## Question 8:

### Part (i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| **Either:** Obtain $\pm\begin{pmatrix}2\\-1\\-15\end{pmatrix}$ for vector $PA$ (where $A$ is point on line) or equivalent | B1 | |
| Use scalar product to find cosine of angle between $PA$ and line | M1 | |
| Obtain $\frac{42}{\sqrt{14 \times 230}}$ or equivalent | A1 | |
| Use trigonometry to obtain $\sqrt{104}$ or $10.2$ or equivalent | A1 | |
| **Or 1:** Obtain $\pm\begin{pmatrix}2n+2\\n-1\\3n-15\end{pmatrix}$ for $PN$ (where $N$ is foot of perpendicular) | B1 | |
| Equate scalar product of $PN$ and line direction to zero, or equate derivative of $PN^2$ to zero, or use Pythagoras' theorem in triangle $PNA$ to form equation in $n$ | M1 | |
| Solve equation and obtain $n = 3$ | A1 | |
| Obtain $\sqrt{104}$ or $10.2$ or equivalent | A1 | |
| **Or 2:** Obtain $\pm\begin{pmatrix}2\\-1\\-15\end{pmatrix}$ for vector $PA$ | B1 | |
| Evaluate vector product of $PA$ and line direction | M1 | |
| Obtain $\pm\begin{pmatrix}12\\-36\\-4\end{pmatrix}$ | A1 | |
| Divide modulus of this by modulus of line direction and obtain $\sqrt{104}$ or $10.2$ or equivalent | A1 | |
| **Or 3:** Obtain $\pm\begin{pmatrix}2\\-1\\-15\end{pmatrix}$ for vector $PA$ | B1 | |
| Evaluate scalar product of $PA$ and line direction to obtain distance $AN$ | M1 | |
| Obtain $3\sqrt{14}$ or equivalent | A1 | |
| Use Pythagoras' theorem in triangle $PNA$ and obtain $\sqrt{104}$ or $10.2$ or equivalent | A1 | |
| **Or 4:** Obtain $\pm\begin{pmatrix}2\\-1\\-15\end{pmatrix}$ for vector $PA$ | B1 | |
| Use a second point $B$ on line and use cosine rule in triangle $ABP$ to find angle $A$, or angle $B$ or use vector product to find area of triangle | M1 | |
| Obtain correct answer (angle $A = 42.25\ldots$) | A1 | |
| Use trigonometry to obtain $\sqrt{104}$ or $10.2$ or equivalent | A1 | [4] |

# Question 8(ii) - Either Method:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use scalar product to obtain a relevant equation in $a, b, c$, e.g. $2a + b + 3c = 0$ | M1 | |
| $2a - b - 15c = 0$ | A1✓ | |
| State two correct equations in $a$, $b$ and $c$ | M1 | |
| Solve simultaneous equations to obtain one ratio | A1 | |
| Obtain $a : b : c = -3 : 9 : -1$ or equivalent | A1 | |
| Obtain equation $-3x + 9y - z = 28$ or equivalent | | |

# Question 8(ii) - Or 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Calculate vector product of two of $\begin{pmatrix}2\\1\\3\end{pmatrix}, \begin{pmatrix}2\\-1\\-15\end{pmatrix}$ and $\begin{pmatrix}8\\2\\-6\end{pmatrix}$ or equiv | M1 | |
| Obtain two correct components of the product | A1✓ | |
| Obtain correct $\begin{pmatrix}-3\\9\\-1\end{pmatrix}$ or equivalent | A1 | |
| Substitute in $-3x + 9y - z = d$ to find $d$ or equivalent | M1 | |
| Obtain equation $-3x + 9y - z = 28$ or equivalent | A1 | |

# Question 8(ii) - Or 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Form a two-parameter equation of the plane | M1 | |
| Obtain $\mathbf{r} = \begin{pmatrix}1\\3\\-4\end{pmatrix} + s\begin{pmatrix}2\\1\\3\end{pmatrix} + t\begin{pmatrix}2\\-1\\-15\end{pmatrix}$ or equivalent | A1✓ | |
| State three equations in $x, y, z, s, t$ | A1 | |
| Eliminate $s$ and $t$ | M1 | |
| Obtain equation $3x - 9y + z = -28$ or equivalent | A1 | [5] |