| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Line-curve intersection conditions |
| Difficulty | Moderate -0.3 This is a standard C1 question on line-curve intersection using the discriminant. Part (a) is routine algebraic manipulation, part (b) applies the standard discriminant condition b²-4ac>0, and part (c) solves a quadratic inequality using factorization or the quadratic formula. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^2 + 7 = k(3x+1) \Rightarrow x^2 - 3kx + 7 - k = 0\) | B1 | AG |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(b^2 - 4ac = (-3k)^2 - 4(7-k)\) (2 distinct roots when) \(b^2 - 4ac > 0\) \(9k^2 + 4k - 28 > 0\) | M1, B1, A1 | Clear attempt at \(b^2 - 4ac\); Condone slip in one term of expression; Must involve \(k\); CSO; AG |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \((9k-14)(k+2)\) Critical points \(-2\) and \(\frac{14}{9}\) Sketch \(\cup\) or sign diagram correct | M1, A1, M1 | Factors or formula correct unsimplified; \( |
| \(k < -2, k > \frac{14}{9}\) | A1 | |
| 4 | ||
| 8 |
## Part (a)
$x^2 + 7 = k(3x+1) \Rightarrow x^2 - 3kx + 7 - k = 0$ | B1 | AG
| | 1 |
## Part (b)
$b^2 - 4ac = (-3k)^2 - 4(7-k)$ (2 distinct roots when) $b^2 - 4ac > 0$ $9k^2 + 4k - 28 > 0$ | M1, B1, A1 | Clear attempt at $b^2 - 4ac$; Condone slip in one term of expression; Must involve $k$; CSO; AG
| | 3 |
## Part (c)
$(9k-14)(k+2)$ Critical points $-2$ and $\frac{14}{9}$ Sketch $\cup$ or sign diagram correct | M1, A1, M1 | Factors or formula correct unsimplified; $| \quad +\text{ve} \quad \mid -2 \mid -\text{ve} \mid \frac{14}{9} \mid +\text{ve}$
$k < -2, k > \frac{14}{9}$ | A1 |
| | 4 |
| | 8 |
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# **TOTAL: 75**7 The curve $C$ has equation $y = x ^ { 2 } + 7$. The line $L$ has equation $y = k ( 3 x + 1 )$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$-coordinates of any points of intersection of the line $L$ with the curve $C$ satisfy the equation
$$x ^ { 2 } - 3 k x + 7 - k = 0$$
\item The curve $C$ and the line $L$ intersect in two distinct points. Show that
$$9 k ^ { 2 } + 4 k - 28 > 0$$
\item Solve the inequality $9 k ^ { 2 } + 4 k - 28 > 0$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2008 Q7 [8]}}