## Question 7:

### Part (a)(i):

| Working/Answer | Mark | Guidance |
|---|---|---|
| $1+\sqrt{3}\,\mathrm{i}$: $r = \sqrt{1+3} = 2$, $\theta = \frac{\pi}{3}$, so $2e^{i\pi/3}$ | B1 | Correct modulus |
| $1-\mathrm{i}$: $r = \sqrt{2}$, $\theta = -\frac{\pi}{4}$, so $\sqrt{2}\,e^{-i\pi/4}$ | M1 A1 | M1 for method, A1 both correct |

### Part (a)(ii):

| Working/Answer | Mark | Guidance |
|---|---|---|
| $(1+\sqrt{3}\,\mathrm{i})^8(1-\mathrm{i})^5 = 2^8 e^{i \cdot 8\pi/3} \cdot (\sqrt{2})^5 e^{-i \cdot 5\pi/4}$ | M1 | Multiply moduli and add arguments |
| Modulus: $2^8 \times 2^{5/2} = 2^{8} \times 4\sqrt{2} = 1024\sqrt{2} \cdot 4 = $ wait: $256 \times 4\sqrt{2} = 1024\sqrt{2}$ | A1 | Correct modulus $1024\sqrt{2}$ |
| Argument: $\frac{8\pi}{3} - \frac{5\pi}{4} = \frac{32\pi - 15\pi}{12} = \frac{17\pi}{12}$, adjusted to $\frac{17\pi}{12} - 2\pi = -\frac{7\pi}{12}$ | A1 | Correct argument $-\frac{7\pi}{12}$ |

### Part (b):

| Working/Answer | Mark | Guidance |
|---|---|---|
| $z^3 = 1024\sqrt{2}\, e^{-7\pi i/12}$, so $|z| = (1024\sqrt{2})^{1/3} = 8\sqrt{2}$ | B1 | Correct modulus $8\sqrt{2}$ |
| General argument: $\frac{1}{3}\left(-\frac{7\pi}{12} + 2k\pi\right)$ for $k = 0, \pm1$ | M1 | Correct method for cube roots |
| $k=0$: $8\sqrt{2}\,e^{-7\pi i/36}$ | A1 | One correct root |
| $k=1$: $8\sqrt{2}\,e^{i \cdot 17\pi/36}$; $k=-1$: $8\sqrt{2}\,e^{-31\pi i/36}$ | A1 | All three correct roots in required form |