AQA C1 2007 June — Question 3 12 marks

Exam BoardAQA
ModuleC1 (Core Mathematics 1)
Year2007
SessionJune
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicDiscriminant and conditions for roots
TypeShow line is tangent, verify
DifficultyModerate -0.8 This is a straightforward C1 question testing standard completing-the-square technique and basic line-curve intersection. Part (a) involves routine algebraic manipulation and reading off graph features from completed square form. Part (b) requires solving a simple quadratic equation. All steps are textbook exercises with no problem-solving insight required, making it easier than average but not trivial due to the multi-part structure.
Spec1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown1.02w Graph transformations: simple transformations of f(x)
3
    1. Express \(x ^ { 2 } + 10 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Write down the coordinates of the vertex (minimum point) of the curve with equation \(y = x ^ { 2 } + 10 x + 19\).
    3. Write down the equation of the line of symmetry of the curve \(y = x ^ { 2 } + 10 x + 19\).
    4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 10 x + 19\).
  2. Determine the coordinates of the points of intersection of the line \(y = x + 11\) and the curve \(y = x ^ { 2 } + 10 x + 19\).
Question 3:
Part (a)(i)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\((x+5)^2\)B1 \(p = 5\)
\(-6\)B1 \(q = -6\)
Part (a)(ii)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(x_{\text{vertex}} = -5\) (or their \(-p\))B1\(\checkmark\) may differentiate but must have \(x = -5\)
\(y_{\text{vertex}} = -6\) (or their \(q\))B1\(\checkmark\) and \(y = -6\). Vertex \((-5, -6)\)
Part (a)(iii)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(x = -5\)B1
Part (a)(iv)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Translation (not shift, move etc)E1 and NO other transformation stated
through \(\begin{bmatrix}-5\\-6\end{bmatrix}\) (or 5 left, 6 down)M1, A1 either component correct; M1, A1 independent of E mark
Part (b)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(x + 11 = x^2 + 10x + 19\) quadratic with all terms on one side
\(\Rightarrow x^2 + 9x + 8 = 0\) or \(y^2 - 13y + 30 = 0\)M1
\((x+8)(x+1) = 0\) or \((y-3)(y-10) = 0\)m1 attempt at formula (1 slip) or to factorise
\(x = -1, \; x = -8\)A1 both \(x\) values correct
\(y = 10\) or \(y = 3\)A1 both \(y\) values correct and linked
SC \((-1, 10)\) B2, \((-8, 3)\) B2 no working 
## Question 3:

### Part (a)(i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(x+5)^2$ | B1 | $p = 5$ |
| $-6$ | B1 | $q = -6$ |

### Part (a)(ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x_{\text{vertex}} = -5$ (or their $-p$) | B1$\checkmark$ | may differentiate but must have $x = -5$ |
| $y_{\text{vertex}} = -6$ (or their $q$) | B1$\checkmark$ | and $y = -6$. Vertex $(-5, -6)$ |

### Part (a)(iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = -5$ | B1 | |

### Part (a)(iv)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Translation (not shift, move etc) | E1 | and NO other transformation stated |
| through $\begin{bmatrix}-5\\-6\end{bmatrix}$ (or 5 left, 6 down) | M1, A1 | either component correct; M1, A1 independent of E mark |

### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x + 11 = x^2 + 10x + 19$ | | quadratic with all terms on one side |
| $\Rightarrow x^2 + 9x + 8 = 0$ or $y^2 - 13y + 30 = 0$ | M1 | |
| $(x+8)(x+1) = 0$ or $(y-3)(y-10) = 0$ | m1 | attempt at formula (1 slip) or to factorise |
| $x = -1, \; x = -8$ | A1 | both $x$ values correct |
| $y = 10$ or $y = 3$ | A1 | both $y$ values correct and linked |
| | | **SC** $(-1, 10)$ B2, $(-8, 3)$ B2 no working |

---
3
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $x ^ { 2 } + 10 x + 19$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
\item Write down the coordinates of the vertex (minimum point) of the curve with equation $y = x ^ { 2 } + 10 x + 19$.
\item Write down the equation of the line of symmetry of the curve $y = x ^ { 2 } + 10 x + 19$.
\item Describe geometrically the transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = x ^ { 2 } + 10 x + 19$.
\end{enumerate}\item Determine the coordinates of the points of intersection of the line $y = x + 11$ and the curve $y = x ^ { 2 } + 10 x + 19$.
\end{enumerate}

\hfill \mbox{\textit{AQA C1 2007 Q3 [12]}}
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