| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Quadratic with equal roots |
| Difficulty | Easy -1.2 This is a straightforward C1 completing the square question with standard parts: finding constants (routine algebra), showing no real roots (direct observation from completed square form), finding k for equal roots (discriminant = 0 or completed square = 0), and sketching a parabola. All parts are textbook exercises requiring only direct application of learned techniques with no problem-solving insight needed. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = 5\) and \((x+5)^2 - 25 + 36\) and \(b = 11\) | B1, M1 A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(b^2 - 4ac = 100 - 144 < 0\), therefore no real roots | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Equal roots if \(b^2 - 4ac = 0\) and \(4k = 100\) and \(k = 25\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Shape, position | B1 B1 | |
| \((-5, 0)\) \((0, 25)\) | B1 B1ft | (4 marks) |
## Part (a)
$a = 5$ and $(x+5)^2 - 25 + 36$ and $b = 11$ | B1, M1 A1 | (3 marks)
## Part (b)
$b^2 - 4ac = 100 - 144 < 0$, therefore no real roots | M1 A1 | (2 marks)
## Part (c)
Equal roots if $b^2 - 4ac = 0$ and $4k = 100$ and $k = 25$ | M1 A1 | (2 marks)
## Part (d)
Shape, position | B1 B1 |
$(-5, 0)$ $(0, 25)$ | B1 B1ft | (4 marks)
**Total: 11 marks**
---Given that
$$x^2 + 10x + 36 = (x + a)^2 + b$$
where $a$ and $b$ are constants,
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$ and the value of $b$. [3]
\item Hence show that the equation $x^2 + 10x + 36 = 0$ has no real roots. [2]
\end{enumerate}
The equation $x^2 + 10x + k = 0$ has equal roots.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the value of $k$. [2]
\item For this value of $k$, sketch the graph of $y = x^2 + 10x + k$, showing the coordinates of any points at which the graph meets the coordinate axes. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q8 [11]}}