## Question 2:

| $\frac{1}{2}r^2\theta = 9 \Rightarrow \frac{1}{2}r^2(0.5) = 9 \Rightarrow r = 6$ | M1 | Attempts to use the area to find a value for $r$ |
|---|---|---|
| $r = 6$ | A1 | Correct value for $r$ |
| Arc length $= r\theta = 6 \times 0.5 = 3$ | M1 | Uses arc length formula to attempt to find the value of $k$ |
| Perimeter $= 6 + 6 + 3 = 15 = 5(3) \Rightarrow k = 5$ | A1 | Correct solution only |

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## Question 3a:

| $\frac{dy}{dx} = Ax^{-\frac{1}{2}} + Bx^{-\frac{3}{2}}$ | M1 | Differentiates to correct form |
|---|---|---|
| $\frac{dy}{dx} = 4x^{-\frac{1}{2}} - 9x^{-\frac{3}{2}}$ | A1 | Achieves correct $\frac{dy}{dx}$ |
| $\frac{d^2y}{dx^2} = -2x^{-\frac{3}{2}} + \frac{27}{2}x^{-\frac{5}{2}}$ | B1 | Achieves correct $\frac{d^2y}{dx^2}$ for their $\frac{dy}{dx}$ |

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## Question 3b:

| Set $\frac{dy}{dx} = 0$ | M1 | Set $\frac{dy}{dx} = 0$ |
|---|---|---|
| $4x^{-\frac{1}{2}} - 9x^{-\frac{3}{2}} = 0 \Rightarrow 4x - 9 = 0 \Rightarrow x = \frac{9}{4}$ | A1 | Rearrange to find $x$, correct answer only |
| Substitute $x = \frac{9}{4}$ into $C$ | M1 | Substitutes their $x$ value(s) into the given equation |
| $y = 8\sqrt{\frac{9}{4}} + \frac{18}{\sqrt{9 \div 4}} - 20 = 4$, stationary point at $\left(\frac{9}{4}, 4\right)$ | A1 | Correct answer only |

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## Question 3c:

| $\frac{d^2y}{dx^2} = -2\left(\frac{9}{4}\right)^{-\frac{3}{2}} + \frac{27}{2}\left(\frac{9}{4}\right)^{-\frac{5}{2}}$ | M1 | Substitutes their $x$ value(s) into their second derivative |
|---|---|---|
| $\frac{d^2y}{dx^2} = \frac{32}{27} > 0$, so is a minimum | A1 | Fully correct solution including correct numerical second derivative (awrt), reference to positive or $> 0$, and a conclusion |

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## Question 4a:

| $f(x) = 2 + \ln(4-x) - x$; $f(2) = 2 + \ln(2) - 2 = 0.693...$ and $f(3) = 2 + \ln(1) - 3 = -1$ | M1 | Attempts $f(2) = \ldots$ and $f(3) = \ldots$ where $f(x) = \pm(2 + \ln(4-x) - x)$ |
|---|---|---|
| Sign change in $[2,3]$, $f(x)$ continuous, therefore $2 < \beta < 3$ | A1 | Both values correct to at least 1 significant figure with correct explanation and conclusion |

## Question 4b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt at using a cobweb diagram starting at $x_1 = 3$ with at least two spirals shown | M1 | Must show at least two spirals on the diagram |
| Correct attempt starting at $x_1 = 3$, deducing that the iteration formula **can** be used to find an approximation for $\beta$ because the cobweb spirals inwards / the cobweb gets closer to the root / the cobweb converges | A1 | Must include correct justification referencing convergent behaviour |

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