| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Challenging +1.2 Part (a) requires setting up areas using standard circle and triangle formulas (sector, triangle with tangent) and algebraic manipulation to reach the given equation - this is a structured geometric derivation with clear steps. Parts (b) and (c) are routine computational tasks: substituting values to verify an interval and applying a given iterative formula. While the geometric setup requires some spatial reasoning, the overall question follows a standard pattern for P3 level work with no novel insights required, making it moderately above average difficulty. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(AT = r\tan x\) or \(BT = r\tan x\) | B1 | |
| Use correct area formula and form an equation in \(r\) and \(x\) | M1 | |
| Rearrange in the given form | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculate the values of a relevant expression or pair of expressions at \(x=1\) and \(x=1.4\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.35\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify \(1.35\) to 2 d.p. or show there is a sign change in the interval \((1.345, 1.355)\) | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $AT = r\tan x$ or $BT = r\tan x$ | B1 | |
| Use correct area formula and form an equation in $r$ and $x$ | M1 | |
| Rearrange in the given form | A1 | |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate the values of a relevant expression or pair of expressions at $x=1$ and $x=1.4$ | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $1.35$ | A1 | |
| Show sufficient iterations to 4 d.p. to justify $1.35$ to 2 d.p. or show there is a sign change in the interval $(1.345, 1.355)$ | A1 | |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{da8dedae-4714-408e-a983-90ece63d9e91-08_501_1086_262_525}
The diagram shows a circle with centre $O$ and radius $r$. The tangents to the circle at the points $A$ and $B$ meet at $T$, and angle $A O B$ is $2 x$ radians. The shaded region is bounded by the tangents $A T$ and $B T$, and by the minor $\operatorname { arc } A B$. The area of the shaded region is equal to the area of the circle.
\begin{enumerate}[label=(\alph*)]
\item Show that $x$ satisfies the equation $\tan x = \pi + x$.
\item This equation has one root in the interval $0 < x < \frac { 1 } { 2 } \pi$. Verify by calculation that this root lies between 1 and 1.4.
\item Use the iterative formula
$$x _ { n + 1 } = \tan ^ { - 1 } \left( \pi + x _ { n } \right)$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q6 [8]}}