The binomial expansion of (1 + px)ⁿ, where n is a positive integer, is 1 - 12x + 54x² + ...

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1 for equating the coefficient of x: np = -12.
2. M1 for equating the coefficient of x²: n(n-1)/2 * p² = 54.
3. A1 for correctly substituting np = -12 into the x² equation, e.g. n(n-1)/2 * (-12/n)² = 54.
4. M1 for solving the resulting equation for n.
5. A1 for n(n-1)/2 * 144/n² = 54 ⇒ 72(n-1)/n = 54 ⇒ 72n - 72 = 54n ⇒ 18n = 72 ⇒ n=4.
6. A1 for substituting n=4 back into np = -12 to find p: 4p = -12 ⇒ p = -3.
7. M1 for finding the general x³ term: n(n-1)(n-2)/ (3!) * (px)³.
8. A1 for substituting n=4, p=-3: 4(3)(2) / 6 * (-3)³ = 4 * (-27) = -108. (So coefficient is -108).