| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Line and curve intersection |
| Difficulty | Moderate -0.3 This is a straightforward line-curve intersection problem requiring setting equations equal, solving a quadratic, then applying distance and midpoint formulas. All techniques are standard AS-level procedures with no novel insight needed, making it slightly easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.10c Magnitude and direction: of vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(3x^2 + x - 2 = 0\) | M1A1 | Eliminates \(x\) or \(y\); sets quadratic to 0 |
| \((x+1)(3x-2) \rightarrow x = -1\) or \(\frac{2}{3}\) | M1 | Attempt to solve their equation |
| \((-1,\ 1),\ (\frac{2}{3},\ 6)\) | A1 | co |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(AB^2 = (5/3)^2 + 5^2\) | M1 | \(\sqrt{}\) their coordinates from (i) |
| \(AB = 5.27(0\ldots)\) | A1 | Or \((5\sqrt{10})/3\) oe |
| mid-point \(= (-1/6,\ 7/2)\) | B1\(\sqrt{}\) | ft from their (i) |
## Question 8:
**Part (i)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $3x^2 + x - 2 = 0$ | M1A1 | Eliminates $x$ or $y$; sets quadratic to 0 |
| $(x+1)(3x-2) \rightarrow x = -1$ or $\frac{2}{3}$ | M1 | Attempt to solve their equation |
| $(-1,\ 1),\ (\frac{2}{3},\ 6)$ | A1 | co |
**Part (ii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $AB^2 = (5/3)^2 + 5^2$ | M1 | $\sqrt{}$ their coordinates from (i) |
| $AB = 5.27(0\ldots)$ | A1 | Or $(5\sqrt{10})/3$ oe |
| mid-point $= (-1/6,\ 7/2)$ | B1$\sqrt{}$ | ft from their (i) |
---8\\
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-3_613_897_1311_623}
The diagram shows part of the curve $y = \frac { 2 } { 1 - x }$ and the line $y = 3 x + 4$. The curve and the line meet at points $A$ and $B$.\\
(i) Find the coordinates of $A$ and $B$.\\
(ii) Find the length of the line $A B$ and the coordinates of the mid-point of $A B$.
\hfill \mbox{\textit{CAIE P1 2010 Q8 [7]}}