# Question 1:

## Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = 2 - 5i$ | B1 | Identifies correct complex conjugate as another root |
| $(z-(2+5i))(z-(2-5i)) = \ldots$ or Sum of roots $= 4$, Product of roots $= 29 \rightarrow z^2 + \ldots$ | M1 | Correct strategy using conjugate pair to find quadratic factor |
| $= z^2 - 4z + 29$ | A1 | Correct quadratic |
| $z^4 - 6z^3 + az^2 + bz + 145 = (z^2 - 4z + 29)(z^2 + cz + 5)$ | M1 | Uses given quartic and quadratic to find other quadratic factor by inspection or long division |
| $z^2 - 2z + 5 = 0$ | A1 | Correct second quadratic |
| $z^2 - 2z + 5 = 0 \Rightarrow z = \ldots$ | M1 | Solves second quadratic |
| $z = 1 \pm 2i$ | A1 | Correct second conjugate pair |

**(7 marks)**

## Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Points $(2,5)$ and $(2,-5)$ plotted correctly | B1ft | Pair of complex roots plotted correctly, either both $2\pm5i$ or follow through second pair. Allow if no labelling, as long as pair is symmetric in real axis |
| All four points $(2,5)$, $(1,2)$, $(1,-2)$, $(2,-5)$ fully correct and labelled | B1 | Fully correct labelled sketch. $(1,\pm2)$ roots must be within sector spanned by $(2,\pm5)$ |

**(2 marks)**

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# Question 1 Alt 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = 2-5i$ | B1 | Correct complex conjugate |
| $\alpha+\beta=4,\ \alpha\beta=2^2+5^2=\ldots$ and $\alpha+\beta+\gamma+\delta=\pm6,\ \alpha\beta\gamma\delta=\pm145$ | M1 | Sum and product approach, attempts formulae for sum and product of known roots and all four roots |
| $\alpha+\beta=4,\ \alpha\beta=29$ and $\alpha+\beta+\gamma+\delta=6,\ \alpha\beta\gamma\delta=145$ | A1 | Correct equations |
| $\Rightarrow \gamma+\delta=2,\ \gamma\delta=\frac{145}{29} \Rightarrow (2-\delta)\delta=\frac{145}{29}$ | M1 | Uses sum and product of known roots to reduce to equations in unknown roots |
| $\delta^2 - 2\delta + 5 = 0$ | A1 | Correct quadratic for remaining roots |
| $\delta^2 - 2\delta + 5 = 0 \Rightarrow \delta = \ldots$ | M1 | Solves quadratic |
| $= 1\pm 2i$ | A1 | Correct second conjugate pair |

**(7 marks)**

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# Question 1 Alt 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = 2-5i$ | B1 | Correct complex conjugate |
| $f(2\pm5i)=0 \Rightarrow -21a+2b+1038\pm(-20a-5b+450)i=0$ | M1 | Factor theorem approach, equates real and imaginary parts |
| $\Rightarrow 21a-2b-1038=0,\ 20a+5b-450=0$ | A1 | Correct equations |
| $\Rightarrow a=\ldots,\ b=\ldots$ | M1 | Solves simultaneous equations |
| $a=42$ and $b=-78$ | A1 | Correct values |
| $z^4-6z^3+\text{"42"}z^2-\text{"78"}z+145=0 \Rightarrow z=\ldots$ | M1 | Solves resulting quartic |
| $z=1\pm2i,\ (2\pm5i)$ | A1 | Correct second conjugate pair from fully correct work |

**(7 marks)**

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