## Question 9:

**Part (a):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{3\sin\theta\cos\theta}{\cos\theta+\sin\theta} = (2+\sec 2\theta)(\cos\theta-\sin\theta)$ $\Rightarrow \frac{3}{2}\sin 2\theta = (2+\sec 2\theta)(\cos\theta-\sin\theta)(\cos\theta+\sin\theta)$ | M1 | Attempts to use $\sin 2\theta = 2\sin\theta\cos\theta$ **or** $\cos 2\theta = \pm\cos^2\theta \pm \sin^2\theta$. Accept going from $(\cos\theta-\sin\theta)(\cos\theta+\sin\theta)$ to $\cos 2\theta$ |
| $\Rightarrow \frac{3}{2}\sin 2\theta = (2+\sec 2\theta)\cos 2\theta$ | dM1 | Attempts to use $\sin 2\theta = 2\sin\theta\cos\theta$ **and** $\cos 2\theta = \pm\cos^2\theta \pm \sin^2\theta$. Depends on previous M mark. |
| $\Rightarrow \frac{3}{2}\sin 2\theta = 2\cos 2\theta + 1 \Rightarrow 3\sin 2\theta - 4\cos 2\theta = 2\ *$ | A1* | Uses $\sec 2\theta = \frac{1}{\cos 2\theta}$ to achieve equation in just $\sin 2\theta$ and $\cos 2\theta$; all previous working shown. Note: accept $6\sin\theta\cos\theta = 4\cos 2\theta + 2$ to given answer. Condone missing $\theta$ or missing bracket but withhold A mark for clear errors such as missing "2" or missing pair of brackets not recovered. |

## Question (b):

**Way 1 – Using $R\sin(2x - \alpha)$ or $R\cos(2x + \alpha)$**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $R = \sqrt{3^2 + 4^2} = 5$ **or** $\alpha = \tan^{-1}\left(\frac{4}{3} \text{ or } \frac{3}{4}\right)$ | M1 | Condone obtaining $\pm 5$; or $\alpha = \tan^{-1}\left(\pm\frac{4}{3}\right)$, $\tan^{-1}\left(\pm\frac{3}{4}\right)$, $\cos^{-1}\left(\pm\frac{3}{5}\right)$, $\sin^{-1}\left(\pm\frac{4}{5}\right)$, $\cos^{-1}\left(\pm\frac{4}{5}\right)$, $\sin^{-1}\left(\pm\frac{3}{5}\right)$ |
| $R = \sqrt{3^2 + 4^2} = 5$ **and** $\alpha = \tan^{-1}\left(\frac{4}{3} \text{ or } \frac{3}{4}\right)$ | dM1 | Both required; depends on previous M1 |
| $5\sin(2x - 0.927) = 2$ **or** $5\cos(2x + 0.644) = -2$ | A1 | Accept awrt 0.93 or 0.64 for $\alpha$; do not allow $\pm 5\sin(2x-0.927)=2$ unless recovered |
| $x = \dfrac{\sin^{-1}\frac{2}{5} + 0.927}{2}$ **or** $x = \dfrac{\cos^{-1}\frac{-2}{5} - 0.644}{2}$ | ddM1 | Correct order of operations; condone $\pm$"0.927" or $\pm$"0.644"; depends on both previous method marks |
| $x = \text{awrt } 3.81$ | A1 | No other values in range; $x = \pi + 0.669$ scores A0 |

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**Way 2 – Squaring**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\sin 2x = 2 + 4\cos 2x \Rightarrow 9\sin^2 2x = 4 + 16\cos 2x + 16\cos^2 2x$ | M1 | Must isolate one trig term then square both sides correctly; do not condone poor squaring |
| $25\cos^2 2x + 16\cos 2x - 5 = 0$ **or** $25\sin^2 2x - 12\sin 2x - 12 = 0$ | dM1, A1 | Uses $\sin^2 2x + \cos^2 2x = 1$; proceeds to 3-term quadratic in $\sin 2x$ or $\cos 2x$ |
| $\cos 2x = \dfrac{-8 \pm 3\sqrt{21}}{25}$ (0.23, $-$0.87) **or** $\sin 2x = \dfrac{6 \pm 4\sqrt{21}}{25}$ (0.97, $-$0.49) | ddM1 | Attempts to solve quadratic; depends on both previous method marks |
| $x = \text{awrt } 3.81$ | A1 | No other values in range |

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**Way 3 – Using double angle formulae**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $6\sin x\cos x - 4\cos^2 x + 4\sin^2 x = 2$ | M1 | Uses $\sin 2x = 2\sin x\cos x$ and $\cos 2x = \pm\sin^2 x \pm \cos^2 x$ |
| $\tan^2 x + 3\tan x - 3 = 0$ | dM1, A1 | Divides by $\cos^2 x$, uses $\sec^2 x = 1 + \tan^2 x$; 3-term quadratic in $\tan x$ |
| $\tan x = \dfrac{-3 \pm \sqrt{21}}{2}$ (0.79, $-$3.79) | ddM1 | Solves quadratic; depends on both previous method marks |
| $x = \text{awrt } 3.81$ | A1 | No other values in range |

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**Way 4 – Divides by $\cos 2x$ and squares**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\tan 2x - 4 = 2\sec 2x \Rightarrow 9\tan^2 2x - 24\tan 2x + 16 = 4\sec^2 2x$ | M1 | Divides by $\cos 2x$ then squares; do not condone poor squaring e.g. $(3\tan 2x-4)^2 = 9\tan^2 2x+16$ |
| $5\tan^2 2x - 24\tan 2x + 12 = 0$ | dM1, A1 | Uses $\sec^2 2x = 1 + \tan^2 2x$; 3-term quadratic in $\tan 2x$ |
| $\tan 2x = \dfrac{12 \pm 2\sqrt{21}}{5}$ (4.23, 0.57) | ddM1 | Solves quadratic; depends on both previous method marks |
| $x = \text{awrt } 3.81$ | A1 | No other values in range; $x = \pi + 0.669$ scores A0 |

**Total: 5 marks for part (b)**