The polynomial p(x) is defined by p(x) = 4x³ + 16x² + 9x – 15.

Mark scheme

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

1. [M1] Carry out division at least as far as 2x² + kx
2. [A1] Obtain quotient 2x² + 5x – 3
3. [A1] Confirm remainder is –6 [AG – necessary detail needed]
4. [M1] Integrate to obtain at least k₁x³ and k₂ ln(2x+3) terms
5. [A1] Obtain (2/3)x³ + (5/2)x² - 3x – 3ln(2x+3) [condone absence of ... +c and modulus signs]
6. [B1 FT] State or imply p(x) + 6 = (2x + 3)(2x² + 5x – 3) [FT their quotient]
7. [B1] Conclude (2x + 3)(2x−1)(x+3)
8. [B1 FT] State or imply sin 2θ = -3/2 or sin 2θ = 1/2 or both [FT their relevant factors]
9. [M1] Carry out correct process to find θ in at least one case
10. [A1] Obtain 99.7 and 110.3 [Or greater accuracy and no others between 0° and 135°]