| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find coordinate from gradient condition |
| Difficulty | Standard +0.8 This question requires understanding that 'touching' means equal gradients, differentiating implicitly to find the condition tan(a) = 2/a, then applying fixed-point iteration (a non-standard A-level technique), and finally using the result. The conceptual leap to the gradient condition and working with the iterative formula elevate this above routine calculus questions, though the individual steps are manageable. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate both equations and equate derivatives | M1* | |
| Obtain equation \(\cos a - a\sin a = -\frac{k}{a^2}\) | A1 + A1 | |
| State \(a\cos a = \frac{k}{a}\) and eliminate \(k\) | DM1 | |
| Obtain the given answer showing sufficient working | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show clearly correct use of iterative formula at least once | M1 | |
| Obtain answer \(1.077\) | A1 | |
| Show sufficient iterations to 5 d.p. to justify \(1.077\) to 3 d.p., or show sign change in interval \((1.0765, 1.0775)\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct method to determine \(k\) | M1 | |
| Obtain answer \(k = 0.55\) | A1 | [2] |
## Question 9:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate both equations and equate derivatives | M1* | |
| Obtain equation $\cos a - a\sin a = -\frac{k}{a^2}$ | A1 + A1 | |
| State $a\cos a = \frac{k}{a}$ and eliminate $k$ | DM1 | |
| Obtain the given answer showing sufficient working | A1 | [5] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show clearly correct use of iterative formula at least once | M1 | |
| Obtain answer $1.077$ | A1 | |
| Show sufficient iterations to 5 d.p. to justify $1.077$ to 3 d.p., or show sign change in interval $(1.0765, 1.0775)$ | A1 | [3] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct method to determine $k$ | M1 | |
| Obtain answer $k = 0.55$ | A1 | [2] |
---9\\
\includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-3_481_483_1434_831}
The diagram shows the curves $y = x \cos x$ and $y = \frac { k } { x }$, where $k$ is a constant, for $0 < x \leqslant \frac { 1 } { 2 } \pi$. The curves touch at the point where $x = a$.\\
(i) Show that $a$ satisfies the equation $\tan a = \frac { 2 } { a }$.\\
(ii) Use the iterative formula $a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { a _ { n } } \right)$ to determine $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
(iii) Hence find the value of $k$ correct to 2 decimal places.
\hfill \mbox{\textit{CAIE P3 2016 Q9 [10]}}