## Question 9:

### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **Way 1:** Divide by $\cos 3\theta$: $\tan 3\theta = \sqrt{3} \Rightarrow 3\theta = \dfrac{\pi}{3}$ **Way 2:** Square both sides, use $\cos^2 3\theta + \sin^2 3\theta=1$, obtain $\cos 3\theta = \pm\dfrac{1}{2}$ or $\sin 3\theta = \pm\dfrac{\sqrt{3}}{2}$, so $3\theta = \dfrac{\pi}{3}$ | M1 | Obtains $\dfrac{\pi}{3}$. Allow $x=\dfrac{\pi}{3}$ or $\theta=\dfrac{\pi}{3}$. Do not allow $\tan 3\theta = -\sqrt{3}$ nor $\tan 3\theta = \pm\dfrac{1}{\sqrt{3}}$ |
| Adds $\pi$ **or** $2\pi$ to previous value (giving $\dfrac{4\pi}{3}$ or $\dfrac{7\pi}{3}$) | M1 | Adding $\pi$ or $2\pi$ to a previously obtained value; not dependent on previous mark |
| $\theta = \dfrac{\pi}{9}, \dfrac{4\pi}{9}, \dfrac{7\pi}{9}$ (all three, no extra in range) | A1 | |

### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4(1-\cos^2 x) + \cos x = 4-k$; applies $\sin^2 x = 1-\cos^2 x$ | M1 | |
| Attempts to solve $4\cos^2 x - \cos x - k = 0$ | dM1 | |
| $\cos x = \dfrac{1 \pm \sqrt{1+16k}}{8}$ **or** $\cos x = \dfrac{1}{8} \pm \sqrt{\dfrac{1}{64}+\dfrac{k}{4}}$ or correct equivalent | A1 | |

### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cos x = \dfrac{1\pm\sqrt{49}}{8} = 1$ and $-\dfrac{3}{4}$ | M1 | |
| Obtains two solutions from $0°, 139°, 221°$ (or $0$ or $2.42$ or $3.86$ in radians) | dM1 | |
| $x = 0°$ and $139°$ and $221°$ (allow awrt $139$ and $221$) must be in degrees | A1 | |

## Question 9 (continued):

### Part (i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| All three correct answers in terms of $\pi$, no extras in range | A1 | Need all three correct answers in terms of $\pi$ and **no extras in range** |
| **NB:** $\theta = 20°, 80°, 140°$ earns M1M1A0 and $0.349, 1.40$ and $2.44$ earns M1M1A0 | — | — |

### Part (ii)(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Applies $\sin^2 x = 1 - \cos^2 x$ | M1 | Allow even if brackets are missing e.g. $4 \times 1 - \cos^2 x$. Must be awarded in (ii)(a) for expression with $k$ not after $k=3$ is substituted |
| Uses formula or completion of square to obtain $\cos x =$ expression in $k$ | dM1 | Factorisation attempt is M0 |
| Final simplified expression | A1 | cao - award for their final simplified expression |

### Part (ii)(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| **Either** attempts to substitute $k=3$ into their answer to obtain two values for $\cos x$, **Or** restarts with $k=3$ to find two values for $\cos x$ | M1 | In both cases they need to have applied $\sin^2 x = 1 - \cos^2 x$ (brackets may be missing) **and** correct method for solving their quadratic (usual rules). Values for $\cos x$ may be $>1$ or $<-1$ |
| Obtains **two correct** values for $x$ | dM1 | — |
| Obtains **all three correct values** in degrees (allow awrt 139 and 221) including 0 | A1 | Ignore excess answers outside range (including 360 degrees). Lose this mark for excess answers in the range or radian answers |