{1}{2}a_n)}$ correctly at least once | M1 | |
| Obtain final answer 2.611 | A1 | |
| Show sufficient iterations to 5 d.p. to justify 2.611 to 3 d.p., or show there is a sign change in the interval $(2.6105, 2.6115)$ | A1 | 2, 2.70272, 2.60285, 2.61152, 2.61070, 2.61077; 2.5, 2.62233, 2.60969, 2.61087, 2.61076; 3, 2.58756, 2.61301, 2.61056, 2.61079. Condone truncation. Accept more than 5 dp |
| **Total** | **3** | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{98ee8d3e-9aba-46a2-aa9c-b1e2093f393e-10_702_597_258_772}
The diagram shows the curves $y = 4 \cos \frac { 1 } { 2 } x$ and $y = \frac { 1 } { 4 - x }$, for $0 \leqslant x < 4$. When $x = a$, the tangents to the curves are perpendicular.\\
(i) Show that $a = 4 - \sqrt { } \left( 2 \sin \frac { 1 } { 2 } a \right)$.\\
(ii) Verify by calculation that $a$ lies between 2 and 3 .\\
(iii) Use an iterative formula based on the equation in part (i) to determine $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2019 Q7 [9]}}