# Question 10:

## Part (a):

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Gradient of $l_1$ is $\frac{7-2}{3-0} = \frac{5}{3}$ | B1 | $m(l_1) = \frac{7-2}{3-0}$. Allow un-simplified. May be implied. |
| $m(l_2) = -1 \div \text{their } \frac{5}{3}$ | M1 | Correct application of perpendicular gradient rule |
| $y - 7 = -\frac{3}{5}(x-3)$ or $y = -\frac{3}{5}x + c$, $7 = -\frac{3}{5}(3) + c \Rightarrow c = \frac{44}{5}$ | M1A1ft | M1: Uses $y-7 = m(x-3)$ with their **changed** gradient or uses $y = mx+c$ with $(3,7)$ and their **changed** gradient. A1ft: Correct ft equation for their perpendicular gradient (dependent on both M marks) |
| $3x + 5y - 44 = 0$ | A1 | Any positive or negative integer multiple. Must be seen in (a) and must include "$= 0$" |

**[5 marks]**

## Part (b):

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| When $y=0$, $x = \frac{44}{3}$ | M1 A1 | M1: Puts $y=0$ and finds a value for $x$. A1: $x = \frac{44}{3}$ (or $14\frac{2}{3}$ or $14.\dot{6}$) or exact equivalent. ($y=0$ not needed) |

*Note: Condone $3x - 5y - 44 = 0$ only leading to the correct answer and condone coordinates written as $(0, 44/3)$ but allow recovery in (c)*

**[2 marks]**

## Part (c):

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Correct attempt at finding area of any one triangle or trapezium (not just one rectangle), using correct pair of base and height | M1 | Note: first three marks apply to their calculated coordinates e.g. their $\frac{44}{3}$, $\frac{44}{5}$, $-\frac{6}{5}$ etc. Given coordinates must be correct e.g. $(0,2)$ and $(3,7)$ |
| Correct numerical **expression** for area of one **triangle** or one **trapezium** for their coordinates | A1ft | |
| Combines correct areas together correctly for chosen method | dM1 | If correct numerical expressions have been incorrectly simplified before combining, M1 may still be given. Dependent on first M mark. |
| Correct numerical expression for area of $ORQP$ — expressions must be **fully correct** | A1 | |
| $54\frac{1}{3}$ | A1 | Any exact equivalent e.g. $\frac{163}{3}$, $\frac{326}{6}$, $54.\dot{3}$ |

**[5 marks]**

### Way 1:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $OPQT = \frac{1}{2}(2+7)\times 3$ or $TRQ = \frac{1}{2}\times 7 \times \left(\frac{44}{3}-3\right)$ | M1A1ft | M1: Correct method for $OPQT$ or $TRQ$ |
| $\frac{1}{2}(2+7)\times 3 + \frac{1}{2}\times 7 \times \left(\frac{44}{3}-3\right)$ | dM1A1 | dM1: Correct numerical combination. A1: Fully correct numerical expression for area $ORQP$ |
| $54\frac{1}{3}$ | A1 | Any exact equivalent e.g. $\frac{163}{3}$, $\frac{326}{6}$, $54.\dot{3}$ |

### Way 2:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $PQR = \frac{1}{2}\times\sqrt{34}\times\frac{7}{3}\times\sqrt{34}$ or $OPR = \frac{1}{2}\times\frac{44}{3}\times 2$ | M1A1ft | M1: Correct method for $PQR$ or $OPR$ |
| $\frac{1}{2}\times\sqrt{34}\times\frac{7}{3}\times\sqrt{34} + \frac{1}{2}\times\frac{44}{3}\times 2$ | dM1A1 | dM1: Correct numerical combination. A1: Fully correct expression for $ORQP$ |
| $54\frac{1}{3}$ | A1 | Any exact equivalent |

