| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Standard +0.3 This is a standard C3 question combining curve sketching, sign change verification, and iterative methods. Part (i) requires recognizing how to rearrange the equation to match the given curve (routine algebraic manipulation). Part (ii)(a) is straightforward substitution to verify a sign change. Part (ii)(b) applies a given iterative formula repeatedly—no derivation or insight needed, just careful calculation. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw inverted parabola roughly symmetrical about the \(y\)-axis and with maximum point more or less on \(y\)-axis | M1 | drawing enough of the parabola that two intersections occur, ignoring their locations at this stage |
| State \(y = 9 - x^2\) and indicate two intersections by marks on diagram or written reference to two intersections | A1 | now needs second curve drawn so that right-hand intersection occurs in first quadrant |
| Total | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Calculate values of quartic expression for \(2.1\) and \(2.2\) | M1 | if no explicit working seen, M1 is implied by at least one correct value; but if no explicit working seen and both values wrong, award M0 |
| Obtain \(-1.9\ldots\) and \(1.6\ldots\) and draw attention to sign change or clear equiv | A1 | |
| Total | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain correct first iterate | B1 | starting anywhere between \(-1\) and \(9\) and showing at least 3 d.p. |
| Carry out process to produce at least three iterates in all | M1 | implied by plausible sequence of values; allow recovery after error. \(2.1 \to 2.15056 \to 2.15531 \to 2.15575 \to 2.15579\); \(2.15 \to 2.15526 \to 2.15574 \to 2.15579\) |
| Obtain at least two more correct iterates | A1 | showing at least 3 decimal places. \(2.2 \to 2.15980 \to 2.15616 \to 2.15583 \to 2.15580\) |
| Obtain \(2.156\) | A1 | final answer needed to exactly 3 d.p.; not given for \(2.156\) as final iterate in sequence, i.e. needs indication (perhaps just underlining) that value of \(\alpha\) found. Answer only: 0/4 |
| Total | [4] |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw inverted parabola roughly symmetrical about the $y$-axis and with maximum point more or less on $y$-axis | M1 | drawing enough of the parabola that two intersections occur, ignoring their locations at this stage |
| State $y = 9 - x^2$ and indicate two intersections by marks on diagram or written reference to two intersections | A1 | now needs second curve drawn so that right-hand intersection occurs in first quadrant |
| **Total** | **[2]** | |
---
## Question 6(ii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Calculate values of quartic expression for $2.1$ and $2.2$ | M1 | if no explicit working seen, M1 is implied by at least one correct value; but if no explicit working seen and both values wrong, award M0 |
| Obtain $-1.9\ldots$ and $1.6\ldots$ and draw attention to sign change or clear equiv | A1 | |
| **Total** | **[2]** | |
---
## Question 6(ii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain correct first iterate | B1 | starting anywhere between $-1$ and $9$ and showing at least 3 d.p. |
| Carry out process to produce at least three iterates in all | M1 | implied by plausible sequence of values; allow recovery after error. $2.1 \to 2.15056 \to 2.15531 \to 2.15575 \to 2.15579$; $2.15 \to 2.15526 \to 2.15574 \to 2.15579$ |
| Obtain at least two more correct iterates | A1 | showing at least 3 decimal places. $2.2 \to 2.15980 \to 2.15616 \to 2.15583 \to 2.15580$ |
| Obtain $2.156$ | A1 | final answer needed to exactly 3 d.p.; not given for $2.156$ as final iterate in sequence, i.e. needs indication (perhaps just underlining) that value of $\alpha$ found. Answer only: 0/4 |
| **Total** | **[4]** | |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-3_524_720_246_676}
The diagram shows the curve $y = x ^ { 4 } - 8 x$.\\
(i) By sketching a second curve on the copy of the diagram, show that the equation
$$x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0$$
has two real roots. State the equation of the second curve.\\
(ii) The larger root of the equation $x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0$ is denoted by $\alpha$.
\begin{enumerate}[label=(\alph*)]
\item Show by calculation that $2.1 < \alpha < 2.2$.
\item Use an iterative process based on the equation
$$x = \sqrt [ 4 ] { 9 + 8 x - x ^ { 2 } } ,$$
with a suitable starting value, to find $\alpha$ correct to 3 decimal places. Give the result of each step of the iterative process.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2014 Q6 [8]}}