Let f(x) = (24x + 13) / ((1 - 2x)(2 + x)²)

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [B1] (a) State or imply the form A/(1-2x) + B/(2+x) + C/(2+x)²
2. [M1] (a) Use a correct method for finding a coefficient A(2+x)² + B(1-2x)(2+x) + C(1-2x) = 24x+13.
3. [A1] (a) Obtain one of A = 4, B = 2 and C = -7 If errors in equating still allow A marks for A and C.
4. [A1] (a) Obtain a second value
5. [A1] (a) Obtain the third value
6. [B1] (a) Mark the form A/(1-2x) + (Dx+E)/(2+x)² where A = 4, D = 2 and E = -3, B1 M1 A1 A1 A1 as above.
7. [M1] (b) Use a correct method to find the first two terms of the expansion of (1-2x)⁻¹, (2+x)⁻¹, (2+x)⁻², (1+x/2)⁻¹ or (1+x/2)⁻² Symbolic coefficients are not sufficient for the M1.
8. [A1FT] (b) Obtain correct un-simplified expansions up to the term in x² of each partial fraction A: 4(1+(-1)(-2x) + (-1)(-2)/2 (-2x)² +..) A = 4.
9. [A1FT] (b) B: 2(1+(-1)(x/2) + (-1)(-2)/2 (x/2)² +...) B = 2.
10. [A1FT] (b) C: -7(1+(-2)(x/2) + (-2)(-3)/2 (x/2)² +...) C = -7.
11. [A1] (b) Obtain final answer (13/4) + (37/4)x + (239/16)x²
12. [B1] (c) |x| < 1/2 OE