| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Graphical equation solving with auxiliary line |
| Difficulty | Moderate -0.8 This is a straightforward C1 question involving basic curve sketching and algebraic manipulation. Part (i) requires reading from a graph, part (ii) is routine quadratic formula application, part (iii) is a simple vertical translation, and part (iv) connects previous results. All techniques are standard with no novel problem-solving required, making it easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02q Use intersection points: of graphs to solve equations1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 2x + 3\) drawn on graph | M1 mark | |
| \(x = 0.2\) to \(0.4\) and \(-1.7\) to \(-1.9\) | A2 mark | 1 each; condone coords; must have line drawn |
| 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 = 2x^2 + 3x\) | M1 mark | for multiplying by \(x\) correctly |
| \(2x^2 + 3x - 1 [= 0]\) | M1 mark | for correctly rearranging to zero (may be earned first) or suitable step re completing square if they go on it, but no ft for factorising |
| attempt at formula or completing square | M1 mark | |
| \(x = \frac{-3 \pm \sqrt{17}}{4}\) | A2 mark | A1 for one soln |
| 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| branch through \((1,3)\), branch through \((-1,1)\) approaching \(y = 2\) from above | 1 mark | and approaching \(y = 2\) from below |
| 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(-1\) and \(\frac{1}{2}\) or ft intersection of their curve and line [tolerance 1 mm] | 2 marks | 1 each; may be found algebraically; ignore \(y\) coords. |
| 2 marks |
**Part (i)**
$y = 2x + 3$ drawn on graph | M1 mark |
$x = 0.2$ to $0.4$ and $-1.7$ to $-1.9$ | A2 mark | 1 each; condone coords; must have line drawn
| | 3 marks |
**Part (ii)**
$1 = 2x^2 + 3x$ | M1 mark | for multiplying by $x$ correctly
$2x^2 + 3x - 1 [= 0]$ | M1 mark | for correctly rearranging to zero (may be earned first) or suitable step re completing square if they go on it, but no ft for factorising
attempt at formula or completing square | M1 mark |
$x = \frac{-3 \pm \sqrt{17}}{4}$ | A2 mark | A1 for one soln
| | 5 marks |
**Part (iii)**
branch through $(1,3)$, branch through $(-1,1)$ approaching $y = 2$ from above | 1 mark | and approaching $y = 2$ from below
| | 2 marks |
**Part (iv)**
$-1$ and $\frac{1}{2}$ or ft intersection of their curve and line [tolerance 1 mm] | 2 marks | 1 each; may be found algebraically; ignore $y$ coords.
| | 2 marks |Answer the whole of this question on the insert provided.
The insert shows the graph of $y = \frac{1}{x}$, $x \neq 0$.
\begin{enumerate}[label=(\roman*)]
\item Use the graph to find approximate roots of the equation $\frac{1}{x} = 2x + 3$, showing your method clearly. [3]
\item Rearrange the equation $\frac{1}{x} = 2x + 3$ to form a quadratic equation. Solve the resulting equation, leaving your answers in the form $\frac{p \pm \sqrt{q}}{r}$. [5]
\item Draw the graph of $y = \frac{1}{x} + 2$, $x \neq 0$, on the grid used for part (i). [2]
\item Write down the values of $x$ which satisfy the equation $\frac{1}{x} + 2 = 2x + 3$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 2006 Q13 [12]}}