## Question 1:

### Part (a)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 8x - 2y\frac{dy}{dx} + 2x\frac{dy}{dx} + 2y = 0$ | M1B1A1 | M1: Differentiating $4x^2 - y^2$ to obtain $Ax + By\frac{dy}{dx}$. B1: Sight of $\frac{d}{dx}(2xy) = 2x\frac{dy}{dx} + 2y$. A1: Fully correct derivative |
| **Way 1:** Sets $\frac{dy}{dx} = 2$ in each term in differentiated expression | dM1 | Depends on previous M and B marks; needs at least two terms in $\frac{dy}{dx}$ and two other terms |
| $\Rightarrow 8x - 4y + 4x + 2y = 0 \Rightarrow y - 6x = 0$* | ddM1, A1* | ddM1: Dependent on both previous M's; obtain unsimplified correct equation in $x$ and $y$. A1*: cso $y - 6x = 0$, no errors seen; accept $y = 6x$ |
| **Way 2:** Obtains $\frac{dy}{dx} = \left(\frac{8x+2y}{2y-2x}\right)$ | dM1 | ft their differentiated expression |
| $\frac{8x+2y}{2y-2x} = 2$, so $y - 6x = 0$* | ddM1, A1* | As above |

### Part (b)

| Answer/Working | Mark | Guidance |
|---|---|---|
| Put $y = 6x$ or $x = \frac{y}{6}$ into $4x^2 - y^2 + 2xy + 5 = 0$, obtains $Ay^2 = B$ or $Ax^2 = B$ | M1 | Substitutes into curve equation to form equation in one variable; reaches two-term quadratic |
| $x = \pm\frac{1}{2}$ or $y = \pm 3$ or $\left(\frac{1}{2}, 3\right)$ or $\left(-\frac{1}{2}, -3\right)$ | A1 | Either one correct pair of coordinates **or** both $x$ or both $y$ values |
| Both $\left(\frac{1}{2}, 3\right)$ and $\left(-\frac{1}{2}, -3\right)$ and no extra solutions | A1 | Both correct with no incorrect working. Unsimplified answers e.g. $\left(\frac{\sqrt{5}}{\sqrt{20}}, 3\right)$ lose final A mark. $x = \pm\frac{1}{2}, y = \pm 3$ not sufficient |

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