Find the quotient and remainder when 2x³ – x² + 6x + 3 is divided by x² + 3.

Mark scheme

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

1. [M1] (a) Commence division and reach quotient of the form 2x + k
2. [A1] (a) Obtain quotient 2x – 1
3. [A1] (a) Obtain remainder 6
4. [B1FT] (b) Obtain terms x² - x (FT on quotient of the form 2x + k)
5. [M1] (b) Obtain term of the form a tan⁻¹ (x/√3)
6. [A1FT] (b) Obtain term (6/√3) tan⁻¹ (x/√3) (FT on a constant remainder)
7. [M1] (b) Use x = 1 and x = 3 as limits in a solution containing a term of the form atan⁻¹ (bx)
8. [A1] (b) Obtain final answer π/√3 + 6, or exact equivalent