Let f(x) = (17x² − 7x + 16) / ((2 + 3x²)(2−x)).

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 (Ax + B)/(2+3x²) + C/(2-x).
2. [M1] (a) Use a correct method for finding a coefficient. e.g. (Ax + B)(2 – x) + C(2 + 3x²) = 17x² - 7x + 16.
3. [A1] (a) Obtain one of A = −2, B = 3 and C = 5.
4. [A1] (a) Obtain a second value.
5. [A1] (a) Obtain the third value.
6. [M1] (b) Use a correct method to find the first two terms of the expansion (2 – x)⁻¹ = 2⁻¹ + (−1) 2⁻²(-x) + [(-1)(-2)2⁻³(-x)²/2!], (1+3x²/2)⁻¹ = 1 - (3x²/2) or (1-x/2)⁻¹ = 1 - (-x/2). Symbolic coefficients are not sufficient for the M1.
7. [A1 FT] (b) Obtain correct un-simplified expansions up to the term in x³ of each partial fraction.
8. [A1 FT] (b) Unsimplified (2 – x)⁻¹ expanded correctly, error in simplifying before their C is involved in the expression, allow A1FT when their C is introduced. The FT is on A, B, C.
9. [M1] (b) Multiply expansion of (1 + 3x²/2)⁻¹ (must reach 1 + 3x²/2) by Ax + B, where AB ≠ 0, up to the term in x³. Allow if used Cx + D (Ax + B miscopied). Allow either ±2 or ±2⁻¹ outside bracket or missing. Allow one error in actual multiplication to acquire the 4 terms [all terms needed]. Ignore errors in higher powers.
10. [A1] (b) Obtain final answer 4 + 1/4 x - 13/8 x² + 29/16 x³, or equivalent.
11. [B1] (c) State answer |x| < 2/√3 or -2/√3 < x < 2/√3 clear conclusion required. Or exact equivalent. Strict inequality.