Find the quotient and remainder when x⁴ + 16 is divided by x² + 4.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. [M1] (a) Divide x⁴ + 16 by x² + 4 to obtain quotient x² + k, or use identity (x²+4)(x² + Bx+C) + D and expand.
2. [A1] (a) Obtain quotient x² - 4 (or C=-4 if using identity method).
3. [A1] (a) Obtain remainder 32 (or D=32 if using identity method).
4. [B1ft] (b) Integrate the quotient part: Obtain ∫(x²-4)dx = x³/3 - 4x. Follow their quotient of form Ax² + B.
5. [M1] (b) Integrate the remainder part: Obtain p tan⁻¹ qx from (remainder)/(x²+4) where q=2 or q=1/2.
6. [A1ft] (b) Obtain 16 tan⁻¹ (x/2) (from (their 32)/(x²+4)). Follow their constant remainder.
7. [M1] (b) Use limits (0 to 2√3) correctly in an expression containing p tan⁻¹ qx and rx³ + sx.
8. [A1] (b) Obtain (4/3)(π+4) from full and correct working. AG.