It is given that in the expansion of (4 + 2x)(2 – ax)⁵, the coefficient of x² is −15. Find the possible values of a.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [B1] Terms required for x²: -5×2⁴×ax + 10×2³×a²x² [= -80ax + 80a²x²]. Can be seen as part of an expansion or in correct products.
2. [*M1] 2× (+their coefficient of x) + 4× (±their coefficient of x²)
3. [DM1] Forming a 3-term quadratic in a, with all terms on the same side or correctly setting up prior to completing the square and solving using factorisation, formula or completing the square. If factorising, factors must expand to give their coefficient of a².
4. [A1] a = 1/8 or a = 3/8 OE.
5. [M1] Forming a 3-term quadratic in a with all terms on the same side. Allow ± sign errors.
6. [M1] Any use of discriminant on a 3-term quadratic.
7. [A1] k = -20
8. [B1] a = 1/4