METHOD 1 attempt to eliminate a variable M1 obtain a pair of equations in two variables

EITHER $-3x+z=-3$ A1 $-3x+z=44$A1

OR $-5x+y=-7$A1 $-5x+y=40$A1

OR $3x-z=3$ A1 $3x-z=-\frac{79}{5}$A1

THEN the two lines are parallel ($-3 \neq 44$ or $-7 \neq 40$ or $3 \neq -\frac{79}{5}$)R1

Note: There are other possible pairs of equations in two variables. To obtain the final R1, at least the initial *M1 *must have been awarded.

hence the three planes do not intersectAG

METHOD 2 vector product of the two normals $=\begin{pmatrix}-1 \\ -5 \\ -3\end{pmatrix}$ (or equivalent)A1 $\mathbf{r}=\begin{pmatrix}1 \\ -2 \\ 0\end{pmatrix}+\lambda\begin{pmatrix}1 \\ 5 \\ 3\end{pmatrix}$ (or equivalent)A1

Note: Award A0 if "$\mathbf{r}=$" is missing. Subsequent marks may still be awarded.

Attempt to substitute $1+\lambda,-2+5\lambda,3\lambda$ in $\prod_3$M1 $-9(1+\lambda)+3(-2+5\lambda)-2(3\lambda)=32$ $-15=32$,a contradictionR1 hence the three planes do not intersectAG

METHOD 3 attempt to eliminate a variable M1 $-3y+5z=6$A1 $-3y+5z=100$A1 $0=94$,a contradictionR1

Note: Accept other equivalent alternatives. Accept other valid methods. To obtain the final R1, at least the initial *M1 *must have been awarded.

hence the three planes do not intersectAG

[4 marks]

${\prod_{1}:2+2+0=4}$ and ${\prod_{2}:1+4+0=5}$A1

[1 mark]

METHOD 1 attempt to find the vector product of the two normals M1 $2 - 1 + 1 \times ( - 2 - 3 )$ $= - 1 - 5 - 3$ A1 $\mathbf{r} = \begin{pmatrix} 1 \\ -2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix}$ A1A1

Note: Award A1A0 if "$\mathbf{r} =$" is missing. Accept any multiple of the direction vector. Working for (b)(ii) may be seen in part (a) Method 2. In this case penalize lack of "$\mathbf{r} =$" only once.

METHOD 2 attempt to eliminate a variable from$\Pi_1$ and$\Pi_2$ M1 $3x - z = 3$ OR$3y - 5z = -6$ OR$5x - y = 7$ Let$x = t$ substituting$x = t$in$3x - z = 3$to obtain $z = -3 + 3t$ and$y = 5t - 7$(for all three variables in parametric form) A1 $\mathbf{r} = \begin{pmatrix} 0 \\ -7 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix}$ A1A1

**Note:**Award A1A0 if "$\mathbf{r} =$" is missing. Accept any multiple of the direction vector.Accept other position vectors which satisfy both the planes$\Pi_1$ and$\Pi_2$.

[4 marks]