## Question 7:

### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| EITHER: AG is $\mathbf{r}=[6,4,8]+tk[1,0,1]$ or $[3,4,5]+tk[1,0,1]$ | B1 | For a correct equation |
| Normal to $BCD$ is $\mathbf{n}=k[1,1,-3]$ | M1, A1 | For finding vector product of any two of $\pm[1,-4,-1]$, $\pm[2,1,1]$, $\pm[1,5,2]$; for correct **n** |
| Equation of $BCD$ is $\mathbf{r}.[1,1,-3]=-6$ | A1 | For correct equation (or in cartesian form) |
| Intersect at $(6+t)+4+(-3)(8+t)=-6$ | M1 | For substituting point on $AG$ into plane |
| $t=-4\ (t=-1$ using $[3,4,5])\Rightarrow\mathbf{OM}=[2,4,4]$ | A1 | For correct position vector of $M$ AG |
| OR: $\mathbf{r}=\mathbf{u}+\lambda\mathbf{v}+\mu\mathbf{w}$ approach | B1 | For a correct equation |
| $\mathbf{u}=[2,1,3]$ or $[1,5,4]$ or $[3,6,5]$; **v**,**w** two of $[1,-4,-1],[1,5,2],[2,1,1]$ | M1, A1 | For a correct parametric equation of $BCD$ |
| Forming 3 equations in $t,\lambda,\mu$ | M1 | For forming 3 equations in $t,\lambda,\mu$ from line and plane, and attempting to solve them |
| $t=-4$ or $\lambda=-\frac{1}{3}$, $\mu=\frac{1}{3}$ | A1 | For correct value of $t$ or $\lambda$, $\mu$ |
| $\Rightarrow\mathbf{OM}=[2,4,4]$ | A1 **6** | For correct position vector of $M$ AG |

### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $A,G,M$ have $t=0,-3,-4$ OR $AG=3\sqrt{2}$, $AM=4\sqrt{2}$ OR $\mathbf{AG}=[-3,0,-3]$, $\mathbf{AM}=[-4,0,-4]$ $\Rightarrow AG:AM=3:4$ | B1 **1** | For correct ratio AEF |

### Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{OP}=\mathbf{OC}+\frac{4}{3}\mathbf{CG}$ | M1 | For using given ratio to find position vector of $P$ |
| $=\left[\frac{11}{3},\frac{11}{3},\frac{16}{3}\right]$ | A1 **2** | For correct vector |

### Part (iv)
| Answer/Working | Mark | Guidance |
|---|---|---|
| EITHER: Normal to $ABD$ is $\mathbf{n}=k[19,3,-17]$ | M1, A1 | For finding vector product of any two of $\pm[4,3,5]$, $\pm[1,5,2]$, $\pm[3,-2,3]$; for correct **n** |
| Equation of $ABD$ is $\mathbf{r}.[19,3,-17]=-10$ | M1 | For finding equation (or in cartesian form) |
| $19\cdot\frac{11}{3}+3\cdot\frac{11}{3}-17\cdot\frac{16}{3}=-10$ | A1 | For verifying that $P$ satisfies equation |
| OR: Equation of $ABD$ is $\mathbf{r}=[6,4,8]+\lambda[4,3,5]+\mu[1,5,2]$ (etc.) | M1 | For finding equation in parametric form |
| $\left[\frac{11}{3},\frac{11}{3},\frac{16}{3}\right]=[6,4,8]+\lambda[4,3,5]+\mu[1,5,2]$ | M1 | For substituting $P$ and solving 2 equations for $\lambda$, $\mu$ |
| $\lambda=-\frac{2}{3}$, $\mu=\frac{1}{3}$ | A1 | For correct $\lambda$, $\mu$ |
| Verifying 3rd equation is satisfied | A1 | |
| OR: $\mathbf{AP}=\left[-\frac{7}{3},-\frac{1}{3},-\frac{8}{3}\right]$ | M1 | For finding 3 relevant vectors in plane $ABDP$ |
| $\mathbf{AB}=[-4,-3,-5]$, $\mathbf{AD}=[-3,2,-3]$ | A1 | For correct **AP** or **BP** or **DP** |
| $\Rightarrow\mathbf{AB}+\mathbf{AD}=[-7,-1,-8]$ | M1 | For finding **AB**, **AD** or **BA**, **BD** or **DB**, **DA** |
| $\Rightarrow\mathbf{AP}=\frac{1}{3}(\mathbf{AB}+\mathbf{AD})$ | A1 **4** | For verifying linear relationship |

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