Moderate -0.8 This is a straightforward quadratic inequality requiring factorisation (or quadratic formula), identification of critical values, and sketch/sign analysis. It's a standard textbook exercise with clear method and no conceptual challenges, making it easier than average but not trivial since it requires multiple routine steps and careful reasoning about inequality direction.
Factorise 3 term quadratic. Need \(a=10\), and either \(b=1\) or \(c=-2\) when expanded. Or solve using a valid method. If using the formula allow one sign slip.
\(x = -\frac{1}{2},\ x = \frac{2}{5}\)
A1 (AO 2.1)
Obtain both correct roots. Could be implied by the two values appearing in an incorrect inequality. SC allow B1 in place of M1A1 if roots are given but with no evidence of solving the quadratic. SC B1 includes \((x+\frac{1}{2})(x-\frac{2}{5})\) unless division by 10 seen prior to factorisation.
\(x < -\frac{1}{2},\ x > \frac{2}{5}\)
M1 (AO 1.1a)
Select outside region. For their two distinct roots. Allow M1A0 for \(x \leq -\frac{1}{2},\ x \geq \frac{2}{5}\). Allow M1A0 for \(\frac{2}{5} < x < -\frac{1}{2}\) or with \(\leq\).
A1 (AO 2.5)
Obtain correct inequalities. Any correct notation, including set notation, but A0 if linked by 'and'. SC Allow B2 for answer only (B1 for sight of correct roots and B1 for correct inequality).
Total: [4]
## Question 1:
**DR** (Detailed Reasoning required)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(5x-2)(2x+1) > 0$ | M1 (AO 1.1a) | Factorise 3 term quadratic. Need $a=10$, and either $b=1$ or $c=-2$ when expanded. Or solve using a valid method. If using the formula allow one sign slip. |
| $x = -\frac{1}{2},\ x = \frac{2}{5}$ | A1 (AO 2.1) | Obtain both correct roots. Could be implied by the two values appearing in an incorrect inequality. **SC** allow B1 in place of M1A1 if roots are given but with no evidence of solving the quadratic. SC B1 includes $(x+\frac{1}{2})(x-\frac{2}{5})$ unless division by 10 seen prior to factorisation. |
| $x < -\frac{1}{2},\ x > \frac{2}{5}$ | M1 (AO 1.1a) | Select outside region. For their two distinct roots. Allow M1A0 for $x \leq -\frac{1}{2},\ x \geq \frac{2}{5}$. Allow M1A0 for $\frac{2}{5} < x < -\frac{1}{2}$ or with $\leq$. |
| | A1 (AO 2.5) | Obtain correct inequalities. Any correct notation, including set notation, but A0 if linked by 'and'. **SC** Allow B2 for answer only (B1 for sight of correct roots and B1 for correct inequality). |
| **Total: [4]** | | |