| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Find discriminant, state roots |
| Difficulty | Easy -1.2 This is a straightforward C1 question testing basic discriminant calculation (b²-4ac) and interpretation. Part (i) requires simple arithmetic substitution, while part (ii) asks students to match curves to equations based on discriminant values (number of x-intercepts). This is standard textbook material requiring only recall and routine application, making it easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(b^2 - 4ac\) | M1 | Uses \(b^2 - 4ac\) |
| (a) \(36 - 9 \times 4 = 0\) | A1 | 1 correct |
| (b) \(100 - 48 = 52\) | A1 | 3 correct |
| (c) \(4 - 20 = -16\) | A1 3 | SR: All 3 values correct but \(\sqrt{\phantom{x}}\) used B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) Fig 3 | B1 | 1 correct matching |
| (b) Fig 2 | B1 | 3 correct matchings |
| (c) Fig 5 | ||
| (a) 1 root, touches \(x\)-axis once, line of symmetry \(x = -3\) or root \(x = -3\) | B1 | 1 correct comment relating to roots/touching/crossing \(x\)-axis or line of symmetry or vertex for one graph |
| (b) 2 roots, meets \(x\)-axis twice, line of symmetry \(x = 5\) | B1 4 | 2 further correct comments for other 2 graphs |
| (c) No real roots, does not meet \(x\)-axis |
## Question 7(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $b^2 - 4ac$ | M1 | Uses $b^2 - 4ac$ |
| (a) $36 - 9 \times 4 = 0$ | A1 | 1 correct |
| (b) $100 - 48 = 52$ | A1 | 3 correct |
| (c) $4 - 20 = -16$ | A1 **3** | SR: All 3 values correct but $\sqrt{\phantom{x}}$ used **B1** |
## Question 7(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| (a) Fig 3 | B1 | 1 correct matching |
| (b) Fig 2 | B1 | 3 correct matchings |
| (c) Fig 5 | | |
| (a) 1 root, touches $x$-axis once, line of symmetry $x = -3$ or root $x = -3$ | B1 | 1 correct comment relating to roots/touching/crossing $x$-axis or line of symmetry or vertex for one graph |
| (b) 2 roots, meets $x$-axis twice, line of symmetry $x = 5$ | B1 **4** | 2 further correct comments for other 2 graphs |
| (c) No real roots, does not meet $x$-axis | | |
---7 (i) Calculate the discriminant of each of the following:
\begin{enumerate}[label=(\alph*)]
\item $x ^ { 2 } + 6 x + 9$,
\item $x ^ { 2 } - 10 x + 12$,
\item $x ^ { 2 } - 2 x + 5$.\\
(ii)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_391_446_628_397}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_449_625_888}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_389_442_630_1384}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_446_1119_644}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_396_447_1119_1137}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{figure}
State with reasons which of the diagrams corresponds to the curve\\
(a) $y = x ^ { 2 } + 6 x + 9$,\\
(b) $y = x ^ { 2 } - 10 x + 12$,\\
(c) $y = x ^ { 2 } - 2 x + 5$.
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2005 Q7 [7]}}