A particle moves in a straight line, starting from a point O. The velocity of the particle at time t s after leaving O is v m s⁻¹. It is given that v = kt½ - 2t - 8, where k is a positive constant. The maximum velocity of the particle is 4.5 m s⁻¹.

Mark scheme

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

1. [*M1] Decrease power by 1 and a change in coefficient in at least one term (which must be the same term). Therefore, must see d/dt (kt^c) = c k t^(c-1) with c ≠ k.
2. [A1] May be unsimplified.
3. [DM1] Equate a to 0 and attempt to solve for t or k. Must be of the correct form, e.g. t = Ak² or k = Bt^(1/2) (possibly implied by forming a correct equation in k).
4. [M1] Or complete method to find k or an equation involving k, e.g. 4t^(1/2) - 2t - 8 = 4.5 ⇒ t = 6.25 therefore k = 4(6.25)^(1/2). Dependent on both previous M marks.
5. [A1] AG-Allow verification. Any errors seen is A0.
6. (M1) Dependent on both previous M marks.
7. (A1) AG
8. [B1] AG Any errors seen is B0.
9. [*M1] Increase power by 1 and a change in coefficient in at least one term (which must be the same term).
10. [A1] Allow unsimplified; allow with k not substituted.
11. [DM1] Attempt at correct use of limits 0 and 1 or 1 and 16 – allow a single slip.
12. [DM1] Allow at most one slip in both.
13. [A1] Allow 47.3 or better (47.3333...)