| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Intersection of two lines |
| Difficulty | Moderate -0.8 This is a straightforward C1 coordinate geometry question testing basic concepts: finding gradient from parallel lines, solving simultaneous linear equations, and substituting a point into a line equation. All three parts are routine procedures with no problem-solving required, making it easier than average but not trivial since it requires multiple standard techniques. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = \pm\frac{5}{3}x + \ldots\) | M1 | |
| \(m = -\frac{5}{3}\) | A1 | Must see \(m = \ldots\) or statement such as "\(AB\) has gradient \(-\frac{5}{3}\) so line parallel to \(AB\) also has gradient \(-\frac{5}{3}\)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(5x+3y+3=0\) & \(3x-2y+17=0\), correct elimination of \(x\) or \(y\), e.g. \(19x+57=0\) or \(19y-76=0\) | M1 | Correct equations used and correct elimination |
| \(x=-3\) or \(x=-\frac{57}{19}\); \(y=4\) or \(y=\frac{76}{19}\) | A1 | Either \(x\) or \(y\) correct in any equivalent form |
| Both \(x=-3\) and \(y=4\), or \((-3,4)\) | A1 | Both coordinates written as integers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(5(2k+3)+3(4-3k)+3=0\), i.e. \(10k+15+12-9k+3=0\) | M1 | Correct substitution into correct equation & correct expansion of brackets |
| \(k=-30\) | A1 |
## Question 1:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = \pm\frac{5}{3}x + \ldots$ | M1 | |
| $m = -\frac{5}{3}$ | A1 | Must see $m = \ldots$ or statement such as "$AB$ has gradient $-\frac{5}{3}$ so line parallel to $AB$ also has gradient $-\frac{5}{3}$" |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $5x+3y+3=0$ & $3x-2y+17=0$, correct elimination of $x$ or $y$, e.g. $19x+57=0$ or $19y-76=0$ | M1 | Correct equations used and correct elimination |
| $x=-3$ or $x=-\frac{57}{19}$; $y=4$ or $y=\frac{76}{19}$ | A1 | Either $x$ or $y$ correct in any equivalent form |
| Both $x=-3$ and $y=4$, or $(-3,4)$ | A1 | Both coordinates written as integers |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $5(2k+3)+3(4-3k)+3=0$, i.e. $10k+15+12-9k+3=0$ | M1 | Correct substitution into correct equation & correct expansion of brackets |
| $k=-30$ | A1 | |
---1 The line $A B$ has equation $5 x + 3 y + 3 = 0$.
\begin{enumerate}[label=(\alph*)]
\item The line $A B$ is parallel to the line with equation $y = m x + 7$.
Find the value of $m$.
\item The line $A B$ intersects the line with equation $3 x - 2 y + 17 = 0$ at the point $B$.
Find the coordinates of $B$.
\item The point with coordinates $( 2 k + 3,4 - 3 k )$ lies on the line $A B$.
Find the value of $k$.\\[0pt]
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2016 Q1 [7]}}