| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Line intersecting general conic |
| Difficulty | Moderate -0.3 This is a standard simultaneous equations problem involving a parabola and a line. Part (i) requires routine substitution and solving a quadratic. Part (ii) uses the discriminant condition for tangency, which is a well-practiced technique. While it requires multiple steps, it's a textbook exercise with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02q Use intersection points: of graphs to solve equations1.07s Parametric and implicit differentiation |
**(i)**
- M1: Complete elimination of $x$ or $y$ (allow multiples) – needs 3 terms
- A1: Solution of quadratic = 0
- DM1: Needs all 4 coordinates
- A1: $(2, 3)$ and $(6, 1)$
**(ii)**
- M1: Complete elimination of $x$ or $y$
- DM1: Use of discriminant $= 0$, $< 0$ or $> 0$
- A1: $k = 8\frac{1}{2}$
- (M1 equating $m$ of line and curve; M1 $x$ to $y$; A1 for $k$)
---4 The equation of a curve is $y ^ { 2 } + 2 x = 13$ and the equation of a line is $2 y + x = k$, where $k$ is a constant.\\
(i) In the case where $k = 8$, find the coordinates of the points of intersection of the line and the curve.\\
(ii) Find the value of $k$ for which the line is a tangent to the curve.
\hfill \mbox{\textit{CAIE P1 2011 Q4 [7]}}