A particle moves in a straight line starting from a point O. The velocity v m s⁻¹ of the particle t s after leaving O is given by v = t³ − 2t² + 1 for 0 ≤ t ≤ 4. You may assume that the velocity of the particle is positive for t < ½, is zero at t = ½ and is negative for t > ½.

Mark scheme

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

1. [*M1] For attempt at integration. Increase power by 1 and a change in coefficient in at least one term (which must be the same term). s = vt is M0.
2. [A1] (s=) t⁴/4 - (2/3)t³ + t [+c]. May be unsimplified.
3. [DM1] Use limits 0 and ½ correctly, or substitute t = ½. M0 if including other regions.
4. [A1] Distance = 21/64 m. Oe, e.g. 0.328125. Condone 0.328.
5. [*M1] Attempt to differentiate [(a =)3t²-9t]. Decrease power by 1 and a change in coefficient in at least one term (which must be the same term). a = v/t is M0.
6. [A1] Solve a=0 to get t=3. Ignore t=0 if not rejected.
7. [DM1] Must be using an expression for s from integration. Allow missing minus sign at start; use limits ½ and their 3 correctly, where ½ < their 3 ≤ 4. Or use limit 3 and find difference from their 21/64 from part (a).
8. [A1FT] 573/32 Oe, e.g. 17.90625. Condone 17.9. FT their positive integration value in part (a), e.g. 17_37/64_ + their (a) or 69/4 + 2× their (a).