| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Line intersecting quadratic curve |
| Difficulty | Standard +0.3 This is a straightforward simultaneous equations problem requiring students to recognize that intersection on the x-axis means y=0, find k=4, then solve the resulting quadratic to find the other intersection point. While it requires multiple steps and some insight about the x-axis condition, the algebraic manipulation is routine and the problem structure is standard for A-level, making it slightly easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(5x - x^2 = x(5-x)\) | M1 | AO 3.1a |
| \([x=0],\ x=5\) | A1 | AO 1.1 |
| The line does not go through the origin so \(x=5\) | E1 | AO 2.4 |
| \(y = 4 - kx\) so \(0 = 4 - 5k\) | M1 | AO 3.2a |
| \(k = \frac{4}{5}\) | A1 | AO 1.1 |
| \(4 - \frac{4}{5}x = 5x - x^2\) | M1 | AO 1.1 |
| \(x^2 - 5\frac{4}{5}x + 4 = 0\) OR \(5x^2 - 29x + 20 = 0\) | ||
| \((5x-4)(x-5) = 0\) | M1 | AO 1.1 |
| \(\left(\frac{4}{5},\ \frac{84}{25}\right)\) o.e. | A1 [8] | AO 2.2a |
## Question 7:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5x - x^2 = x(5-x)$ | M1 | AO 3.1a | DR Factorisation |
| $[x=0],\ x=5$ | A1 | AO 1.1 | Finding 5 |
| The line does not go through the origin so $x=5$ | E1 | AO 2.4 | Rejection of origin as a point where they cross; May be later |
| $y = 4 - kx$ so $0 = 4 - 5k$ | M1 | AO 3.2a | |
| $k = \frac{4}{5}$ | A1 | AO 1.1 | |
| $4 - \frac{4}{5}x = 5x - x^2$ | M1 | AO 1.1 | |
| $x^2 - 5\frac{4}{5}x + 4 = 0$ OR $5x^2 - 29x + 20 = 0$ | | | |
| $(5x-4)(x-5) = 0$ | M1 | AO 1.1 | |
| $\left(\frac{4}{5},\ \frac{84}{25}\right)$ o.e. | A1 [8] | AO 2.2a | $\frac{84}{25} = 3\frac{9}{25} = 3.36$ |7 In this question you must show detailed reasoning.\\
Fig. 7 shows the curve $y = 5 x - x ^ { 2 }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-7_511_684_383_694}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}
The line $y = 4 - k x$ crosses the curve $y = 5 x - x ^ { 2 }$ on the $x$-axis and at one other point.\\
Determine the coordinates of this other point.
\hfill \mbox{\textit{OCR MEI Paper 3 2018 Q7 [8]}}