A2 June 2020 Paper 1 Q7
7 Three planes have equations
\[\begin{alignedat}{3} (4k + 1)x &\;-\; & 3y &\;+\; & (k - 5)z &= 3 \\ (k - 1)x &\;+\; & (3 - k)y &\;+\; & 2z &= 1 \\ 7x &\;-\; & 3y &\;+\; & 4z &= 2 \end{alignedat}\](a) The planes do not meet at a unique point.
Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\). [3 marks]
(b) For each value of \(k\) found in part (a), identify the configuration of the given planes.
In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Shows correctly that \(k = 4.5\) gives a determinant of 0 or shows \(k = 4.5\) from solving \(\det M = 0\) | B1 | 1.1b |
| Correctly expands the determinant of the matrix and equates to 0 Condone misread of \(-3y\) as \(+3y\) | M1 | 1.1a |
| Obtains \(k = -1\) | A1 | 1.1b |
Typical solution
\[\begin{aligned} 0 = (4k + 1)\big[4(3 - k) + 6\big] &+ 3\big[4(k - 1) - 14\big] \\ &+ (k - 5)\big[-3(k - 1) - 7(3 - k)\big] \end{aligned}\]\[0 = -12k^2 + 42k + 54\]\[k = 4.5,\ k = -1\]| Scheme | Marks | AO |
|---|---|---|
| When \(k = 4.5\), clearly shows or explains that the system of equations is consistent, using equations of planes or augmented matrix form. Must state the system is consistent. | B1 | 3.1a |
| States that two planes are the same and intersect the third plane. | B1 | 3.2a |
| When \(k = -1\), completes appropriate working to find the consistency of the system using their \(k\). | M1 | 3.1a |
| States that the system is inconsistent and that the three planes form a prism. CSO | A1 | 3.2a |
| (7 marks) |
Typical solution
When \(k = 4.5\) matrix becomes:
\[\begin{array}{ccc|c} 19 & -3 & -0.5 & 3 \\ 3.5 & -1.5 & 2 & 1 \\ 0 & 0 & 0 & 0 \end{array}\]The system of equations is consistent.
Two planes are the same and intersect the third plane in a line.
When \(k = -1\) matrix becomes:
\[\begin{array}{ccc|c} 0 & 0 & 0 & 66 \\ -2 & 4 & 2 & 1 \\ 11 & -11 & 0 & 0 \end{array}\]The system of equations is inconsistent.
The three planes form a prism.