| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Quadratic inequalities |
| Difficulty | Standard +0.3 This is a standard multi-part C1 question covering completing the square, line of symmetry, and discriminant conditions. Part (a) is routine manipulation, part (b)(i) is straightforward algebraic rearrangement (shown), and parts (b)(ii-iii) apply standard discriminant theory and quadratic inequality solving. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| \((-)\) \((x + 5)^2\) | M1 | |
| \(29 - (x + 5)^2\) | A1 | \(q = 5\); condone \((-x - 5)^2\); \(p = 29\) and \(q = 5\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = -5\) is line of symmetry | B1√ | FT; \(x = -\) their \(q\) or correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(4 - 10x - x^2 = k(4x - 13)\) | Must see both these lines OE | |
| \(\Rightarrow x^2 + 4kx + 10x - 13k - 4 = 0\) | ||
| \(\Rightarrow x^2 + 2(2k + 5)x - (13k + 4) = 0\) | B1 | AG all correct working and = 0 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 distinct roots \(\Rightarrow b^2 - 4ac > 0\) | B1 | stated or used (must be > 0) |
| Discriminant \(= 4(2k + 5)^2 + 4(13k + 4)\) | M1 | condone one slip (may be within formula) |
| \(4(4k^2 + 20k + 25 + 13k + 4) > 0\) | or \(16k^2 + 132k + 116 > 0\) | |
| \(\Rightarrow 4k^2 + 33k + 29 > 0\) | A1 | AG > 0 must appear before final line |
| Answer | Marks | Guidance |
|---|---|---|
| \((4k + 29)(k + 1)\) | M1 | correct factors or correct unsimplified quadratic equation formula \(\frac{-33 \pm \sqrt{33^2 - 4 \times 4 \times 29}}{8}\) |
| \(k = -\frac{29}{4}, k = -1\) | A1 | condone \(k = -\frac{58}{8}, -7.25\) etc but not left with square roots etc as above |
| [sketch or sign diagram including values] | M1 | sketch or sign diagram including values; \(\begin{matrix} + & - & + \\ -\frac{29}{4} & -1 \end{matrix}\) |
| \(k < -\frac{29}{4}, k > -1\) | A1 | condone use of OR but not AND |
| Total | 11 | |
| TOTAL | 75 |
**7(a)(i)**
$(-)$ $(x + 5)^2$ | M1 |
$29 - (x + 5)^2$ | A1 | $q = 5$; condone $(-x - 5)^2$; $p = 29$ and $q = 5$ | 2
**7(a)(ii)**
$x = -5$ is line of symmetry | B1√ | FT; $x = -$ their $q$ or correct | 1
**7(b)(i)**
$4 - 10x - x^2 = k(4x - 13)$ | | Must see both these lines OE
$\Rightarrow x^2 + 4kx + 10x - 13k - 4 = 0$ | |
$\Rightarrow x^2 + 2(2k + 5)x - (13k + 4) = 0$ | B1 | AG all correct working and = 0
**7(b)(ii)**
2 distinct roots $\Rightarrow b^2 - 4ac > 0$ | B1 | stated or used (must be > 0)
Discriminant $= 4(2k + 5)^2 + 4(13k + 4)$ | M1 | condone one slip (may be within formula)
$4(4k^2 + 20k + 25 + 13k + 4) > 0$ | | or $16k^2 + 132k + 116 > 0$
$\Rightarrow 4k^2 + 33k + 29 > 0$ | A1 | AG > 0 must appear before final line
**7(b)(iii)**
$(4k + 29)(k + 1)$ | M1 | correct factors or correct unsimplified quadratic equation formula $\frac{-33 \pm \sqrt{33^2 - 4 \times 4 \times 29}}{8}$
$k = -\frac{29}{4}, k = -1$ | A1 | condone $k = -\frac{58}{8}, -7.25$ etc but not left with square roots etc as above
[sketch or sign diagram including values] | M1 | sketch or sign diagram including values; $\begin{matrix} + & - & + \\ -\frac{29}{4} & -1 \end{matrix}$
$k < -\frac{29}{4}, k > -1$ | A1 | condone use of OR but not AND | 4
| | **Total** | **11** |
| | **TOTAL** | **75**7
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $4 - 10 x - x ^ { 2 }$ in the form $p - ( x + q ) ^ { 2 }$.
\item Hence write down the equation of the line of symmetry of the curve with equation $y = 4 - 10 x - x ^ { 2 }$.
\end{enumerate}\item The curve $C$ has equation $y = 4 - 10 x - x ^ { 2 }$ and the line $L$ has equation $y = k ( 4 x - 13 )$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinates of any points of intersection of the curve $C$ with the line $L$ satisfy the equation
$$x ^ { 2 } + 2 ( 2 k + 5 ) x - ( 13 k + 4 ) = 0$$
\item Given that the curve $C$ and the line $L$ intersect in two distinct points, show that
$$4 k ^ { 2 } + 33 k + 29 > 0$$
\item Solve the inequality $4 k ^ { 2 } + 33 k + 29 > 0$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2011 Q7 [11]}}