### Way 3:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $SQR = \frac{1}{2}\times 7 \times \frac{238}{15}$ or $SPO = \frac{1}{2}\times\frac{6}{5}\times 2$ | M1A1ft | M1: Correct method for $SQR$ or $SPO$ |
| $\frac{1}{2}\times 7 \times \frac{238}{15} - \frac{1}{2}\times\frac{6}{5}\times 2$ | dM1A1 | dM1: Correct numerical combination. A1: Fully correct expression for $ORQP$ |
| $54\frac{1}{3}$ | A1 | Any exact equivalent |

### Way 4:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $PVQ = \frac{1}{2}\times 5 \times 3$ or $QUR = \frac{1}{2}\times 7 \times \frac{35}{3}$ | M1A1ft | M1: Correct method for $PVQ$ or $QUR$ |
| $OVUR\ 7\times\frac{44}{3} - \frac{1}{2}\times 5\times 3 - \frac{1}{2}\times 7\times\frac{35}{3}$ | dM1A1 | dM1: Correct numerical combination. A1: Fully correct expression for $ORQP$ |
| $54\frac{1}{3}$ | A1 | Any exact equivalent |

### Way 5:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $OWR = \frac{1}{2}\times\frac{44}{3}\times\frac{44}{5}$ or $PWQ = \frac{1}{2}\times\left(\frac{44}{5}-2\right)\times 3$ | M1A1ft | M1: Correct method for $OWR$ or $PWQ$ |
| $\frac{1}{2}\times\frac{44}{3}\times\frac{44}{5} - \frac{1}{2}\times\left(\frac{44}{5}-2\right)\times 3$ | dM1A1 | dM1: Correct numerical combination. A1: Fully correct expression for $ORQP$ |
| $54\frac{1}{3}$ | A1 | Any exact equivalent |

## Question 10 (Area ORQP - Ways 6, 7, 8):

**WAY 6:**

| Answer/Working | Marks | Guidance |
|---|---|---|
| $OPNR = \frac{1}{2} \times \left(\frac{34}{3} + \frac{44}{3}\right) \times 2$ **or** $PNQ = \frac{1}{2} \times \frac{34}{3} \times 5$ | M1A1ft | Correct method for $OPNR$ or $PNQ$ |
| $\frac{1}{2}\left(\frac{34}{3} + \frac{44}{3}\right) \times 2 + \frac{1}{2} \times \frac{34}{3} \times 5$ | dM1A1 | Correct numerical combination of areas calculated correctly; Fully Correct numerical expression for area $ORQP$ |
| $54\frac{1}{3}$ | A1 | Any exact equivalent e.g. $\frac{163}{3}$, $\frac{326}{6}$, $54.\dot{3}$ |

**WAY 7:**

| Answer/Working | Marks | Guidance |
|---|---|---|
| $OPQ = \frac{1}{2} \times 3 \times 2$ **or** $OQR = \frac{1}{2} \times \frac{44}{3} \times 7$ | M1A1ft | Correct method for $OPQ$ or $OQR$ |
| $\frac{1}{2} \times 3 \times 2 + \frac{1}{2} \times \frac{44}{3} \times 7$ | dM1A1 | Correct numerical combination; Fully Correct numerical expression for area $ORQP$ |
| $54\frac{1}{3}$ | A1 | Any exact equivalent e.g. $\frac{163}{3}$, $\frac{326}{6}$, $54.\dot{3}$ |

**WAY 8:**

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}\begin{vmatrix} 0 & \frac{44}{3} & 3 & 0 & 0 \\ 0 & 0 & 7 & 2 & 0 \end{vmatrix}$ | M1A1ft | Uses vertices of quadrilateral to form a determinant |
| $\frac{1}{2}\left(\frac{44}{3} \times 7 + 3 \times 2\right)$ | dM1A1 | Fully correct determinant method with no errors; Fully Correct numerical expression for area $ORQP$ |
| $54\frac{1}{3}$ | A1 | Any exact equivalent e.g. $\frac{163}{3}$, $\frac{326}{6}$, $54.\dot{3}$ |

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