| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Line intersecting quadratic curve |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing basic algebraic skills: discriminant to show no real roots, substitution to solve simultaneous equations, and graphical interpretation. All parts are routine textbook exercises requiring standard techniques with no problem-solving insight needed. The multi-part structure adds length but not conceptual difficulty. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(b^2 - 4ac = 36 - 40 < 0\) | M1 | |
| so no real roots. | E1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^2 - 6x + 10 = 7 - 2x \Rightarrow x^2 - 4x + 3 = 0\) | M1 A1 | |
| \(\Rightarrow (x-1)(x-3) = 0 \Rightarrow (1,5),\ (3,1)\) | A1 A1 | Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| [Sketch of parabola] | M1 | Quadratic shape and orientation |
| A1 | Fully correct | |
| B1 | Correct plot for the line. Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| The intersection of this curve with the \(x\) axis is the same as the intersection of the line and curve above. | B1 | |
| [Sketch showing curve and \(x\)-axis intersections] | B1 B1 | Curve; Indication on graph. Total: 3 |
# Question 11:
## Part (i):
$b^2 - 4ac = 36 - 40 < 0$ | M1 |
so no real roots. | E1 | **Total: 2**
## Part (ii):
$x^2 - 6x + 10 = 7 - 2x \Rightarrow x^2 - 4x + 3 = 0$ | M1 A1 |
$\Rightarrow (x-1)(x-3) = 0 \Rightarrow (1,5),\ (3,1)$ | A1 A1 | **Total: 4**
## Part (iii):
[Sketch of parabola] | M1 | Quadratic shape and orientation
| A1 | Fully correct
| B1 | Correct plot for the line. **Total: 3**
## Part (iv):
The intersection of this curve with the $x$ axis is the same as the intersection of the line and curve above. | B1 |
[Sketch showing curve and $x$-axis intersections] | B1 B1 | Curve; Indication on graph. **Total: 3**
---11 (i) Show algebraically that the equation $x ^ { 2 } - 6 x + 10 = 0$ has no real roots.\\
(ii) Solve algebraically the simultaneous equations $y = x ^ { 2 } - 6 x + 10$ and $y + 2 x = 7$.\\
(iii) Plot the graph of the function $y = x ^ { 2 } - 6 x + 10$ on graph paper, taking $1 \mathrm {~cm} = 1$ unit on each axis, with the $x$ axis from 0 to 6 and the $y$ axis from - 2 to 10 .\\
On the same axes plot the line with equation $y + 2 x = 7$ showing clearly where the line cuts the quadratic curve.\\
(iv) Explain why these $x$ coordinates satisfy the equation $x ^ { 2 } - 4 x + 3 = 0$.
Plot a graph of the function $y = x ^ { 2 } - 4 x + 3$ on the same axes to illustrate your answer.
\hfill \mbox{\textit{OCR MEI C1 Q11 [12]}}