Particles P and Q leave a fixed point A at the same time and travel in the same straight line. The velocity of P after t seconds is 6t(t − 3) m s⁻¹ and the velocity of Q after t seconds is (10 – 2t) m s⁻¹.

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

1. [B1] Straight line, reaching positive v-axis and positive t-axis (negative gradient)
2. [B1] Quadratic (U shape, through (0,0) and cutting t-axis at t < 5)
3. [B1] Fully correct graphs with correct labelling with t = 3, t = 5, v = 10, v = 60 seen
4. [B1] Use either integration to find s for Q or use a correct formula to find the area under the relevant triangle
5. [M1] Use integration to find the displacement for P
6. [A1] Correct integration for P (unsimplified)
7. [B1] Either evaluation of s(P) at t = 5 and show that at t = 5, s(P) = s(Q) = 25 or show that t = 5 is a solution of the cubic by solving or verify t = 5 is a solution of the cubic by substitution.
8. [M1] Find the distance between P and Q Allow either sign s_P and s_Q must have been found by integration
9. [M1] Differentiate to obtain an equation in t and attempt to solve
10. [A1]
11. [A1] Maximum Distance PQ = (-)48.4 m