AS June 2018 Paper 1 Q12
12
(a) Show that the matrix \(\begin{bmatrix} 5 - k & 2 \\ k^3 + 1 & k \end{bmatrix}\) is singular when \(k = 1\). [1 mark]
(b) Find the values of \(k\) for which the matrix \(\begin{bmatrix} 5 - k & 2 \\ k^3 + 1 & k \end{bmatrix}\) has a negative determinant.
Fully justify your answer. [5 marks]
Fully justify your answer. [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(k = 1\) and correctly calculates the determinant, and concludes that the matrix is singular. | B1 | 2.2a |
Typical solution
\[\text{determinant} = 4 \times 1 - 2 \times 2 = 0\]\(\therefore\) the matrix is singular
AG
| Scheme | Marks | AO |
|---|---|---|
| Finds determinant in terms of \(k\). Allow one error. | M1 | 1.1a |
| Obtains a correct inequality in \(k\). | A1 | 1.1b |
| Obtains three correct critical values. | A1 | 1.1b |
| Deduces one correct region. FT their three real distinct critical values if given as \(a \lt k \lt b\), \(k \gt c\) (o.e.) where \(a \lt b \lt c\) | A1F | 2.2a |
| Deduces the other correct region. FT their three real distinct critical values if given as \(a \lt k \lt b\), \(k \gt c\) (o.e.) where \(a \lt b \lt c\) Condone the use of ‘and’. | A1F | 2.2a |
| (6 marks) |
Typical solution
\[\text{determinant} = k(5 - k) - 2(k^3 + 1)\]\[5k - k^2 - 2k^3 - 2 \lt 0\]\[2k^3 + k^2 - 5k + 2 \gt 0\]\[(k - 1)(2k^2 + 3k - 2) \gt 0\]\[(k - 1)(2k - 1)(k + 2) \gt 0\]