| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find intersection points |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on inverse functions requiring standard techniques: finding an inverse algebraically, applying the reflection property y=x, and using a given iteration formula. All steps are routine C3 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt correct process for finding inverse | M1 | maybe in terms of \(y\) so far |
| Obtain \(2x^3 - 4\) | A1 | or equiv; in terms of \(x\) now |
| State \(\sqrt[3]{2}\) or 1.26 | B1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State reflection in \(y = x\) | B1 | or clear equiv |
| Refer to intersection of \(y = x\) and \(y = f(x)\) and hence confirm \(x = \sqrt[3]{\frac{1}{2}x + 2}\) | B1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain correct first iterate | B1 | |
| Show correct process for iteration | M1 | with at least one more step |
| Obtain at least 3 correct iterates in all | A1 | allowing recovery after error |
| Obtain 1.39 | A1 | 4 |
# Question 6:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt correct process for finding inverse | M1 | maybe in terms of $y$ so far |
| Obtain $2x^3 - 4$ | A1 | or equiv; in terms of $x$ now |
| State $\sqrt[3]{2}$ or 1.26 | B1 | **3** | |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State reflection in $y = x$ | B1 | or clear equiv |
| Refer to intersection of $y = x$ and $y = f(x)$ and hence confirm $x = \sqrt[3]{\frac{1}{2}x + 2}$ | B1 | **2** | AG; or equiv |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain correct first iterate | B1 | |
| Show correct process for iteration | M1 | with at least one more step |
| Obtain at least 3 correct iterates in all | A1 | allowing recovery after error |
| Obtain 1.39 | A1 | **4** | following at least 3 steps; answer required to exactly 2 d.p. |
Iterates shown: $[0 \to 1.259921 \to 1.380330 \to 1.390784 \to 1.391684$, $1 \to 1.357209 \to 1.388789 \to 1.391512 \to 1.391747$, $1.26 \to 1.380337 \to 1.390784 \to 1.391684 \to 1.391761$, $1.5 \to 1.401020 \to 1.392564 \to 1.391837 \to 1.391775$, $2 \to 1.442250 \to 1.396099 \to 1.392141 \to 1.391801]$
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_627_689_264_726}
The function f is defined for all real values of $x$ by
$$f ( x ) = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$
The graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ meet at the point $P$, and the graph of $y = \mathrm { f } ^ { - 1 } ( x )$ meets the $x$-axis at $Q$ (see diagram).\\
(i) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and determine the $x$-coordinate of the point $Q$.\\
(ii) State how the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ are related geometrically, and hence show that the $x$-coordinate of the point $P$ is the root of the equation
$$x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$
(iii) Use an iterative process, based on the equation $x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$, to find the $x$-coordinate of $P$, giving your answer correct to 2 decimal places.
\hfill \mbox{\textit{OCR C3 2009 Q6 [9]}}