| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Iterative formula with graphical justification |
| Difficulty | Standard +0.3 This is a standard C3 question combining curve sketching with iterative methods. Part (i) requires sketching two familiar curves and making a straightforward observation about intersections. Part (ii) involves routine integer-bounding by substitution and applying a given iterative formula—both are textbook procedures requiring no novel insight, making this slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch (more or less) correct \(y = 14 - x^2\) | B1 | Assessed separately; must exist in all four quadrants; ignore any intercepts given |
| Sketch (more or less) correct \(y = k\ln x\) | B1 | Must exist in first and fourth quadrants; if clearly meets \(y\)-axis award B0; if maximum point in first quadrant award B0 |
| Indicate one root ('blob' on sketch or written reference to one intersection) | B1 | Dependent on both curves being correct in first quadrant and there being no possibility of further points of intersection elsewhere |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Calculate values for at least 2 integers | M1 | |
| Obtain correct values for \(x = 3\) and \(x = 4\): \(14 - x^2 - 3\ln x\): \(1.7\), \(-6.2\); \(14-x^2\): \(5, 3.3\); \(3\ln x\): \(-2, 4.2\) | A1 | |
| State 3 and 4 | A1 | Following correct calculations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain correct first iterate | B1 | Having started with any positive value; B1 available if 'iteration' never goes beyond a first iterate |
| Attempt iteration process | M1 | Implied by plausible sequence of values |
| Obtain at least 3 correct iterates in all | A1 | Showing at least 2 d.p. |
| Obtain 3.24 | A1 | Answer required to exactly 2 d.p.; not given for 3.24 as the final iterate; needs an indication (perhaps underlining) that \(\alpha\) is found. \([3 \to 3.27172 \to 3.23173 \to 3.23743 \to 3.23661]\); \([3.5 \to 3.20027 \to 3.24196 \to 3.23596 \to 3.23682]\); \([4 \to 3.13706 \to 3.25118 \to 3.23465 \to 3.23701]\) |
# Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch (more or less) correct $y = 14 - x^2$ | B1 | Assessed separately; must exist in all four quadrants; ignore any intercepts given |
| Sketch (more or less) correct $y = k\ln x$ | B1 | Must exist in first and fourth quadrants; if clearly meets $y$-axis award B0; if maximum point in first quadrant award B0 |
| Indicate one root ('blob' on sketch or written reference to one intersection) | B1 | Dependent on both curves being correct in first quadrant and there being no possibility of further points of intersection elsewhere |
---
# Question 5(ii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Calculate values for at least 2 integers | M1 | |
| Obtain correct values for $x = 3$ and $x = 4$: $14 - x^2 - 3\ln x$: $1.7$, $-6.2$; $14-x^2$: $5, 3.3$; $3\ln x$: $-2, 4.2$ | A1 | |
| State 3 and 4 | A1 | Following correct calculations |
---
# Question 5(ii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain correct first iterate | B1 | Having started with any positive value; B1 available if 'iteration' never goes beyond a first iterate |
| Attempt iteration process | M1 | Implied by plausible sequence of values |
| Obtain at least 3 correct iterates in all | A1 | Showing at least 2 d.p. |
| Obtain 3.24 | A1 | Answer required to exactly 2 d.p.; not given for 3.24 as the final iterate; needs an indication (perhaps underlining) that $\alpha$ is found. $[3 \to 3.27172 \to 3.23173 \to 3.23743 \to 3.23661]$; $[3.5 \to 3.20027 \to 3.24196 \to 3.23596 \to 3.23682]$; $[4 \to 3.13706 \to 3.25118 \to 3.23465 \to 3.23701]$ |
---5 (i) It is given that $k$ is a positive constant. By sketching the graphs of
$$y = 14 - x ^ { 2 } \text { and } y = k \ln x$$
on a single diagram, show that the equation
$$14 - x ^ { 2 } = k \ln x$$
has exactly one real root.\\
(ii) The real root of the equation $14 - x ^ { 2 } = 3 \ln x$ is denoted by $\alpha$.
\begin{enumerate}[label=(\alph*)]
\item Find by calculation the pair of consecutive integers between which $\alpha$ lies.
\item Use the iterative formula $x _ { n + 1 } = \sqrt { 14 - 3 \ln x _ { n } }$, with a suitable starting value, to find $\alpha$. Show the result of each iteration, and give $\alpha$ correct to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2012 Q5 [10]}}