AS June 2024 Paper 1 Q10
10 The curve \(C\) has equation
\[y = \frac{2x - 10}{3x - 5}\]Figure 1 shows the curve \(C\) with its asymptotes.
(a) Write down the equations of the asymptotes of \(C\) [2 marks]
(b) The line \(L\) has equation\[y = -\frac{2}{5}x + 2\]
(i) Draw the line \(L\) on Figure 1 [2 marks]
(ii) Hence, or otherwise, solve the inequality\[\frac{2x - 10}{3x - 5} \leqslant -\frac{2}{5}x + 2\] [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| States \(x = \dfrac{5}{3}\) or \(y = \dfrac{2}{3}\) | M1 | 2.2a |
| States \(x = \dfrac{5}{3}\) and \(y = \dfrac{2}{3}\) and no incorrect equations seen. | A1 | 2.2a |
| (2) |
Typical solution
\[x = \frac{5}{3}\]\[y = \frac{2}{3}\]| Scheme | Marks | AO |
|---|---|---|
| (i) Draws a straight line with negative gradient passing through \((0, 2)\) Accept freehand if the intention is clear. | M1 | 1.1a |
| Draws a straight line passing through \((0, 2)\) and \((5, 0)\) | A1 | 1.1b |
| (2) | ||
| (ii) Deduces one of the ranges \(x \leqslant 0\) or \(\dfrac{5}{3} \lt x \leqslant 5\) Condone \(\dfrac{5}{3} \leqslant x \leqslant 5\) or \(\dfrac{5}{3} \lt x \lt 5\) for this mark only. Ignore any incorrect ranges. | M1 | 2.2a |
| Deduces the solution \(x \leqslant 0\), \(\dfrac{5}{3} \lt x \leqslant 5\) | A1 | 2.2a |
| (2) | ||
| (6 marks) |
Typical solution
(i)
(ii)
\[x \leqslant 0\]\[\frac{5}{3} \lt x \leqslant 5\]