## Question 5:

### Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{\Delta y}{\Delta x} = \frac{9-6}{-2-10} = \left(-\frac{1}{4}\right)$ | M1 | Must be "correct way up" $\frac{\Delta y}{\Delta x}$ with attempt at differences on numerator or denominator |
| Uses gradient and a point to form equation of $l_1$: $y - 9 = -\frac{1}{4}(x+2)$ | dM1 | Uses gradient and one point to form straight line. Coordinates must be in correct place. Dependent on previous M |
| $y = -\frac{1}{4}x + \frac{17}{2}$ | A1 | Must be in form $y = mx + c$. ISW after sight of correct answer |

### Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Equation of $l_2$ is $y = 4x$ | B1ft | Correct follow through normal equation for their $y = -\frac{1}{4}x + \frac{17}{2}$. Implied by $y=4x+0$ following correct (a) |
| Attempts to solve $y = 4x$ and $y = -\frac{1}{4}x + \frac{17}{2}$ simultaneously | M1 | Non-calculator route required |
| $R = (2, 8)$ | A1, A1 | |

### Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts $OR = \sqrt{"2"^2 + "8"^2}$ or $PQ = \sqrt{3^2 + 12^2}$ | M1 | |
| Full attempt at area $OPQ = \frac{1}{2} \times \sqrt{3^2+12^2} \times \sqrt{2^2+8^2}$ | dM1 | |
| $= \frac{1}{2} \times 3\sqrt{17} \times 2\sqrt{17} = 51$ | A1 | |

# Question 5 (coordinate geometry):

## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = 4x$ and $y = -\frac{1}{4}x + \frac{17}{2} \Rightarrow 4x = -\frac{1}{4}x + \frac{17}{2}$ | M1 | Look for two highlighted equations with $x$ terms collected; allow "4"$x$ = "−0.25$x$ + 8.5" |
| $4.25x = 8.5$ | — | Implied by correct working |
| $R = (2, 8)$ | A1 | One correct coordinate (usually $x=2$); may be written separately |

## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sqrt{2^2 + 8^2}$ using coordinates of $R$, or $\sqrt{3^2 + 12^2}$ using given $P$ and $Q$ | M1 | Can be implied by exact answer or 3sf answer |
| Full attempt at area using $\frac{1}{2} \times OR \times PQ$ | dM1 | Both $PQ$ and $OR$ via correct method; dependent on previous M |
| Look for $\frac{1}{2} \times 2\sqrt{17} \times 3\sqrt{17}$ before seeing 51 | A1 | Correct answer via correct non-calculator method |
| **ALT 1 (cosine rule):** $\cos POQ = \frac{136+85-153}{2\sqrt{85}\sqrt{136}} = \frac{1}{\sqrt{10}}$, Area $= \frac{1}{2}\sqrt{85}\sqrt{136} \times \frac{3}{\sqrt{10}} = 51$ | M1, dM1, A1 | Decimals likely → A0 |
| **ALT 2 (shoelace):** Area $= \frac{1}{2}\begin{vmatrix}0 & -2 & 10 & 0\\ 0 & 9 & 6 & 0\end{vmatrix} = \frac{1}{2}\|(0\times9)+(-2\times6)+(10\times0)-(-2\times0)-(10\times9)-(0\times6)\| = \frac{1}{2}\times102 = 51$ | M2, A1 | Score M2 for full attempt at expression |
| **ALT 3 (trapezium):** $\frac{12}{2}(9+6)-\frac{1}{2}\times10\times6-\frac{1}{2}\times2\times9 = 90-30-9=51$ | M2, A1 | |
| **ALT 4 (two scalene triangles):** When $x=0, y=\frac{17}{2}$; Area $= \frac{1}{2}\times\frac{17}{2}\times10+\frac{1}{2}\times\frac{17}{2}\times2 = \frac{85}{2}+\frac{17}{2}=51$ | M1, dM1, A1 | M1: attempts $y$-intercept of $l_1$; dM1: full attempt at area |

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