In this question, all lengths are in metres and time is in seconds. A particle P moves in a straight line such that its displacement s from a fixed point O at time t is given by s = (t-4)²(t-1) for t≥ 0.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [B1] for correct cubic shape with maximum in the first quadrant, a point of tangency to the t-axis at the minimum point; must meet the s-axis
2. [B1] for correct intercepts; must have attempted the correct shape
3. [M1] v: 2(t-4)(t-1)+(t-4)² (1) or S: t³ - 9t² + 24t - 16 and v: 3t² - 18t + 24
4. [A1] v: 3(t-4)(t-2)
5. [B2] B1 for correct quadratic shape with minimum in the fourth quadrant; must meet the v-axis
6. [B1] for correct intercepts; must have attempted the correct shape
7. [B1] 6t-18 or 3(t-4)+(3t-6) oe (FT their 3-term quadratic expression oe for v)
8. [B1] for a single ruled line with positive gradient in fourth and first quadrants; must meet the a-axis and cross the t-axis
9. [B1] FT for a line with t-intercept 3 (FT -their(-18)÷ their 6 providing their a is a linear expression of the form mt + c)
10. [B1] FT for a line with a-intercept -18 (FT their (-18) providing their a is a linear expression of the form mt + c)