## Question 9:

**Given:** $A(-1,5,1),\; B(1,0,3),\; C(2,-1,2),\; D(3,6,-1)$

### Part (a):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{AB}=\begin{pmatrix}2\\-5\\2\end{pmatrix},\; \mathbf{AC}=\begin{pmatrix}3\\-6\\1\end{pmatrix},\; \mathbf{AD}=\begin{pmatrix}4\\1\\-2\end{pmatrix}$; $\mathbf{DB}=\begin{pmatrix}2\\6\\-4\end{pmatrix},\; \mathbf{DC}=\begin{pmatrix}1\\7\\-3\end{pmatrix},\; \mathbf{BC}=\begin{pmatrix}1\\-1\\-1\end{pmatrix}$ | M1 | Attempts 3 edges with a common vertex; method shown or at least 1 correct |
| $\begin{vmatrix}2&-5&2\\3&-6&1\\4&1&-2\end{vmatrix}$ or $\begin{pmatrix}2\\-5\\2\end{pmatrix}\cdot\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\-5&3&-6\\2&4&1\end{vmatrix}$ | dM1 | Attempt appropriate triple product with their edges (M0 if a vector is obtained) |
| $=\frac{1}{6}(22-50+54)=\frac{13}{3}\left(4\frac{1}{3}\text{ or }4.3\text{ rec}\right)$ | ddM1A1 | dM1: Completes including $\frac{1}{6}$, depends on both M marks; A1: Correct volume (allow equivalents); Total: (4) |

**Cartesian method:**

| Answer/Working | Marks | Guidance |
|---|---|---|
| Find equation of plane containing a face of the tetrahedron | M1 | |
| Then find area of triangle and perpendicular height | dM1 | |
| Complete by using $\text{Vol}=\frac{1}{3}\times\text{base area}\times\text{height}=\frac{13}{3}$ | ddM1A1 | Total: (4) |

### Part (b):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{AB}\times\mathbf{AC}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\2&-5&2\\3&-6&1\end{vmatrix}=\begin{pmatrix}7\\4\\3\end{pmatrix}$ | M1A1 | M1: Attempt cross product between two sides of $ABC$, min one element correct; A1: Correct normal vector (any multiple) |
| $\begin{pmatrix}7\\4\\3\end{pmatrix}\cdot\begin{pmatrix}-1\\5\\1\end{pmatrix}(=16)$ | dM1 | Attempt scalar product using their normal vector; answer correct for their vectors or method shown |
| $7x+4y+3z=16$ | A1 | Correct equation (any multiple); Total: (4) |

**Alternative for (b):**

| Answer/Working | Marks | Guidance |
|---|---|---|
| $-a+5b+c=d,\; a+3c=d,\; 2a-b+2c=d$ | M1 | Uses $A$, $B$ and $C$ to form 3 equations |
| $a=7,\; b=4,\; c=3$ | A1 | Correct values |
| $\begin{pmatrix}7\\4\\3\end{pmatrix}\cdot\begin{pmatrix}-1\\5\\1\end{pmatrix}(=16)$ | dM1 | |
| $7x+4y+3z=16$ | A1 | Correct equation (any multiple) |

### Part (c):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{DT}=\begin{pmatrix}3\\6\\-1\end{pmatrix}+\lambda\begin{pmatrix}7\\4\\3\end{pmatrix}$ | M1 | Attempt parametric form of $\mathbf{DT}$ using their normal vector |
| $7(3+7\lambda)+4(6+4\lambda)+3(-1+3\lambda)=16 \Rightarrow \lambda=\ldots$ | dM1 | Substitutes parametric form of $\mathbf{DT}$ into their plane equation and solves for $\lambda$ |
| $\lambda=-\frac{13}{37}\Rightarrow T$ is $\left(\frac{20}{37},\frac{170}{37},-\frac{76}{37}\right)$ | ddM1A1 | M1: Uses their $\lambda$ in their $\mathbf{DT}$ equation (indicated by any coordinate correct for their $\mathbf{DT}$ and $\lambda$); A1: Correct exact coordinates or correct vector $\overrightarrow{OT}$; Total: (4) |