| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2004 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Line intersecting reciprocal curve |
| Difficulty | Moderate -0.3 This is a straightforward simultaneous equations problem requiring substitution to form a quadratic, solving it, and then finding a perpendicular bisector using midpoint and negative reciprocal gradient. All techniques are standard P1 content with no conceptual challenges, making it slightly easier than average but not trivial due to the algebraic manipulation and two-part structure. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02q Use intersection points: of graphs to solve equations1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
| Answer | Marks |
|---|---|
| M1 A1 DM1 A1 [4] | Needs \(x\) or \(y\) removed completely; Correct only (no need for = 0); Equation must = 0; Everything ok. |
| Answer | Marks |
|---|---|
| B1 M1 | For his two points in (i); Use of \(m_1m_2 = -1\) |
| Answer | Marks |
|---|---|
| M1 A1 [4] | Use of \(m_1m_2 = -1\); Any form – needs the M marks. |
**(i)** eliminates $x$ (or $y$) completely → $x^2 + x - 6 = 0$ or $y^2 - 17y + 66 = 0$ Solution of quadratic = 0 → $(2, 6)$ and $(-3, 11)$
| M1 A1 DM1 A1 [4] | Needs $x$ or $y$ removed completely; Correct only (no need for = 0); Equation must = 0; Everything ok. |
**(ii)** Midpoint = $(-\frac{1}{2}, 8\frac{1}{2})$ Gradient of line = $-1$ Gradient of perpendicular = $1$
| B1 M1 | For his two points in (i); Use of $m_1m_2 = -1$ |
$\to y - 8\frac{1}{2} = 1(x + \frac{1}{2})$ (or $y = x + 9$)
| M1 A1 [4] | Use of $m_1m_2 = -1$; Any form – needs the M marks. |
---6 The curve $y = 9 - \frac { 6 } { x }$ and the line $y + x = 8$ intersect at two points. Find\\
(i) the coordinates of the two points,\\
(ii) the equation of the perpendicular bisector of the line joining the two points.
\hfill \mbox{\textit{CAIE P1 2004 Q6 [8]}}