October 2021 Paper 1 Q3
3
(a) The diagram shows the line \(y = x + 5\) and the curve \(y = 8 - 2x - x^2\). The shaded region is the finite region between the line and the curve. The curved part of the boundary is included in the region but the straight part is not included.
Write down the inequalities that define the shaded region. [2]
Write down the inequalities that define the shaded region. [2]
(b) In this question you must show detailed reasoning.
Solve the inequality \(8 - 2x - x^2 > x + 5\) giving your answer in exact form. [3]
Solve the inequality \(8 - 2x - x^2 > x + 5\) giving your answer in exact form. [3]
| Scheme | Marks | AO |
|---|---|---|
| \(y > x + 5\) | B1 | 2.5 |
| \(y \leqslant 8 - 2x - x^2\) | B1 | 2.5 |
| [2] |
Notes
B1: Allow interchange of \(>\) and \(\geqslant\) or \(<\) and \(\leqslant\) for one inequality as long as the direction is correct
B1: Both inequalities fully correct. Allow \(x + 5 < y \leqslant 8 - 2x - x^2\) oe
| Scheme | Marks | AO |
|---|---|---|
| DR Boundary values when \(8 - 2x - x^2 = x + 5\) \(x^2 + 3x - 3 = 0\) | M1 | 2.1 |
| giving \(x = \dfrac{-3 \pm \sqrt{21}}{2}\) | A1 | 2.1 |
| From the graph, the line lies below the curve for \(\dfrac{-3-\sqrt{21}}{2} < x < \dfrac{-3+\sqrt{21}}{2}\) or \(\left\{x : x > \dfrac{-3-\sqrt{21}}{2}\right\} \cap \left\{x : x < \dfrac{-3+\sqrt{21}}{2}\right\}\) | B1 | 2.1 |
| [3] |
Notes
M1: A correct three term quadratic equation (or inequality) must be seen
Any method including BC is acceptable for solving the quadratic equation clearly seen in the form \(ax^2 + bx + c = 0\).
A1: Correct roots of the equation soi
Must be exact
B1: Shows correct inequality FT their roots
Allow B1 for \(-3.79 < x < 0.79\) www