## Question 4:

### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Assume true for $n=k$: $z^k = r^k(\cos k\theta + \mathrm{i}\sin k\theta)$ | | |
| $n=k+1$: $z^{k+1} = (z^k \times z =) r^k(\cos k\theta + \mathrm{i}\sin k\theta) \times r(\cos\theta + \mathrm{i}\sin\theta)$ | M1 | |
| $= r^{k+1}(\cos k\theta\cos\theta - \sin k\theta\sin\theta + \mathrm{i}(\sin k\theta\cos\theta + \cos k\theta\sin\theta))$ | M1 | |
| $= r^{k+1}(\cos(k+1)\theta + \mathrm{i}\sin(k+1)\theta)$ | M1dep A1 cso | |
| $\therefore$ if true for $n=k$, also true for $n=k+1$ | | |
| $k=1$: $z^1 = r^1(\cos\theta + \mathrm{i}\sin\theta)$; True for $n=1$, $\therefore$ true for all $n$ | A1 cso (5) | |

### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $w = 3\left(\cos\dfrac{3\pi}{4} + \mathrm{i}\sin\dfrac{3\pi}{4}\right)$ | | |
| $w^5 = 3^5\left(\cos\dfrac{15\pi}{4} + \mathrm{i}\sin\dfrac{15\pi}{4}\right)$ | M1 | |
| $w^5 = 243\left(\dfrac{1}{\sqrt{2}} - \dfrac{1}{\sqrt{2}}\mathrm{i}\right)$ $\left[= \dfrac{243\sqrt{2}}{2} - \dfrac{243\sqrt{2}}{2}\mathrm{i}\right]$ or equivalent | A1 (2) | |

# Question 4:

## Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $z^{k+1} = z^k \times z = r^k(\cos k\theta + i\sin k\theta) \times r(\cos\theta + i\sin\theta)$ | M1 | Using result for $n=k$ to write $z^{k+1}$ |
| Multiplying out and collecting real and imaginary parts using $i^2 = -1$, OR using sum of arguments and product of moduli to get $r^{k+1}(\cos(k\theta+\theta)+i\sin(k\theta+\theta))$ | M1 | |
| Using addition formulae to obtain single $\cos$ and $\sin$ terms, OR factorise argument $r^{k+1}(\cos\theta(k+1)+i\sin\theta(k+1))$ | M1dep | Dependent on second M mark |
| $r^{k+1}(\cos(k+1)\theta + i\sin(k+1)\theta)$ | A1cso | Only if all previous steps fully correct |
| All 5 underlined statements seen | A1cso | |

**Alternative (Euler's form):**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ | M1 | May not be seen explicitly |
| $z^{k+1} = z^k \times z = (re^{i\theta})^k \times re^{i\theta} = r^k e^{ik\theta} \times re^{i\theta}$ | M1 | |
| $= r^{k+1}e^{i(k+1)\theta}$ | M1dep | Dependent on 2nd M mark |
| $= r^{k+1}(\cos(k+1)\theta + i\sin(k+1)\theta)$ | A1cso | |
| $k=1$: $z^1 = r^1(\cos\theta + i\sin\theta)$; True for $n=1$ $\therefore$ true for all $n$ etc | A1cso | All 5 underlined statements must be seen |

## Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempting to apply de Moivre to $w$ or attempting to expand $w^5$ and collecting real and imaginary parts | M1 | No need to simplify |
| $243\left(\dfrac{1}{\sqrt{2}} - \dfrac{1}{\sqrt{2}}i\right) \left[= \dfrac{243\sqrt{2}}{2} - \dfrac{243\sqrt{2}}{2}i\right]$ | A1cao | oe e.g. $3^5$ instead of 243 |

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