| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Range of parameter for intersection |
| Difficulty | Standard +0.3 Part (a) is routine completion of the square to find centre and radius. Part (b) requires substituting the line equation into the circle, forming a quadratic in x, and applying the discriminant condition b²-4ac > 0 for two distinct intersections. This is a standard technique but involves careful algebraic manipulation across multiple steps, making it slightly above average difficulty for AS level. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Attempts to complete the square \((x \pm 3)^2 + (y \pm 5)^2 = ...\) | M1 | May be implied by centre \((\pm 3, \pm 5)\) or \(r^2 = 25\) |
| Centre \((3, -5)\) | A1 | |
| Radius \(5\) | A1 | Do not accept \(\sqrt{25}\); answers only (no working) scores all three marks |
| Answer | Marks | Guidance |
|---|---|---|
| Uses a sketch or otherwise to deduce \(k = 0\) is a critical value | B1 | May award for correct \(k < 0\); award if \(k \leqslant 0\) |
| Substitute \(y = kx\) in \(x^2 + y^2 - 6x + 10y + 9 = 0\) | M1 | Or substitute \(x = \frac{y}{k}\) into circle equation |
| Collects terms to form correct 3TQ: \((1+k^2)x^2 + (10k-6)x + 9 = 0\) | A1 | |
| Attempts \(b^2 - 4ac...0\) for their \(a\), \(b\), \(c\) leading to values for \(k\): \(``(10k-6)^2 - 36(1+k^2)...0" \rightarrow k = ...\), giving \(0\) and \(\frac{15}{8}\) | M1 | |
| Uses \(b^2 - 4ac > 0\) and chooses outside region for their critical values (both \(a\) and \(b\) must have been expressions in \(k\)) | dM1 | |
| Deduces \(k < 0, k > \frac{15}{8}\) oe | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Uses a sketch or otherwise to deduce \(k = 0\) is a critical value | B1 | 2.2a |
| Distance from \((a, ka)\) to \((0,0)\) is \(3 \Rightarrow a^2(1+k^2) = 9\) | M1 | 3.1a |
| Tangent and radius are perpendicular \(\Rightarrow k \times \frac{ka+5}{a-3} = -1 \Rightarrow a(1+k^2) = 3-5k\) | M1 | 3.1a |
| Solve simultaneously (dependent upon both M's) | dM1 | 1.1b |
| \(k = \frac{15}{8}\) | A1 | 1.1b |
| Deduces \(k < 0, k > \frac{15}{8}\) | A1 | 2.2a |
| Total | (6) |
## Question 14(a):
Attempts to complete the square $(x \pm 3)^2 + (y \pm 5)^2 = ...$ | M1 | May be implied by centre $(\pm 3, \pm 5)$ or $r^2 = 25$
Centre $(3, -5)$ | A1 |
Radius $5$ | A1 | Do not accept $\sqrt{25}$; answers only (no working) scores all three marks
---
## Question 14(b):
Uses a sketch or otherwise to deduce $k = 0$ is a critical value | B1 | May award for correct $k < 0$; award if $k \leqslant 0$
Substitute $y = kx$ in $x^2 + y^2 - 6x + 10y + 9 = 0$ | M1 | Or substitute $x = \frac{y}{k}$ into circle equation
Collects terms to form correct 3TQ: $(1+k^2)x^2 + (10k-6)x + 9 = 0$ | A1 |
Attempts $b^2 - 4ac...0$ for their $a$, $b$, $c$ leading to values for $k$: $``(10k-6)^2 - 36(1+k^2)...0" \rightarrow k = ...$, giving $0$ and $\frac{15}{8}$ | M1 |
Uses $b^2 - 4ac > 0$ and chooses outside region for **their** critical values (both $a$ and $b$ must have been expressions in $k$) | dM1 |
Deduces $k < 0, k > \frac{15}{8}$ oe | A1 |
## Question 14 (Alt b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses a sketch or otherwise to deduce $k = 0$ is a critical value | B1 | 2.2a |
| Distance from $(a, ka)$ to $(0,0)$ is $3 \Rightarrow a^2(1+k^2) = 9$ | M1 | 3.1a |
| Tangent and radius are perpendicular $\Rightarrow k \times \frac{ka+5}{a-3} = -1 \Rightarrow a(1+k^2) = 3-5k$ | M1 | 3.1a |
| Solve simultaneously (dependent upon both M's) | dM1 | 1.1b |
| $k = \frac{15}{8}$ | A1 | 1.1b |
| Deduces $k < 0, k > \frac{15}{8}$ | A1 | 2.2a |
| **Total** | **(6)** | |
---\begin{enumerate}
\item The circle $C$ has equation
\end{enumerate}
$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0$$
(a) Find\\
(i) the coordinates of the centre of $C$\\
(ii) the radius of $C$
The line with equation $y = k x$, where $k$ is a constant, cuts $C$ at two distinct points.\\
(b) Find the range of values for $k$.
\hfill \mbox{\textit{Edexcel AS Paper 1 2018 Q14 [9]}}