## Question 11:

| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i) EITHER:** Obtain a vector in the plane e.g. $\overrightarrow{PQ} = -3\mathbf{i} + 4\mathbf{j} + \mathbf{k}$ | B1 | |
| Use scalar product to obtain relevant equation in $a, b, c$ e.g. $-3a + 4b + c = 0$ or $6a - 2b + c = 0$ or $3a + 2b + 2c = 0$ | M1 | |
| State two correct equations in $a, b, c$ | A1 | |
| Solve simultaneous equations to obtain one ratio e.g. $a : b$ | M1 | |
| Obtain $a : b : c = 2 : 3 : -6$ or equivalent | A1 | |
| Obtain equation $2x + 3y - 6z = 8$ or equivalent | A1 | |
| **OR:** Substitute for $P, Q, R$ in equation of plane and state 3 equations in $a, b, c, d$ | B1 | |
| Eliminate one unknown, e.g. $d$, entirely | M1 | |
| Obtain 2 equations in 3 unknowns | A1 | |
| Solve to obtain one ratio e.g. $a : b$ | M1 | |
| Obtain $a : b : c = 2 : 3 : -6$ or equivalent | A1 | |
| Obtain equation $2x + 3y - 6z = 8$ or equivalent | A1 | |
| **OR:** Obtain a vector in plane e.g. $\overrightarrow{QR} = 6\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ | B1 | |
| Find second vector in plane and form correctly a 2-parameter equation for the plane | M1 | |
| Obtain equation in any correct form e.g. $\mathbf{r} = \lambda(-3\mathbf{i}+4\mathbf{j}+\mathbf{k}) + \mu(6\mathbf{i}-2\mathbf{j}+\mathbf{k}) + \mathbf{i} - \mathbf{k}$ | A1 | |
| State 3 equations in $x, y, z, \lambda$, and $\mu$ | A1 | |
| Eliminate $\lambda$ and $\mu$ | M1 | |
| Obtain equation $2x + 3y - 6z = 8$ or equivalent | A1 | |
| **OR:** Obtain a vector in plane e.g. $\overrightarrow{PR} = 3\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ | B1 | |
| Obtain second vector in plane and calculate vector product of the two vectors, e.g. $(-3\mathbf{i}+4\mathbf{j}+\mathbf{k}) \times (3\mathbf{i}+2\mathbf{j}+2\mathbf{k})$ | M1 | |
| Obtain 2 correct components of the product | A1 | |
| Obtain correct product e.g. $6\mathbf{i} + 9\mathbf{j} - 18\mathbf{k}$ or equivalent | A1 | |
| Substitute in $2x + 3y - 6z = d$ and find $d$ or equivalent | M1 | |
| Obtain equation $2x + 3y - 6z = 8$ or equivalent | A1 | **Total: 6** |

## Question (ii) EITHER:

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| State equation of $SN$ is $\mathbf{r} = 3\mathbf{i} + 5\mathbf{j} - 6\mathbf{k} + \lambda(2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k})$ or equivalent | $B1\sqrt{}$ | |
| Express $x, y, z$ in terms of $\lambda$ e.g. $(3 + 2\lambda,\ 5 + 3\lambda,\ -6 - 6\lambda)$ | $B1\sqrt{}$ | |
| Substitute in the equation of the plane and solve for $\lambda$ | $M1$ | |
| Obtain $\overrightarrow{ON} = \mathbf{i} + 2\mathbf{j}$, or equivalent | $A1$ | |
| Carry out method for finding $SN$ | $M1$ | |
| Show that $SN = 7$ correctly | $A1$ | |

## Question (ii) OR (Method 2):

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Letting $\overrightarrow{ON} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$, obtain two equations in $x, y, z$ by equating scalar product of $\overrightarrow{NS}$ with two of $\overrightarrow{PQ}, \overrightarrow{QR}, \overrightarrow{RP}$ to zero | $B1\sqrt{} + B1\sqrt{}$ | |
| Using the plane equation as third equation, solve for $x$, $y$, and $z$ | $M1$ | |
| Obtain $\overrightarrow{ON} = \mathbf{i} + 2\mathbf{j}$, or equivalent | $A1$ | |
| Carry out method for finding $SN$ | $M1$ | |
| Show that $SN = 7$ correctly | $A1$ | |

## Question (ii) OR (Method 3):

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Use Cartesian formula or scalar product of $\overrightarrow{PS}$ with a normal vector to find $SN$ | $M1$ | |
| Obtain $SN = 7$ | $A1$ | |
| State a unit normal $\hat{\mathbf{n}}$ to the plane | $B1\sqrt{}$ | |
| Use $\overrightarrow{ON} = \overrightarrow{OS} \pm 7\hat{\mathbf{n}}$ | $M1$ | |
| Obtain an unsimplified expression e.g. $3\mathbf{i} + 5\mathbf{j} - 6\mathbf{k} \pm 7\left(\frac{2}{7}\mathbf{i} + \frac{3}{7}\mathbf{j} - \frac{6}{7}\mathbf{k}\right)$ | $A1\sqrt{}$ | |
| Obtain $\overrightarrow{ON} = \mathbf{i} + 2\mathbf{j}$, or equivalent, only | $A1$ | **[6]** |