| Exam Board | WJEC |
|---|---|
| Module | Unit 1 (Unit 1) |
| Year | 2023 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Find parameter values for tangency using discriminant |
| Difficulty | Moderate -0.3 This is a standard AS-level question on tangency conditions using discriminants. Part (a)(i) requires setting equations equal and using discriminant = 0 for tangency (routine technique), part (a)(ii) involves solving a quadratic and finding coordinates (straightforward), and part (b) uses discriminant > 0 (immediate application). The question is slightly easier than average because it's entirely procedural with no problem-solving insight required, though the multi-part structure and algebraic manipulation provide some substance. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.07a Derivative as gradient: of tangent to curve1.07m Tangents and normals: gradient and equations |
The curve $C$ has equation $y = 2x^2 + 5x - 12$ and the line $L$ has equation $y = mx - 14$, where $m$ is a real constant.
\begin{enumerate}[label=(\alph*)]
\item Given that $L$ is a tangent to $C$,
\begin{enumerate}[label=(\roman*)]
\item show that $m$ satisfies the equation
$$m^2 - 10m + 9 = 0,$$ [5]
\item find the coordinates of the two possible points of contact of $C$ and $L$. [6]
\end{enumerate}
\item Given instead that $L$ intersects $C$ at two distinct points, find the range of values of $m$. [2]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 1 2023 Q7 [13]}}