| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Sketch quadratic curve |
| Difficulty | Moderate -0.8 This is a straightforward completing-the-square exercise followed by a routine sketch. Part (a) requires standard algebraic manipulation to convert to vertex form, and part (b) asks for a basic sketch showing intercepts—both are textbook exercises requiring only recall and routine application of C1 techniques with no problem-solving insight needed. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a=4\) | B1 | States \(a=4\) or obtains \(4(x+b)^2+c\) |
| \(b=1\) | B1 | States \(b=1\) or obtains \(a(x+1)^2+c\) |
| All three: \(a=4\), \(b=1\) and \(c=-1\), i.e. \(4(x+1)^2-1\) | B1 | Needs all 3 correct for final mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| U-shaped quadratic graph | M1 | U shape (V shape is M0) |
| Curve correctly positioned: minimum in third quadrant, crosses \(x\)-axis twice on negative \(x\)-axis, \(y\)-axis once on positive \(y\)-axis | A1 | |
| Curve cuts \(y\)-axis at \((0, 3)\) only | B1 | Allow \(y=3\) or \((0,3)\) in text |
| Curve cuts \(x\)-axis at \(\left(-\frac{3}{2}, 0\right)\) and \(\left(-\frac{1}{2}, 0\right)\) | B1 | Allow \(-3/2\) and \(-1/2\) if given on \(x\)-axis |
## Question 10:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a=4$ | B1 | States $a=4$ or obtains $4(x+b)^2+c$ |
| $b=1$ | B1 | States $b=1$ or obtains $a(x+1)^2+c$ |
| All three: $a=4$, $b=1$ and $c=-1$, i.e. $4(x+1)^2-1$ | B1 | Needs all 3 correct for final mark |
**[3 marks]**
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| U-shaped quadratic graph | M1 | U shape (V shape is M0) |
| Curve correctly positioned: minimum in third quadrant, crosses $x$-axis twice on negative $x$-axis, $y$-axis once on positive $y$-axis | A1 | |
| Curve cuts $y$-axis at $(0, 3)$ only | B1 | Allow $y=3$ or $(0,3)$ in text |
| Curve cuts $x$-axis at $\left(-\frac{3}{2}, 0\right)$ and $\left(-\frac{1}{2}, 0\right)$ | B1 | Allow $-3/2$ and $-1/2$ if given on $x$-axis |
**[4 marks] — Total: 7 marks**
---10.
$$4 x ^ { 2 } + 8 x + 3 \equiv a ( x + b ) ^ { 2 } + c$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $a , b$ and $c$.
\item On the axes on page 27, sketch the curve with equation $y = 4 x ^ { 2 } + 8 x + 3$, showing clearly the coordinates of any points where the curve crosses the coordinate axes.
\includegraphics[max width=\textwidth, alt={}, center]{099016ad-e742-4679-9669-47dcd1d9cc5f-15_1283_1284_319_322}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2013 Q10 [7]}}