A2 October 2021 Paper 1 Q15
15 The equations of three planes are
\(\begin{aligned} -4x + ky + 7z &= 4, \\ x - 2y + 5z &= l, \\ 2x + 3y + z &= 2. \end{aligned}\)
Given that the planes form a sheaf, determine the values of \(k\) and \(l\). [6]
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{vmatrix} -4 & k & 7 \\ 1 & -2 & 5 \\ 2 & 3 & 1 \end{vmatrix} = -4 \times (-17) - k \times (-9) + 7 \times 7\) | M1 | 3.1a |
| \(= 117 + 9k\) | M1 | 1.1 |
| so det = 0 when \(k = -13\) | A1 | 1.1 |
| \(\begin{aligned} -4x - 13y + 7z &= 4 && (1) \\ x - 2y + 5z &= l && (2) \\ 2x + 3y + z &= 2 && (3) \end{aligned}\) \((1) + 2 \times (3):\ -7y + 9z = 8\) | M1 | 1.1 |
| \((3) - 2 \times (2):\ 7y - 9z = 2 - 2l\) | M1 | 1.1 |
| so \(2l - 2 = 8 \Rightarrow l = 5\) | A1 | 1.1 |
| [6] |
Notes
M1: finding det of matrix of coeffs
M1: setting det = 0
A1: \(k = -13\)
or B2 for use of linear dependency to find \(k\)
M1: finding eqn in 2 variables
M1: finding 2nd eqn in 2 variables
or M2 for \(2 \times (2) - 3 \times (3)\)
A1: \(l = 5\)