| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Solving quadratics and applications |
| Type | Finding quadratic constants from algebraic conditions |
| Difficulty | Moderate -0.3 This question involves standard techniques: identifying parabola orientation, reading roots from a graph, forming a quadratic from factors, and completing the square. While multi-part, each step is routine for C1 level with no novel problem-solving required. The final part about graphs not meeting requires simple reasoning about discriminants or comparing completed square forms, making it slightly easier than average overall. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks |
|---|---|
| Because it is "upside down" or words to that effect | B1 |
| Answer | Marks |
|---|---|
| \((x - 1)\) and \((x - 3)\) | B1, B1 |
| Answer | Marks |
|---|---|
| \(\Rightarrow a = -2, b = 8, c = -6\) | M1, A1, A1, A1 |
| Answer | Marks |
|---|---|
| \(\Rightarrow y = (x - 2)^2 + 6\) | M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Minimum value of this function is 6 | B1 | Minimum |
| and it is "right way up" | B1, B1 | 6; right way up |
**Part (i)**
Because it is "upside down" or words to that effect | B1 |
**Part (ii)**
$(x - 1)$ and $(x - 3)$ | B1, B1 |
**Part (iii)**
$y = k(x - 1)(x - 3)$
Through $(0, -6) \Rightarrow k = -2$
$\Rightarrow a = -2, b = 8, c = -6$ | M1, A1, A1, A1 |
**Part (iv)**
$y = x^2 - 4x + 10 = x^2 - 4x + 4 + 6$
$\Rightarrow y = (x - 2)^2 + 6$ | M1, A1 |
**Part (v)**
Minimum value of this function is 6 | B1 | Minimum
and it is "right way up" | B1, B1 | 6; right way up11 Fig. 11 shows the graph of $y = a x ^ { 2 } + b x + c$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-4_572_1509_465_285}
\captionsetup{labelformat=empty}
\caption{Fig. 11}
\end{center}
\end{figure}
(i) Explain why a must be negative.\\
(ii) State two factors of $y = a x ^ { 2 } + b x + c$.\\
(iii) Hence, or otherwise, find the values of $a , b$ and $c$.
Another function is given by $y = x ^ { 2 } - 4 x + 10$.\\
(iv) Write this in completed square form.\\
(v) Explain why the graphs of these two functions never meet.
\hfill \mbox{\textit{OCR MEI C1 Q11 [12]}}