## Question 11:

### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\cos^2 x + 1 = 4(1-\cos^2 x)$ or equivalent using $\sin^2 x$ or $\tan^2 x$ or $\cos 2x$ | M1 | Uses $\sin^2 x = 1-\cos^2 x$ or $\cos^2 x = 1-\sin^2 x$ or divides by $\cos^2 x$ or uses $\cos 2x$ forms; condone missing brackets |
| $\cos^2 x = \frac{3}{7}$ or $\sin^2 x = \frac{4}{7}$ or $\tan^2 x = \frac{4}{3}$ or $\cos 2x = -\frac{1}{7}$ | A1 | Correct value; may be implied by $\cos x = \sqrt{\frac{3}{7}}$ or $\sin x = \sqrt{\frac{4}{7}}$ or $\tan x = \sqrt{\frac{4}{3}}$ |
| $\cos x = \pm\sqrt{\frac{3}{7}} \Rightarrow x = \cos^{-1}\!\left(\sqrt{\frac{3}{7}}\right)$ | M1 | Correct order of operations to obtain expression for $x$ |
| $x = \text{awrt } 0.86, 2.28, 4.00, 5.43$ | A2,1,0 | A1 for any two of awrt $0.86, 2.28, 4.00, 5.43$; A1 for all four with no additional solutions in range |

### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $5\sin(\theta+10°) = \cos(\theta+10°) \Rightarrow \tan(\theta+10°) = 0.2$ | M1A1 | M1: reaches $\tan(\ldots) = \alpha$ where $\alpha$ is a constant including zero; A1: $\tan(\ldots)=0.2$ |
| $\theta = \tan^{-1}(0.2) - 10°$ | dM1 | Correct order of operations to produce one value for $\theta$; accept $\theta = \tan^{-1}(\alpha)-10$, $\alpha \neq 0$, or one correct answer as evidence; dependent on first M |
| $\theta = \text{awrt } 1.3°, 181.3°$ | A1A1 | First A1: one of awrt $\theta=1.3, 181.3$; second A1: both awrt $\theta=1.3, 181.3$ and no other solutions in range |

#### Alternative for (ii) by squaring:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $25\sin^2(\ldots) = \cos^2(\ldots) \Rightarrow 25(1-\cos^2(\ldots)) = \cos^2(\ldots)$ leading to $\sin^2(\ldots)=\ldots$ or $\cos^2(\ldots)=\ldots$ | M1 | Squares both sides, replaces $\sin^2(\ldots)$ by $1-\cos^2(\ldots)$ or replaces $\cos^2(\ldots)$ by $1-\sin^2(\ldots)$ |
| $\sin^2(\ldots) = \frac{1}{26}$ or $\cos^2(\ldots) = \frac{25}{26}$ | A1 | Correct value; may be implied by $\sin(\ldots)=\frac{1}{\sqrt{26}}$ or $\cos(\ldots)=\sqrt{\frac{25}{26}}$ |
| $\theta = \sin^{-1}\frac{1}{\sqrt{26}}-10°$ or $\theta = \cos^{-1}\frac{5}{\sqrt{26}}-10°$ | dM1 | Correct order of operations to produce one value; dependent on first M |
| $\theta = 1.3°, 181.3°$ | A1A1 | First A1: one of awrt $\theta=1.3, 181.3$; second A1: both with no other solutions in range |

## Alternative 2 for (ii) — Using the Addition Formulae

| Answer/Working | Mark | Guidance |
|---|---|---|
| $5\sin\theta\cos10 + 5\cos\theta\sin10 = \cos\theta\cos10 - \sin\theta\sin10$ — uses correct addition formulae on both sides and rearranges to $\tan(\ldots) =$ | M1 | Uses correct addition formulae on both sides and rearranges to $\tan(\ldots) =$ |
| $\tan\theta = \dfrac{\cos10 - 5\sin10}{5\cos10 + \sin10} = (0.0229)$ | A1 | Correct value for $\tan\theta$ |
| $\tan\theta = 0.0229 \Rightarrow \theta = \ldots$ | dM1 | Uses arctan to produce one value for $\theta$. **Dependent on the first M.** |
| $\Rightarrow \theta = 1.3°, 181.3°$ | A1A1 | A1: One of awrt $\theta = 1.3, 181.3$; A1: Both awrt $\theta = 1.3, 181.3$ and no other solutions in range; ignore solutions outside the range. |

> Note: Final answers in **radians** in (ii) cannot score the final 2 A marks but the earlier marks are available (maximum 11100)

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