A particle P moves in a straight line starting from a point O. The velocity v m s⁻¹ of P at time t s is given by v = 12t - 4t² for 0 ≤ t ≤ 2, v = 16 - 4t for 2 ≤ t ≤ 4.

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

1. [M1] dv/dt = 12-8t For attempted differentiation of 12t – 4t² (or for alternative e.g. completing the square)
2. [M1] [Maximum v when t=1.5 ⇒ v=12×1.5-4×1.5² ] For finding and using t
3. [A1] Maximum velocity is 9 (m s⁻¹)
4. [M1] dv/dt = 12-8t = -4 Finding acceleration for 0 ≤ t ≤ 2 when t = 2
5. [A1] Acceleration for 2 < t < 4 is -4 Both values correct, with correct statement No instantaneous change
6. [B1] Quadratic shape (with max) for 0 ≤ t ≤ 2
7. [B1] Line with negative gradient from (2, ...) to (4,0)
8. [B1] All correct, smooth join and key values indicated
9. [B1] Area of triangle is 8 (May be obtained by integrating 16 – 4t or use of uvast)
10. [M1] [∫ (12t - 4t²) dt = 6t² – 4/3 t³] Integration attempt for 0 ≤ t ≤ 2
11. [DM1] [6×2² – 4/3 × 2³ – 6×0² + 4/3 × 0³] Use of limits 0 and 2; condone absence of zero terms
12. [A1] Area under curve is 40/3 or 13.3
13. [A1] Distance travelled is 64/3 (m) or 21.3 (m)