A particle X moves along a straight track, starting from a point O at time t = 0. The displacement of X from O at time ts is sm, where s = 3t³/² - 6t.

Mark scheme

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

1. [*M1] For an attempt at differentiation. The power of t must decrease by 1 in at least 1 term with a change of coefficient in the same term. (v=9/2 t½ - 6)
2. [A1] Allow un-simplified.
3. [M1] For attempt to solve for t. Condone sign errors and transposing fractions. Correct squaring of √t. (v=0 => t=16/9)
4. [DM1] Use of limits 0 and their 16/9 or their 16/9 and 16 correctly. For both i.e. F(16)-F(their) - F(their)-[F(0)] OR F(16)-F(their) + F(their)-[F(0)] OR F(16)-2xF(their)
5. [A1] AWRT 103 m (928/9 m)
6. [*M1] For integration. The power of t must increase by 1 in at least 1 term with a change of coefficient in the same term. (v=0.8t-0.3t² +7.5)
7. [A1]
8. [DM1] For using v=-9.6 where v= pt+qt² +r where p≠0andq≠0 (no restriction for r) and attempt to solve for t. (v=-9.6 => 0.3t² -0.8t-17.1=0)
9. [A1] (t = 9 [or -19/3])
10. [B1FT] FT their v which is a 3 term quadratic in t. (s = 0.4t² – 0.1t³ +7.5t(+c))
11. [B1] At t=9 sx = 3×9³/² -6×9=27
12. [B1] At t=9 sy = 0.4×9² -0.1×9³ +7.5×9=27