June 2024 Paper 3 Q7
7 The graphs with equations
\[y = 2 + 3x - 2x^2 \quad \text{and} \quad x + y = 1\]are shown in the diagram below.
The graphs intersect at the points \(A\) and \(B\)
(a) On the diagram above, shade and label the region, \(R\), that is satisfied by the inequalities\[0 \leqslant y \leqslant 2 + 3x - 2x^2\]
and
\[x + y \geqslant 1\] [2 marks](b) Find the exact coordinates of \(A\) [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Shades two of regions 1, 2 or 3 only or shades one of region 1 or 2 or 3 only or shades regions 1, 2 and 3 only  | M1 | 1.1a |
| Shades the correct regions 1 and 2 only Condone missing label \(R\) | R1 | 2.2a |
| (2) |
Typical solution
| Scheme | Marks | AO |
|---|---|---|
| Eliminates \(y\) or \(x\) correctly to obtain a quadratic in \(x\) or \(y\) | M1 | 1.1a |
| Obtains \(x = \dfrac{2 - \sqrt{6}}{2}\) or \(\dfrac{2 \pm \sqrt{6}}{2}\) Accept AWFW [\(-0.225\), \(-0.22\)] or obtains \(y = \dfrac{\sqrt{6}}{2}\) or \(\pm\dfrac{\sqrt{6}}{2}\) Accept AWFW [1.22, 1.225] May be unsimplified | A1 | 1.1b |
| Obtains \(\left(\dfrac{2 - \sqrt{6}}{2}, \dfrac{\sqrt{6}}{2}\right)\) Accept \(x = \dfrac{2 - \sqrt{6}}{2}\) and \(y = \dfrac{\sqrt{6}}{2}\) ISW Must be simplified | A1 | 1.1b |
| (3) | ||
| (5 marks) |
Typical solution
\[1 - x = 2 + 3x - 2x^2\]\[2x^2 - 4x - 1 = 0\]\[x = \frac{2 - \sqrt{6}}{2}\]\[y = \frac{\sqrt{6}}{2}\]So \(A\left(\dfrac{2 - \sqrt{6}}{2}, \dfrac{\sqrt{6}}{2}\right)\)