| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find coordinate from gradient condition |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on fixed-point iteration. Part (i) requires equating derivatives of two standard functions (exponential and logarithm), part (ii) is routine sign-change verification, and part (iii) is mechanical iteration with a given formula. All steps are standard A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07l Derivative of ln(x): and related functions1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State the correct derivatives \(2e^{2x-3}\) and \(2/x\) | B1 | |
| Equate derivatives and use a law of logarithms on an equation equivalent to \(ke^{2x-3} = m/x\) | M1 | |
| Obtain the given result correctly (or work vice versa) | A1 | [3 marks] |
| (ii) Consider the sign of \(a - \frac{1}{3}(3 - \ln a)\) when \(a = 1\) and \(a = 2\), or equivalent | M1 | |
| Complete the argument with correct calculated values | A1 | [2 marks] |
| (iii) 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 | [3 marks total] |
**(i)** State the correct derivatives $2e^{2x-3}$ and $2/x$ | B1 |
Equate derivatives and use a law of logarithms on an equation equivalent to $ke^{2x-3} = m/x$ | M1 |
Obtain the given result correctly (or work vice versa) | A1 | [3 marks]
**(ii)** Consider the sign of $a - \frac{1}{3}(3 - \ln a)$ when $a = 1$ and $a = 2$, or equivalent | M1 |
Complete the argument with correct calculated values | A1 | [2 marks]
**(iii)** 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 | [3 marks total]
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-2_435_597_1516_776}
The diagram shows the curves $y = \mathrm { e } ^ { 2 x - 3 }$ and $y = 2 \ln x$. When $x = a$ the tangents to the curves are parallel.\\
(i) Show that $a$ satisfies the equation $a = \frac { 1 } { 2 } ( 3 - \ln a )$.\\
(ii) Verify by calculation that this equation has a root between 1 and 2 .\\
(iii) Use the iterative formula $a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)$ to calculate $a$ correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2013 Q6 [8]}}