## Question 7(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Express general point of $l$ or $m$ in component form e.g. $(a+\lambda, 2-2\lambda, 3+3\lambda)$ or $(2+2\mu, 1-\mu, 2+\mu)$ | B1 | |
| Equate at least two pairs of corresponding components and solve for $\lambda$ or for $\mu$ | M1 | |
| Obtain either $\lambda=-2$ or $\mu=-5$, **or** $\lambda=\frac{1}{3}a$ or $\mu=\frac{2}{3}a-1$, **or** $\lambda=\frac{1}{5}(a-4)$ or $\mu=\frac{1}{5}(3a-7)$ | A1 | |
| Obtain $a=-6$ | A1 | |

## Question 7(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Use scalar product to obtain a relevant equation in $a$, $b$ and $c$, e.g. $a-2b+3c=0$ | B1 | |
| Obtain a second equation, e.g. $2a-b+c=0$ and solve for one ratio | M1 | |
| Obtain $a:b:c=1:5:3$ | A1 | OE |
| Substitute a relevant point and values of $a$, $b$, $c$ in general equation and find $d$ | M1 | |
| Obtain correct answer $x+5y+3z=13$ | A1FT | OE. The FT is on $a$ from part (i), if used |

**Alternative method for question 7(ii):**

| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to calculate vector product of relevant vectors | M1 | e.g. $(\mathbf{i}-2\mathbf{j}+3\mathbf{k})\cdot(2\mathbf{i}-\mathbf{j}+\mathbf{k})$ |
| Obtain two correct components | A1 | |
| Obtain correct answer, e.g. $\mathbf{i}+5\mathbf{j}+3\mathbf{k}$ | A1 | |
| Substitute a relevant point and find $d$ | M1 | |
| Obtain correct answer $x+5y+3z=13$ | A1FT | OE. The FT is on $a$ from part (i), if used |

**Second alternative method for question 7(ii):**

| Answer | Marks | Guidance |
|--------|-------|----------|
| Using a relevant point and relevant vectors, form a 2-parameter equation for the plane | M1 | |
| State a correct equation, e.g. $\mathbf{r}=2\mathbf{i}+\mathbf{j}+2\mathbf{k}+\lambda(\mathbf{i}-2\mathbf{j}+3\mathbf{k})+\mu(2\mathbf{i}-\mathbf{j}+\mathbf{k})$ | A1FT | |
| State three correct equations in $x$, $y$, $z$, $\lambda$ and $\mu$ | A1FT | |
| Eliminate $\lambda$ and $\mu$ | M1 | |
| Obtain correct answer $x+5y+3z=13$ | A1FT | OE. The FT is on $a$ from part (i), if used |