The coefficient of x² in the expansion of (2 + x/2)⁶ + (a + x)⁵ is 330. Find the value of the constant a.

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

1. [B2,1,0] Coefficient of x² in (2 + x/2)⁶ is ⁶C₂ × 2⁴ × (1/2)² (x²) (= 60). 3 things wanted –1 each incorrect component, must be multiplied together. Allow ⁶C₄, (6/4) and factorial equivalents. Marks can be awarded for correct term in an expansion.
2. [B1] Coefficient of x² in (a + x)⁵ is ⁵C₂ × a³ (x²) (= 10a³). Marks can be awarded for correct term in an expansion.
3. [M1] Forms an equation ‘their 60' + ‘their 10a³ = 330, OK with x² in all three terms initially. This can be recovered by a correct answer.
4. [A1] a = 3. Condone ±3 as long as +3 is selected.