A2 June 2024 Q2
2. Use algebra to determine the values of \(x\) for which
\[\left|x^2 - 2x\right| \leqslant x\](4)
| Scheme | Marks | AO |
|---|---|---|
| A complete method to form equations in an attempt to find the critical values Method 1 Uses \(x^2 - 2x = x \Rightarrow x^2 - 3x = 0\) and \(x^2 - 2x = -x \Rightarrow x^2 - x = 0\) Method 2 Squares both sides \(\left(x^2 - 2x\right)^2 = x^2 \Rightarrow x^4 - 4x^3 + 3x^2 = 0\) | M1 | 1.1b |
| Solves their equations to find at least all the non-zero the critical values | dM1 | 3.1a |
| \(x = 0, 1, 3\) | A1 | 1.1b |
| \(x = 0,\ 1 \leqslant x \leqslant 3\) | A1 | 2.2a |
| (4) | ||
| (4 marks) |
Notes
M1: A complete method to form equations/an equation that will give all the critical values. For method 1, both equations must be attempted. Allow with inequalities in place of equals throughout and do not be concerned if the inequalities are correct or not. This is for the method to find an equation or equations for the CVs.
dM1: Dependent on the previous method mark. Solve their equation(s) to find at least all the non-zero critical values. If using method 1 both equations must be solved, in method 2 the quadratic after factoring out or dividing by \(x^2\) must be solved. Allow for attempts by dividing through by \(x\) or \(x^2\).
A1: Correct critical values, including 0 (although this may be later rejected), seen stated or used in the solution. Must have come from correct work.
A1: Deduces the correct range following correct work. Both parts must clearly be identified for the answer (e.g. underlined during working if not on the final line).
Accept alternative notation, e.g. set or interval notation. In words “and” or “or” may be used, but if using mathematical notation, must use union and not intersection.
E.g. \(\{0\} \cup \{x : 1 \leqslant x \leqslant 3\}\) or \([0] \cup [1, 3]\) (if intervals given, must be square brackets not round) are acceptable answers.