A particle P moves in a straight line such that its velocity v m/s at time t seconds is given by v = 6t² - 10t + 3. The particle is initially at the origin O. (a) Find the acceleration of P when t=2. (b) Find the displacement of P from O when t=3.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. (a)
2. M1 for differentiating v with respect to t to find the acceleration a = dv/dt.
3. A1 for a = 12t - 10.
4. A1 for substituting t=2 into a: a = 12(2) - 10 = 14 m/s². (cao)
5. (b)
6. M1 for integrating v with respect to t to find the displacement s = ∫ v dt.
7. A1 for s = (6t³)/3 - (10t²)/2 + 3t + C (or 2t³ - 5t² + 3t + C).
8. B1 for using the initial condition s=0 when t=0 to find C=0. (May be implied if C is not shown).
9. M1 for substituting t=3 into their expression for s (with C=0).
10. A1 for s = 2(3)³ - 5(3)² + 3(3) = 2(27) - 5(9) + 9 = 54 - 45 + 9 = 18 m. (cao)