Let f(x) = (8 + 5x + 12x²)/((1-x)(2 + 3x)²).

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

1. State or imply the form A/(1-x) + B/(2+3x) + C/(2+3x)²
2. Use a correct method for finding a coefficient
3. Obtain one of A = 1, B = −1, C = 6
4. Obtain a second value
5. Obtain the third value
6. Alternative form: In the form A/(1-x) + (Dx+E)/(2+3x)² where A = 1, D = − 3 and E = 4 can score B1 M1 A1 A1 A1 as above.
7. Use a correct method to find the first two terms of the expansion of (1-x)⁻¹, (2+3x)⁻¹, (1+(3/2)x)⁻¹, (2+3x)⁻² or (1+(3/2)x)⁻² (Symbolic coefficients are not sufficient for the M1)
8. Obtain correct un-simplified expansions up to the term in x² of each partial fraction: for A: (1+x+x²); for B: (1-3x/2+9x²/4)/2; for C: (1-3x+27x²/4)/4 (The FT is on A, B, C)
9. Obtain final answer 2 - 11/4 x + 10x², or equivalent (Allow unsimplified fractions)