| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Line intersecting quadratic curve |
| Difficulty | Moderate -0.8 Part (a) is a routine two-variable linear system solvable by elimination in 2-3 steps. Part (b) requires substitution into a quadratic, then solving the resulting quadratic equation—standard GCSE/AS-level technique with minimal conceptual challenge. The parametric element (k) adds slight complexity but this remains a straightforward textbook exercise requiring only recall of standard methods. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02p Interpret algebraic solutions: graphically |
| Answer | Marks | Guidance |
|---|---|---|
| \(3x = k+1\) or \(2k - 3y = 1\) | M1 (1.1a) | Elimination of one of \(x\), \(y\) |
| \(x = \frac{k+1}{3}\) oe isw | A1 (1.1) | May be as part of a pair of coordinates |
| \(y = \frac{2k-1}{3}\) oe isw | A1 (2.2a) | May be as part of a pair of coordinates; Allow correct unsimplified answers |
| Answer | Marks | Guidance |
|---|---|---|
| \(k - x = x^2 + k\) | M1 (3.1a) | Substituting for \(x\) gets M0 unless it leads to correct values of \(y\) |
| \(x^2 + x = 0 \Rightarrow x = 0,\ x = -1\) | A1 (1.1) | |
| \((0,\ k)\) | A1 (1.1) | \((0, k)\) unsupported earns B1 SC; Need not be as coordinates but must be clear which \(y\) goes with which \(x\) |
| \((-1,\ k+1)\) | A1 (2.1) |
## Question 3:
**Part (a):**
$3x = k+1$ or $2k - 3y = 1$ | M1 (1.1a) | Elimination of one of $x$, $y$
$x = \frac{k+1}{3}$ oe isw | A1 (1.1) | May be as part of a pair of coordinates
$y = \frac{2k-1}{3}$ oe isw | A1 (2.2a) | May be as part of a pair of coordinates; Allow correct unsimplified answers
[3 marks]
**Part (b):**
$k - x = x^2 + k$ | M1 (3.1a) | Substituting for $x$ gets M0 unless it leads to correct values of $y$
$x^2 + x = 0 \Rightarrow x = 0,\ x = -1$ | A1 (1.1) |
$(0,\ k)$ | A1 (1.1) | $(0, k)$ unsupported earns B1 SC; Need not be as coordinates but must be clear which $y$ goes with which $x$
$(-1,\ k+1)$ | A1 (2.1) |
[4 marks]
---3
\begin{enumerate}[label=(\alph*)]
\item Determine, in terms of $k$, the coordinates of the point where the lines with the following equations intersect.
$$\begin{array} { r }
x + y = k \\
2 x - y = 1
\end{array}$$
\item Determine, in terms of $k$, the coordinates of the points where the line $\mathrm { x } + \mathrm { y } = \mathrm { k }$ crosses the curve $y = x ^ { 2 } + k$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 3 2021 Q3 [7]}}