| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Distance from centre to line |
| Difficulty | Standard +0.3 This is a straightforward application of the tangent-circle discriminant condition. Part (a) likely involves finding intersection points (routine algebra), and part (b) requires using perpendicular distance from center to line or substituting into the circle equation and setting discriminant to zero—both standard techniques covered extensively in P1. Slightly above average due to multi-step nature but no novel insight required. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.03f Circle properties: angles, chords, tangents |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((x+1)^2 + (3x-22)^2 = 85\) | M1 | OE. Substitute equation of line into equation of circle |
| \(10x^2 - 130x + 400\ [=0]\) | A1 | Correct 3-term quadratic |
| \([10](x-8)(x-5)\) leading to \(x=8\) or \(5\) | A1 | Dependent on factors or formula or completing of square seen |
| \((8,4),\ (5,-5)\) | A1 | If M1A1A0A0 scored, then SC B1 for correct final answer only |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Mid-point of \(AB = \left(6\frac{1}{2}, -\frac{1}{2}\right)\) | M1 | Any valid method |
| Use of \(C = (-1, 2)\) | B1 | SOI |
| \(r^2 = \left(-1-6\frac{1}{2}\right)^2 + \left(2+\frac{1}{2}\right)^2\) | M1 | Attempt to find \(r^2\). Expect \(r^2 = 62\frac{1}{2}\) |
| Equation of circle is \((x+1)^2 + (y-2)^2 = 62\frac{1}{2}\) | A1 | OE |
| 4 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(x+1)^2 + (3x-22)^2 = 85$ | M1 | OE. Substitute equation of line into equation of circle |
| $10x^2 - 130x + 400\ [=0]$ | A1 | Correct 3-term quadratic |
| $[10](x-8)(x-5)$ leading to $x=8$ or $5$ | A1 | Dependent on factors or formula or completing of square seen |
| $(8,4),\ (5,-5)$ | A1 | If M1A1A0A0 scored, then SC B1 for correct final answer only |
| | **4** | |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Mid-point of $AB = \left(6\frac{1}{2}, -\frac{1}{2}\right)$ | M1 | Any valid method |
| Use of $C = (-1, 2)$ | B1 | SOI |
| $r^2 = \left(-1-6\frac{1}{2}\right)^2 + \left(2+\frac{1}{2}\right)^2$ | M1 | Attempt to find $r^2$. Expect $r^2 = 62\frac{1}{2}$ |
| Equation of circle is $(x+1)^2 + (y-2)^2 = 62\frac{1}{2}$ | A1 | OE |
| | **4** | |\begin{enumerate}[label=(\alph*)]
\item Find, by calculation, the coordinates of $A$ and $B$.
\item Find an equation of the circle which has its centre at $C$ and for which the line with equation $y = 3 x - 20$ is a tangent to the circle.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q6 [8]}}