In this question all lengths are in kilometres and time is in hours. A particle P moves in a straight line such that its displacement, s, from a fixed point at time t is given by s = (t+2)(t−5)², for t ≥ 0.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [M1] for attempt at differentiation of a product or expansion and differentiation with at least two out of three terms of their expansion differentiated correctly
2. [A1] ds/dt = 3t² - 16t + 5 oe (all correct, allow factorised)
3. [M1 Dep] for attempt to solve their ds/dt = 0, must be in quadratic form
4. [A1] t = 1/3, t = 5 (for both)
5. [B1] for a correct curve in the first quadrant only. There must be an indication of a max point in the correct position and a min point at (5, 0)
6. [B1] for (0, 50) provided basic curve shape is correct
7. [B1] for a quadratic curve in the first and fourth quadrants
8. [B1] for (0, 5), (1/3, 0) or (0.333, 0) marked and passing through (5, 0) on the x-axis
9. [B1] Acceleration = 6t - 16
10. [B1] for a straight line with a positive gradient in the first and fourth quadrants, meeting the vertical axis
11. [Dep B1] for (8/3, 0) or (2.67, 0) and (0, -16)