AS June 2023 Paper 1 Q7
7 In this question you must show detailed reasoning.
Matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & -6 & a - 3 \\ a + 9 & a & 4 \\ 0 & -13 & a - 1 \end{pmatrix}\) where \(a\) is a constant.
Find all possible values of \(a\) for which \(\det\mathbf{A}\) has the same value as it has when \(a = 2\). [6]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\det\mathbf{A} = a(a(a - 1) - 4 \times (-13)) - (-6)((a + 9)(a - 1) - 0) + (a - 3)((a + 9)(-13) - 0)\) | M1 | 3.1a |
| \(= a^3 - 8a^2 + 22a + 297\) | A1 | 1.1 |
| \(a^3 - 8a^2 + 22a + 297 = 2^3 - 8 \times 2^2 + 22 \times 2 + 297\) (or \(a^3 - 8a^2 + 22a + 297 = 317\)) | M1 | 2.2a |
| \(a^3 - 8a^2 + 22a - 20 = 0\) \(a^2(a - 2) - 6a(a - 2) + 10(a - 2) = 0\) \(a^2 - 6a + 10 = 0\) | M1 | 1.1 |
| \((a - 3)^2 - 9 + 10 = 0\) \((a - 3)^2 = -1\) \(a - 3 = \pm\mathrm{i}\) | M1 | 1.1 |
| \((a =)\ 3 \pm \mathrm{i}\) (and \(a = 2\)) | A1 | 1.1 |
| [6] |
Notes
M1: Attempt to expand the determinant. If using standard method must see at least two terms, at least one of which comprises \(\pm\) a number multiplied by the residual determinant. eg
\(a(a(a - 1) - 4 \times (-13)) - (a + 9)((-6)(a - 1) - (a - 3)(-13))\)
This mark can be awarded with \(a = 2\) substituted (ie attempt to find \(\begin{vmatrix} 2 & -6 & -1 \\ 11 & 2 & 4 \\ 0 & -13 & 1 \end{vmatrix}\)) but working must be shown. eg \(2(2 + 52) - 11(-6 - 13)\) \((= 317)\)
Allow other correct methods.
M1: Setting up equation equating their determinant to their specific determinant value for \(a = 2\)
M1: Rearranging to = 0 and use of factor theorem to derive a quadratic equation in \(a\) by dividing by \((a - 2)\)
Need some evidence of how to solve cubic (Not just cubic written and then roots BC). Need to see =0 on one side of cubic (but it might disappear after this, e.g. when dividing by \((a - 2)\))
M1: Attempt to solve quadratic involving \(\sqrt{-1} = \mathrm{i}\) oe
\(\dfrac{6 \pm \sqrt{6^2 - 4 \times 1 \times 10}}{2 \times 1} = \dfrac{6 \pm \sqrt{-4}}{2} = \dfrac{6 \pm 2\mathrm{i}}{2}\)
A1: Both. No need to mention \(a = 2\)
Don’t need to see “\(a =\)” explicitly
SC – if third and/or fourth method mark not awarded then allow SC B1 for sight of \(3 \pm \mathrm{i}\).