## Question 9(i):

| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain a vector parallel to the plane, e.g. $\overrightarrow{CB} = 2\mathbf{i}+\mathbf{j}$ | B1 | |
| Use scalar product to obtain an equation in $a$, $b$, $c$ | M1 | e.g. $2a+b=0$, $a+5c=0$, $a+b-5c=0$ |
| Obtain two correct equations in $a$, $b$, $c$ | A1 | |
| Solve to obtain $a:b:c$ | M1 | or equivalent |
| Obtain $a:b:c = 5:-10:-1$ | A1 | or equivalent |
| Obtain equation $5x-10y-z=-25$ | A1 | or equivalent |
| **Alternative method 1:** | | |
| Obtain a vector parallel to the plane, e.g. $\overrightarrow{CD}=\mathbf{i}+5\mathbf{k}$ | B1 | $\overrightarrow{BD}=-\mathbf{i}-\mathbf{j}+5\mathbf{k}$ |
| Obtain a second such vector and calculate their vector product, e.g. $(2\mathbf{i}+\mathbf{j})\times(\mathbf{i}+5\mathbf{k})$ | M1 | |
| Obtain two correct components | A1 | |
| Obtain correct answer, e.g. $5\mathbf{i}-10\mathbf{j}-\mathbf{k}$ | A1 | |
| Substitute to find $d$ | M1 | |
| Obtain equation $5x-10y-z=-25$ | A1 | or equivalent |
| **Alternative method 2:** | | |
| Obtain a vector parallel to the plane, e.g. $\overrightarrow{DB}=\mathbf{i}+\mathbf{j}-5\mathbf{k}$ | B1 | |
| Obtain a second such vector and form correctly a 2-parameter equation for the plane | M1 | |
| State a correct equation, e.g. $\mathbf{r}=3\mathbf{i}+4\mathbf{j}+\lambda(\mathbf{i}+5\mathbf{k})+\mu(\mathbf{i}+\mathbf{j}-5\mathbf{k})$ | A1 | |
| State three equations in $x$, $y$, $z$, $\lambda$ and $\mu$ | A1 | |
| Eliminate $\lambda$ and $\mu$ | M1 | |
| Obtain equation $5x-10y-z=-25$ | A1 | or equivalent |
| **Alternative method 3:** | | |
| Substitute for $B$ and $C$ and obtain $3a+4b=d$ and $a+3b=d$ | B1 | |
| Substitute for $D$ to obtain a third equation and eliminate one unknown ($a$, $b$, or $d$) entirely | M1 | |
| Obtain two correct equations in two unknowns, e.g. $a$, $b$, $c$ | A1 | |
| Solve to obtain their ratio, e.g. $a:b:c$ | M1 | |
| Obtain $a:b:c=5:-10:-1$, $a:c:d=5:-1:-25$, or $b:c:d=10:1:25$ | A1 | or equivalent |
| Obtain equation $5x-10y-z=-25$ | A1 | or equivalent |
| **Alternative method 4:** | | |
| Substitute for $B$ and $C$ and obtain $3a+4b=d$ and $a+3b=d$ | B1 | |
| Solve to obtain $a:b:d$ | M2 | or equivalent |
| Obtain $a:b:d=1:-2:-5$ | A1 | or equivalent |
| Substitute for $C$ to obtain $c$ | M1 | |
| Obtain equation $5x-10y-z=-25$ | A1 | or equivalent |
| **Total** | **6** | |

## Question 9(ii):

| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply a normal vector for the plane $OABC$ is $\mathbf{k}$ | B1 | |
| Carry out correct process for evaluating a scalar product of two relevant vectors, e.g. $(5\mathbf{i}-10\mathbf{j}-\mathbf{k})\cdot(\mathbf{k})$ | M1 | i.e. correct process using $\mathbf{k}$ and their normal |
| Using correct process for calculating the moduli, divide scalar product by product of moduli and evaluate the inverse cosine | M1 | Allow M1M1 for clear use of an incorrect vector stated to be the normal to $OABC$ |
| Obtain answer $84.9°$ or $1.48$ radians | A1 | |
| **Total** | **4** | |