AS June 2025 Q4
4.
In this question you must show all stages of your working.
Solutions based on calculator technology are not acceptable.
Determine the values of \(x\) for which
\[\frac{x - 8}{x} \leqslant \frac{7}{x(x - 2)}\]giving your answer in set notation.
(6)
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{x - 8}{x} \leqslant \dfrac{7}{x(x - 2)}\) | ||
| \(\dfrac{(x - 2)(x - 8) - 7}{x(x - 2)} \leqslant 0\) or \(x(x - 8)(x - 2)^2 - 7x(x - 2) \leqslant 0\) Or \(x^3(x - 8)(x - 2)^2 - 7x^3(x - 2) \leqslant 0\) | M1 | 2.1 |
| \(\dfrac{(x - 9)(x - 1)}{x(x - 2)} \leqslant 0\) or \(x(x - 1)(x - 2)(x - 9) \leqslant 0\) Or \(x^3(x - 1)(x - 2)(x - 9) \leqslant 0\) | dM1 | 1.1b |
| Critical values 0 and 2 | B1 | 1.1b |
| All 4 critical values 0, 1, 2, 9 | A1 | 1.1b |
| \(\{x \in \mathbb{R} : 0 \lt x \leqslant 1\} \cup \{x \in \mathbb{R} : 2 \lt x \leqslant 9\}\) | ddM1 A1 | 2.2a 2.5 |
| (6) | ||
| (6 marks) |
Notes
M1: Gathers terms on one side and puts over a common denominator, or multiplies by \(x^2(x - 2)^2\) or \(x^4(x - 2)^2\) and gathers terms on one side
dM1: Factorises numerator or factorises into 4 factors
B1: Identifies the critical values 0 and 2, may be seen in an inequality
A1: All 4 correct critical values, must have s cored dM1
ddM1: Deduces 2 “inside” inequalities are required with critical values in ascending order as shown
A1: Exactly 2 correct intervals using correct notation minimum \(0 \lt x \leqslant 1 \cup 2 \lt x \leqslant 9\)
(Corrected from the printed mark scheme: in the final answer and in the last note, the upper end of the first interval is printed with a 9 and a 1 overprinted on each other; the correct value is 1.)