The polynomial p(x) is defined by p(x) = 2x³ + 5x² + ax + 2a, where a is an integer.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [1] Carry out algebraic long division at least as far as 2x² + kx
2. [1] Obtain quotient 2x²+x+a-2
3. [1] Confirm remainder is 4 AG – necessary detail needed SC B1 for use of remainder theorem to obtain 4
4. [1] Identify integrand as 2x²+x+a-2+4/(x+2) B1 FT Following their quotient, may be implied
5. [1] Integrate to obtain at least 2 terms from the form of k₁x³ + k₂x² + k₃ ln(x+2) for non-zero k₁,k₂, k₃
6. [1] Obtain correct (2/3)x³ + (1/2)x² + (a - 2)x + 4ln(x + 2)
7. [1] Apply limits correctly and attempt correct process to find a or b M1 Must have the correct form
8. [1] Obtain -8/3 + 2a = 22/3 or equivalent and hence a = 5
9. [1] Obtain 4ln 3 and hence b=81