# Question 7(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $w = \frac{z-3}{2\text{i}-z} \Rightarrow 2\text{i}w - wz = z-3 \Rightarrow z = \ldots$ | M1 | Attempts to make $z$ the subject and obtains any $f(w)$ |
| $z = \frac{3+2\text{i}w}{w+1}$ or $\frac{-3-2\text{i}w}{-w-1}$ | A1 | Any correct expression for $z$ in terms of $w$ |
| $= \frac{3+2\text{i}u-2v}{u+\text{i}v+1} \times \frac{u+1-\text{i}v}{u+1-\text{i}v}$ | M1 | Applies $w = u + \text{i}v$ and correct multiplier for their $z$; denominator must have had a "$w$" |
| Equates imaginary part to $x+3$ using $y = x+3$ | M1 | Multiplies, extracts real and imaginary parts and uses in $y = x+3$ to produce equation in $u$ and $v$ only |
| $u^2 + 7u + v^2 + v + 6 = 0$ | dddM1 | Expands and simplifies to obtain circle equation with 4 or 5 real unlike terms; all previous Ms required |
| $\left(u+\frac{7}{2}\right)^2 + \left(v+\frac{1}{2}\right)^2 = \frac{13}{2}$ | M1 | Extracts centre and/or radius from their circle equation with 4 or 5 real unlike terms |
| Centre: $\left(-\frac{7}{2}, -\frac{1}{2}\right)$ | A1 | Correct centre from correct circle equation |
| Radius: $\frac{\sqrt{26}}{2}$ or $\sqrt{\frac{13}{2}}$ | A1 | Correct radius from correct circle equation |

# Question 7(a):

**Way 2:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $w = \frac{z-3}{2i-z} = \frac{x+iy-3}{2i-x-iy} = \frac{x-3+i(x+3)}{2i-x-i(x+3)}$ [Note: replace $x$ with $y-3$] | M1 A1 | M1: Uses $z=x+iy$ and $y=x+3$; A1: Correct expression for $w$ in terms of $x$ |
| $\frac{x-3+i(x+3)}{-x-i(x+1)} = u+iv \Rightarrow x-3+i(x+3) = -xu+v(x+1)-iu(x+1)-ivx$ | M1 | Applies $w=u+iv$ and multiplies |
| $x-3 = -ux+vx+v$, $x+3=-ux-u-vx$; $x=\frac{3+v}{1+u-v}$, $x=\frac{-3-u}{1+u+v}$ | M1 | Equates real and imaginary parts and makes $x$ the subject twice |
| $3+3u+3v+v+uv+v^2 = -3-3u+3v-u-u^2+uv \Rightarrow u^2+v^2+7u+v+6=0$ | dddM1 | Equates expressions for $x$ to obtain circle equation with 4 or 5 real unlike terms. **All previous Ms required.** |
| $\left(u+\frac{7}{2}\right)^2+\left(v+\frac{1}{2}\right)^2 = \frac{13}{2}$; centre: $\left(-\frac{7}{2},-\frac{1}{2}\right)$; radius: $\frac{\sqrt{26}}{2}$ or $\sqrt{\frac{13}{2}}$ | M1 A1 A1 | M1: Correct process to extract centre/radius. A1: Correct centre. A1: Correct radius |

**Way 3:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. 3 points on line: $(0,3)$, $(1,4)$, $(2,5)$; or $z_1=3i$, $z_2=1+4i$, $z_3=2+5i$ | M1 | Attempts three points/complex numbers on $y=x+3$ with 2 correct |
| $w_1=\frac{3i-3}{-i}$, $w_2=\frac{-2+4i}{-1-2i}$, $w_3=\frac{-1+5i}{-2-3i}$ | A1 | Correct transformed complex numbers |
| Multiply to remove $i$ from denominator | M1 | At least two correct multipliers, requires 2 correct points on line |
| $w_1=-3-3i$, $w_2=\frac{6}{5}-\frac{8}{5}i$, $w_3=-1-i$ | M1 | Two correct complex numbers in $a+ib$ form |
| $6g+6f-c=18$; e.g. $\frac{12}{5}g+\frac{16}{5}f-c=0$; $2g+2f-c=0$ | dddM1 | Uses general circle equation to form three simultaneous equations. **All previous Ms required.** |
| $g=\frac{7}{2}$, $f=\frac{1}{2}$, $c=6$; centre $\left(-\frac{7}{2},-\frac{1}{2}\right)$; radius $\sqrt{g^2+f^2-c}=\frac{\sqrt{26}}{2}$ or $\sqrt{\frac{13}{2}}$ | M1 A1 A1 | M1: Solves for at least one correct coordinate or radius. A1: Correct centre. A1: Correct centre and radius |

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# Question 7(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Circle drawn with whole interior indicated | M1 (B1 on ePen) | Any circle with whole interior indicated. Ignore inconsistencies with stated centre, radius or circle equation |
| Correct circle in correct position (entirely in quadrants 2 & 3, centre in Q3), whole interior shaded | A1 (B1 on ePen) | Must be shaded, does not require label. Condone dotted circumference. **Requires full marks in (a).** |

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