Standard +0.8 This requires systematic case analysis of modulus inequalities (splitting at critical points x = ±3 and x = 0.5), solving multiple quadratic inequalities, and carefully combining solution sets. More demanding than routine modulus equations but standard for Further Maths F2 with clear methodology.
Attempts to solve \(x^2-9=1-2x\) OR \(x^2-9=-1+2x\) to obtain two non-zero values of \(x\)
\(x = \frac{-2 \pm \sqrt{44}}{2}\) OR \(x=-2, 4\)
A1
One correct pair of values. Allow irrational roots as given, or \(-1\pm\sqrt{11}\), or awrt 2.32, -4.32, or truncated 2.3, -4.3
\(x^2-9=-1+2x \Rightarrow x^2-2x-8=0\) AND \(x^2-9=1-2x\)
M1
Attempts to solve both equations to obtain four non-zero values of \(x\)
\(x = \frac{-2\pm\sqrt{44}}{2}\) AND \(x=-2, 4\)
A1
Both pairs of values correct. Allow irrational roots as given or \(-1\pm\sqrt{11}\) or awrt 2.32, -4.32 or truncated 2.3, -4.3
\(-1+\sqrt{11} < x < 4\) or \(-1-\sqrt{11} < x < -2\)
B1
One correct inequality. Allow alternative notation e.g. \(\left(-1+\sqrt{11}, 4\right)\), \(\left(-1-\sqrt{11}, -2\right)\)
\(-1+\sqrt{11} < x < 4\) and \(-1-\sqrt{11} < x < -2\)
B1
Both inequalities correct. In an otherwise fully correct solution, if any extra incorrect regions are given, deduct the final B mark.
# Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| $x^2 - 9 = 1-2x \Rightarrow x^2+2x-10=0$ **OR** $x^2-9 = -1+2x \Rightarrow x^2-2x-8=0$ | M1 | Attempts to solve $x^2-9=1-2x$ **OR** $x^2-9=-1+2x$ to obtain two non-zero values of $x$ |
| $x = \frac{-2 \pm \sqrt{44}}{2}$ **OR** $x=-2, 4$ | A1 | One correct pair of values. Allow irrational roots as given, or $-1\pm\sqrt{11}$, or awrt 2.32, -4.32, or truncated 2.3, -4.3 |
| $x^2-9=-1+2x \Rightarrow x^2-2x-8=0$ **AND** $x^2-9=1-2x$ | M1 | Attempts to solve **both** equations to obtain four non-zero values of $x$ |
| $x = \frac{-2\pm\sqrt{44}}{2}$ **AND** $x=-2, 4$ | A1 | Both pairs of values correct. Allow irrational roots as given or $-1\pm\sqrt{11}$ or awrt 2.32, -4.32 or truncated 2.3, -4.3 |
| $-1+\sqrt{11} < x < 4$ **or** $-1-\sqrt{11} < x < -2$ | B1 | One correct inequality. Allow alternative notation e.g. $\left(-1+\sqrt{11}, 4\right)$, $\left(-1-\sqrt{11}, -2\right)$ |
| $-1+\sqrt{11} < x < 4$ **and** $-1-\sqrt{11} < x < -2$ | B1 | Both inequalities correct. In an otherwise fully correct solution, if any extra incorrect regions are given, deduct the final B mark. |
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