(i) EITHER: Obtain a vector parallel to the plane, e.g. $\vec{AB} = -\mathbf{i} + 2\mathbf{j}$ | B1 |
Use scalar product of perpendicular vectors to obtain an equation in $a, b, c$, e.g. $-a + 2b = 0$, or $-a + b + 2c = 0$, or $-b + 2c = 0$ | M1 |
Obtain two correct equations in $a, b, c$ | A1 |
Solve to obtain ratio $a : b : c$, or equivalent | M1 |
Obtain $a : b : c = 4 : 2 : 1$, or equivalent | A1 |
Obtain equation $4x + 2y + z = 8$, or equivalent | A1 |

OR1:

Substitute for $A$ and $B$ and obtain $2a = d$ and $a + 2b = d$ | B1 |
Substitute for $C$ to obtain a third equation and eliminate one unknown ($a$, $b$, or $d$) entirely | M1 |
Obtain two correct equations in three unknowns, e.g. $a, b, c$ | A1 |
Solve to obtain their ratio, e.g. $a : b : c$, or equivalent | M1 |
Obtain $a : b : c = 4 : 2 : 1$, or $a : c : d = 4 : 1 : 8$, or $b : c : d = 2 : 1 : 8$, or equivalent | A1 |
Obtain equation $4x + 2y + z = 8$, or equivalent | A1 |

OR2:

Substitute for $A$ and $B$ and obtain $2a = d$ and $a + 2b = d$ | B1 |
Solve to obtain ratio $a : b : d$, or equivalent | M2 |
Obtain $a : b : d = 2 : 1 : 4$, or equivalent | A1 |
Substitute for $C$ to find $c$ | M1 |
Obtain equation $4x + 2y + z = 8$, or equivalent | A1 |

OR3:

Obtain a vector parallel to the plane, e.g. $\vec{BC} = -\mathbf{j} + 2\mathbf{k}$ | B1 |
Obtain a second such vector and calculate their vector product, e.g. $(-\mathbf{i}+2\mathbf{j}) \times (-\mathbf{j}+2\mathbf{k})$ | M1 |
Obtain two correct components of the product | A1 |
Obtain correct answer, e.g. $4\mathbf{i} + 2\mathbf{j} + \mathbf{k}$ | A1 |
Substitute in $4x + 2y + z = d$ to find $d$ | M1 |
Obtain equation $4x + 2y + z = 8$, or equivalent | A1 |

OR4:

Obtain a vector parallel to the plane, e.g. $\vec{AC} = -\mathbf{i} + \mathbf{j} + 2\mathbf{k}$ | B1 |
Obtain a second such vector and form correctly a 2-parameter equation for the plane | M1 |
Obtain a correct equation, e.g. $\mathbf{r} = 2\mathbf{i} + \lambda(-\mathbf{i}+2\mathbf{j}) + \mu(-\mathbf{i} + \mathbf{j} + 2\mathbf{k})$ | A1 |
State three equations in $x, y, z, \lambda, \mu$ | A1 |
Eliminate $\lambda$ and $\mu$ | M1 |
Obtain equation $4x + 2y + z = 8$, or equivalent | A1 | **Total: 6 marks**

(ii) State or imply a normal vector for plane $OAB$ is $\mathbf{k}$, or equivalent | B1 |
Carry out correct process for evaluating a scalar product of two relevant vectors, e.g. $(4\mathbf{i} + 2\mathbf{j} + \mathbf{k})(\mathbf{k})$ | M1 |
Using the correct process for calculating the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result | M1 |
Obtain answer $77.4°$ or 1.35 radians | A1 | **Total: 4 marks**

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