## Question 6(i):

| Answer | Mark | Guidance |
|--------|------|----------|
| Verify that the point with position vector $\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$ lies in the plane | B1 | |
| **EITHER:** Find a second point on $l$ and substitute its coordinates in the equation of $p$ | (M1 | |
| Verify that the second point, e.g. $(3, 1, -2)$, lies in the plane | A1) | |
| **OR:** Expand scalar product of a normal to $p$ and the direction vector of $l$ | (M1 | |
| Verify scalar product is zero | A1) | |

**Total: 3**

## Question 6(ii):

**EITHER method:**

| Answer | Mark | Guidance |
|--------|------|----------|
| Use scalar product to obtain relevant equation in $a$, $b$ and $c$, e.g. $2a - b + c = 0$ | B1 | |
| Obtain second equation, e.g. $3a + b - 5c = 0$, and solve for one ratio e.g. $a:b$ | M1 | |
| Obtain $a:b:c = 4:13:5$, or equivalent | A1 | |
| Substitute $(3,-1,2)$ and values of $a$, $b$, $c$ in general equation to find $d$ | M1 | |
| Obtain $4x + 13y + 5z = 9$, or equivalent | A1 | |

**OR1:**

| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to calculate vector product of relevant vectors, e.g. $(2\mathbf{i}-\mathbf{j}+\mathbf{k})\times(3\mathbf{i}+\mathbf{j}-5\mathbf{k})$ | M1 | |
| Obtain two correct components | A1 | |
| Obtain correct answer, e.g. $4\mathbf{i}+13\mathbf{j}+5\mathbf{k}$ | A1 | |
| Substitute $(3,-1,2)$ in $4x+13y+5z=d$, and find $d$ | M1 | |
| Obtain $4x+13y+5z=9$, or equivalent | A1 | |

**OR2:**

| Answer | Mark | Guidance |
|--------|------|----------|
| Using relevant point and vectors form a 2-parameter equation for the plane | M1 | |
| State correct equation, e.g. $\mathbf{r} = 3\mathbf{i}-\mathbf{j}+2\mathbf{k}+\lambda(2\mathbf{i}-\mathbf{j}+\mathbf{k})+\mu(3\mathbf{i}+\mathbf{j}-5\mathbf{k})$ | A1 | |
| State three correct equations in $x, y, z, \lambda$ and $\mu$ | A1 | |
| Eliminate $\lambda$ and $\mu$ | M1 | |
| Obtain $4x+13y+5z=9$, or equivalent | A1 | |

**OR3:**

| Answer | Mark | Guidance |
|--------|------|----------|
| Using relevant point and vectors form a determinant equation for the plane | M1 | |
| State correct equation, e.g. $\begin{vmatrix} x-3 & y+1 & z-2 \\ 2 & -1 & 1 \\ 3 & 1 & -5 \end{vmatrix} = 0$ | A1 | |
| Attempt to expand the determinant | M1 | |
| Obtain or imply two correct cofactors | A1 | |
| Obtain $4x+13y+5z=9$, or equivalent | A1 | |

**Total: 5**

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