# Question 9:

## Part (i):

**EITHER Method:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain a vector parallel to the plane, e.g. $\overrightarrow{AB} = \mathbf{i} - 2\mathbf{j} - 3\mathbf{k}$ | B1 | |
| Use scalar product to obtain an equation in $a$, $b$, $c$ e.g. $a - 2b - 3c = 0$, $a + b - c = 0$, or $3b + 2c = 0$ | M1 | |
| State two correct equations | A1 | |
| Solve to obtain ratio $a:b:c$ | M1 | |
| Obtain $a:b:c = 5:-2:3$ | A1 | |
| Obtain equation $5x - 2y + 3z = 5$, or equivalent | A1 | |

**OR1 Method:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute for two points, e.g. $A$ and $B$, and obtain $a + 3b + 2c = d$ and $2a + b - c = d$ | (B1) | |
| Substitute for another point, e.g. $C$, to obtain a third equation and eliminate one unknown entirely from all three equations | M1 | |
| Obtain two correct equations in three unknowns, e.g. in $a$, $b$, $c$ | A1 | |
| Solve to obtain their ratio | M1 | |
| Obtain $a:b:c = 5:-2:3$, $a:c:d = 5:3:5$, $a:b:d = 5:-2:5$, or $b:c:d = -2:3:5$ | A1 | |
| Obtain equation $5x - 2y + 3z = 5$, or equivalent | A1) | |

**OR2 Method:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain a vector parallel to the plane, e.g. $\overrightarrow{AC} = \mathbf{i} + \mathbf{j} - \mathbf{k}$ | (B1) | |
| Obtain a second such vector and calculate their vector product, e.g. $(\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}) \times (\mathbf{i} + \mathbf{j} - \mathbf{k})$ | M1 | |
| Obtain two correct components of the product | A1 | |
| Obtain correct answer e.g. $5\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$ | A1 | |
| Substitute in $5x - 2y + 3z = d$ to find $d$ | M1 | |
| Obtain equation $5x - 2y + 3z = 5$, or equivalent | A1) | |

**OR3 Method:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain a vector parallel to the plane, e.g. $\overrightarrow{BC} = 3\mathbf{j} + 2\mathbf{k}$ | (B1) | |
| Obtain a second such vector and form correctly a 2-parameter equation for the plane | M1 | |
| Obtain a correct equation, e.g. $\mathbf{r} = \mathbf{i} + 3\mathbf{j} + 2\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}) + \mu(3\mathbf{j} + 2\mathbf{k})$ | A1 | |
| State three correct equations in $x$, $y$, $z$, $\lambda$, $\mu$ | A1 | |
| Eliminate $\lambda$ and $\mu$ | M1 | |
| Obtain equation $3x - 2y + 3z = 5$, or equivalent | A1) | [6] |

## Part (ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Correctly form an equation for the line through $D$ parallel to $OA$ | M1 | |
| Obtain a correct equation e.g. $\mathbf{r} = -3\mathbf{i} + \mathbf{j} + 2\mathbf{k} + \lambda(\mathbf{i} + 3\mathbf{j} + 2\mathbf{k})$ | A1 | |
| Substitute components in the equation of the plane and solve for $\lambda$ | M1 | |
| Obtain $\lambda = 2$ and position vector $-\mathbf{i} + 7\mathbf{j} + 6\mathbf{k}$ for $P$ | A1 | |
| Obtain the given answer correctly | A1 | [5] |

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