| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle from diameter endpoints |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring substitution to find intersection points (standard algebraic manipulation), then applying the diameter formula for a circle. Both techniques are routine for P1 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((x-1)^2 + (x - 9 + 4)^2 = 40\) | M1 | Substitute line into circle |
| \(x^2 - 6x - 7 [= 0]\) leading to \((x+1)(x-7) [= 0]\) | M1 | Simplify to 3-term quadratic and factorise OE |
| \((-1, -10)\), \((7, -2)\) or \(x = -1\) and \(7\), \(y = -10\) and \(-2\) | A1 A1 | Answers only SC B1, SC B1 but must see a correct quadratic equation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([C \text{ is mid-point} =] \left(\frac{\text{their } x_1 + \text{their } x_2}{2}, \frac{\text{their } y_1 + \text{their } y_2}{2}\right)\) | M1 | Expect \((3, -6)\) |
| Radius \(= \sqrt{(\text{their } x - \text{their } 3)^2 + (\text{their } y - \text{their } (-6))^2}\) OR their \(\sqrt{\left((7-(-1))^2 + (-2-(-10))^2\right)}/2\) | M1 | Expect \(\sqrt{32}\) |
| \((x-3)^2 + (y+6)^2 = 32\) | A1 | OE |
## Question 5:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(x-1)^2 + (x - 9 + 4)^2 = 40$ | M1 | Substitute line into circle |
| $x^2 - 6x - 7 [= 0]$ leading to $(x+1)(x-7) [= 0]$ | M1 | Simplify to 3-term quadratic and factorise OE |
| $(-1, -10)$, $(7, -2)$ or $x = -1$ and $7$, $y = -10$ and $-2$ | A1 A1 | Answers only SC B1, SC B1 but must see a correct quadratic equation |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[C \text{ is mid-point} =] \left(\frac{\text{their } x_1 + \text{their } x_2}{2}, \frac{\text{their } y_1 + \text{their } y_2}{2}\right)$ | M1 | Expect $(3, -6)$ |
| Radius $= \sqrt{(\text{their } x - \text{their } 3)^2 + (\text{their } y - \text{their } (-6))^2}$ OR their $\sqrt{\left((7-(-1))^2 + (-2-(-10))^2\right)}/2$ | M1 | Expect $\sqrt{32}$ |
| $(x-3)^2 + (y+6)^2 = 32$ | A1 | OE |
---5 A circle has equation $( x - 1 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 40$. A line with equation $y = x - 9$ intersects the circle at points $A$ and $B$.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the two points of intersection.
\item Find an equation of the circle with diameter $A B$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q5 [7]}}