**Answer:** $2\sin\theta - 3\cos\theta = R\sin(\theta - \alpha)$
$= R\sin\theta\cos\alpha - R\cos\theta\sin\alpha$

$\Rightarrow$ Rcosα =2, Rsinα=3

$\Rightarrow R^2 = 2^2 + 3^2 = 13, R = \sqrt{13}$

tan α = 3/2.

$\Rightarrow \alpha = 0.983$

minimum $1 - \sqrt{13}$, maximum $1 + \sqrt{13}$

| M1 | correct pairs |
| B1 | $R = \sqrt{13}$ or 3.61 or better |
| M1 | |
| A1 | 0.98 or better accept multiples of π that round to 0.98 |
| B1 B1 | or -2.61, 4.61 or better |
| | **condone wrong sign at this stage** **correct division, ft from first M1 radians only accept multiples of π that round to 0.98** |
| | **allow B1, B1ft for l-√R and 1+√R for their R to 2dp or better** |

**Total: [6]**

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## Question 4(i):

**Answer:** $x = 2\sin\theta, y = \cos 2\theta$

When $\theta = \pi/3$, $x = 2\sin\pi/3 = \sqrt{3}$

$y = \cos 2\pi/3 = -\frac{1}{2}$

| B1 | $x = \sqrt{3}$ |
| B1 | $y = -\frac{1}{2}$ |
| | exact only (isw all dec answers following exact ans ) |

**Total: [2]**

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## Question 4 (EITHER):

**Answer:** $\frac{dv}{d\theta} = 2\cos\theta, \frac{dy}{d\theta} = -2\sin 2\theta$

$\Rightarrow \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$ used

$\frac{dy}{dx} = \frac{-\sin 2\theta}{\cos\theta}$

$= \frac{-\sin 2\pi/3}{-\cos\pi/3} = \frac{-\sqrt{3}/2}{-1/2} = -\sqrt{3}$

| M1 | $\frac{dy}{dx} = (\frac{dy}{d\theta})/(\frac{dx}{d\theta})$ used |
| A1 | any correct equivalent form |
| A1 | exact www |
| ....... | .............................................. |
| M1 | |
| A1 | |
| A1 | exact www |
| | ft their derivatives if right way up (condone one further minor slip if intention clear) condone poor notation can isw if incorrect simplification |

**Total: [5]**

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## Question 4 (OR):

**Answer:** expressing $y$ in terms of $x$, $y=1-x^2/2$

$\frac{dy}{dx} = -x$ or $-2\sin\theta$

$= -\sqrt{3}$

| M1 | |
| A1 | |
| A1 | exact www |

**Total: [3]**

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## Question 4(ii):

**Answer:** $y = 1 - 2\sin^2\theta = 1 - 2(x/2)^2 = 1 - \frac{1}{2}x^2$

| M1A1 | or reference to (i) if used there |
| [2] | for M1, need correct trig identity and attempt to substitute for x allow SC B1 for y=cos 2arcsin(x/2) or equivalent |

**Total: [2]**

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