| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Tangency condition for line and curve |
| Difficulty | Moderate -0.3 This is a standard C1 tangency question requiring substitution to form a quadratic, then applying the discriminant condition b²-4ac=0 for equal roots. While it involves multiple steps and connecting algebra to geometry, it follows a well-established template that students practice extensively. Slightly easier than average due to its predictable structure and straightforward algebraic manipulation. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(mx - 1 = x^2 - 5x + 3\) | Strict mark — no trailing equals signs | |
| \(\Rightarrow x^2 - 5x - mx + 4 = 0\) | ||
| \(\Rightarrow x^2 - (5+m)x + 4 = 0\) | B1 | AG (be convinced about algebra and \(= 0\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \((5+m)^2 - 16 = 0\) | M1 | Equal roots when "\(b^2 - 4ac\)" \(= 0\) used |
| \(5 + m = (\pm)4\) or \((m+1)(m+9) = 0\) | m1 | Square root or factor/formula attempt |
| \(m = -1\) | A1 | |
| \(m = -9\) | A1 | SC B1, B1 only for \(-1\), \(-9\) (no working) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Line is a tangent to the curve | E1 | Line touches curve, cuts at one point etc |
## Question 8:
**Part (a)**
| Working | Marks | Guidance |
|---------|-------|----------|
| $mx - 1 = x^2 - 5x + 3$ | | Strict mark — no trailing equals signs |
| $\Rightarrow x^2 - 5x - mx + 4 = 0$ | | |
| $\Rightarrow x^2 - (5+m)x + 4 = 0$ | B1 | **AG** (be convinced about algebra and $= 0$) |
**Part (b)**
| Working | Marks | Guidance |
|---------|-------|----------|
| $(5+m)^2 - 16 = 0$ | M1 | Equal roots when "$b^2 - 4ac$" $= 0$ used |
| $5 + m = (\pm)4$ or $(m+1)(m+9) = 0$ | m1 | Square root or factor/formula attempt |
| $m = -1$ | A1 | |
| $m = -9$ | A1 | **SC** B1, B1 only for $-1$, $-9$ (no working) |
**Part (c)**
| Working | Marks | Guidance |
|---------|-------|----------|
| Line is a tangent to the curve | E1 | Line touches curve, cuts at one point etc |
**Total: 6 marks**
**Overall Total: 75 marks**8 A line has equation $y = m x - 1$, where $m$ is a constant.\\
A curve has equation $y = x ^ { 2 } - 5 x + 3$.
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$-coordinate of any point of intersection of the line and the curve satisfies the equation
$$x ^ { 2 } - ( 5 + m ) x + 4 = 0$$
\item Find the values of $m$ for which the equation $x ^ { 2 } - ( 5 + m ) x + 4 = 0$ has equal roots.\\
(4 marks)
\item Describe geometrically the situation when $m$ takes either of the values found in part (b).\\
(1 mark)
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2005 Q8 [9]}}