| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Line-circle intersection points |
| Difficulty | Moderate -0.3 This is a standard circle question requiring completion of the square to find the centre, using perpendicularity of radius and chord to find the line equation, then solving simultaneous equations. All techniques are routine for P1 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks |
|---|---|
| \((-2, 1)\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Gradient of \(CD = \frac{1}{2} \div 3\frac{1}{2} = \frac{1}{7}\) | B1 | |
| Gradient of \(AB = -7\) | M1 | With gradient \(-1\)/their \(m\) |
| Equation of \(AB\) is \(y - 1\frac{1}{2} = -7(x - 1\frac{1}{2})\) | M1 | |
| \(y = -7x + 12\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^2 + (12-7x)^2 + 4x - 2(12-7x) - 20 = 0\) | M1 | Substituting their \(AB\) equation into circle equation |
| \((50)(x^2 - 3x + 2) = 0\) | A1 | |
| \(x = 1, 2\) | A1 | Dependent on method seen for solving quadratic equation |
# Question 10:
## Part 10(a):
| $(-2, 1)$ | B1 | |
## Part 10(b):
| Gradient of $CD = \frac{1}{2} \div 3\frac{1}{2} = \frac{1}{7}$ | B1 | |
| Gradient of $AB = -7$ | M1 | With gradient $-1$/their $m$ |
| Equation of $AB$ is $y - 1\frac{1}{2} = -7(x - 1\frac{1}{2})$ | M1 | |
| $y = -7x + 12$ | A1 | |
## Part 10(c):
| $x^2 + (12-7x)^2 + 4x - 2(12-7x) - 20 = 0$ | M1 | Substituting their $AB$ equation into circle equation |
| $(50)(x^2 - 3x + 2) = 0$ | A1 | |
| $x = 1, 2$ | A1 | Dependent on method seen for solving quadratic equation |
---10 The circle $x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 20 = 0$ has centre $C$ and passes through points $A$ and $B$.\\
(a) State the coordinates of $C$.\\
It is given that the midpoint, $D$, of $A B$ has coordinates $\left( 1 \frac { 1 } { 2 } , 1 \frac { 1 } { 2 } \right)$.\\
(b) Find the equation of $A B$, giving your answer in the form $y = m x + c$.\\
(c) Find, by calculation, the $x$-coordinates of $A$ and $B$.\\
\hfill \mbox{\textit{CAIE P1 2020 Q10 [8]}}