FP1 June 2013 Q1
1. \[\mathbf{M} = \begin{pmatrix} x & x - 2 \\ 3x - 6 & 4x - 11 \end{pmatrix}\]
Given that the matrix \(\mathbf{M}\) is singular, find the possible values of \(x\). (4)
| Scheme | Marks |
|---|---|
| \(\mathbf{M} = \begin{pmatrix} x & x - 2 \\ 3x - 6 & 4x - 11 \end{pmatrix}\) | |
| \(\det\mathbf{M} = x(4x - 11) - (3x - 6)(x - 2)\) Correct attempt at determinant | M1 |
| \(x^2 + x - 12\ (= 0)\) Correct 3 term quadratic | A1 |
| \((x + 4)(x - 3)\ (= 0) \to x = \ldots\) Their 3TQ = 0 and attempts to solve relevant quadratic using factorisation or completing the square or correct quadratic formula leading to \(x = \) | M1 |
| \(x = -4,\ x = 3\) Both values correct | A1 |
| (4) | |
| Total 4 |
Notes
\(x(4x - 11) = (3x - 6)(x - 2)\) award first M1
\(\pm(x^2 + x - 12)\) seen award first M1A1
Method mark for solving 3 term quadratic:
1. Factorisation
\((x^2 + bx + c) = (x + p)(x + q)\), where \(|pq| = |c|\), leading to x =
\((ax^2 + bx + c) = (mx + p)(nx + q)\), where \(|pq| = |c|\) and \(|mn| = |a|\), leading to x =
2. Formula
Attempt to use correct formula (with values for \(a\), \(b\) and \(c\)).
3. Completing the square
Solving \(x^2 + bx + c = 0\): \(\left(x \pm \dfrac{b}{2}\right)^2 \pm q \pm c,\quad q \neq 0,\quad\) leading to x =...
Both correct with no working 4/4, only one correct 0/4