A2 June 2019 Paper 1 Q12
12 Three planes have equations
\[\begin{aligned} 4x - 5y + z &= 8 \\ 3x + 2y - kz &= 6 \\ (k - 2)x + ky - 8z &= 6 \end{aligned}\]where \(k\) is a real constant.
The planes do not meet at a unique point.
(a) Find the possible values of \(k\). [3 marks]
(b) For each value of \(k\) found in part (a), identify the configuration of the given planes.
Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| Recognises the need to set the determinant = 0 | M1 | 3.1a |
| Obtains and solves a three-term quadratic equation in \(k\) | M1 | 1.1a |
| Obtains the correct values of \(k\) | A1 | 1.1b |
Typical solution
\[9k^2 - 9k - 180 = 0\]\[k = 5 \text{ and } k = -4\]| Scheme | Marks | AO |
|---|---|---|
| Selects an appropriate method and substitutes their first value of \(k\) | M1 | 3.1a |
| For \(k = 5\) (\(k\) must be correct): Deduces that equations are consistent – must have sufficient working to justify comment. | M1 | 2.2a |
| Gives correct geometrical description with full working. | A1 | 3.2a |
| For \(k = -4\) (\(k\) must be correct): Deduces that equations are inconsistent by comparing eqs 2 & 3 – must have comment. | B1 | 2.2a |
| Gives correct geometrical description. | B1 | 3.2a |
| (8 marks) |
Typical solution
For \(k = 5\)
\[\begin{bmatrix} 4 & -5 & 1 & 8 \\ 0 & 23 & -23 & 0 \\ 0 & 35 & -35 & 0 \end{bmatrix}\]Consistent
Line of intersection (sheaf)
For \(k = -4\)
\[\begin{aligned} 3x + 2y + 4z &= 6 \\ -6x - 4y - 8z &= 6 \end{aligned}\]Inconsistent
Two planes parallel and distinct with third plane crossing both