AS October 2020 Paper 1 Q9
9 Three planes have equations
\[\begin{aligned} kx + y - 2z &= 0 \\ 2x + 3y - 6z &= -5 \\ 3x - 2y + 5z &= 1 \end{aligned}\]where \(k\) is a constant.
Investigate the arrangement of the planes for each of the following cases. If in either case the planes meet at a unique point, find the coordinates of that point.
(a) \(k = -1\) [3]
(b) \(k = \tfrac{2}{3}\) [4]
| Scheme | Marks | AO |
|---|---|---|
| when \(k = -1\), \(\det\mathbf{M} \ne 0\) [so meet at a point] | M1 | 1.1 |
| \(\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \mathbf{M}^{-1}\begin{pmatrix} 0 \\ -5 \\ 1 \end{pmatrix}\) | M1 | 1.1 |
| by calculator, point of intersection is \((-1, 3, 2)\) | A1 | 1.1 |
| [3] |
Notes
M1: (1st) calculating determinant
or finding \(\mathbf{M}^{-1}\)
M1: (2nd) solving (soi)
or by sim equations
A1: BC, allow \(\begin{pmatrix} -1 \\ 3 \\ 2 \end{pmatrix}\)
If correct ans found by solving simultaneously SCB3
| Scheme | Marks | AO |
|---|---|---|
| when \(k = 2/3\), \(\det\mathbf{M} = 0\), | M1 | 3.1a |
| so no unique point of intersection | A1 | 1.1 |
| [coeffts of 2nd plane are 3 times those of first] | 3.2a | |
| so first two planes are parallel | B1 | |
| and intersected by third plane | B1 | 3.2a |
| [4] |
Notes
M1: or 2 planes are parallel