O-LevelAdditional MathematicsDifferentiation and integration4037/21 · May/June 2025 · Q1011 Marks
A particle P moves in a straight line. t seconds after passing a fixed point, O, the acceleration of P, a m s⁻², is given by a = t²-2 for 0 ≤ t ≤ 4 a = 19-5e^(8-2t) for 4 < t ≤ 10. When t = 3, the velocity of P is 1/3 m s⁻¹ and its displacement from O is -1/4 m.
- (a)(i)Find the velocity of P when t = 4.[3]
- (a)(ii)Find the displacement of P from O when t = 4.[3]
- (b)Find the displacement of P from O when t = 10.[5]
Mark scheme
Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.
- [M1] [v=] t³/3 - 2t (+c)
- [A1] Or substituting in limits [v=] t³/3 - 2t - 10/3
- [A1] v = 10
- [M1] FT an expression in form at³+bt
- [A1] Or 2 correct terms in [s=] t⁴/12 - 10t/3
- [A1] s = 4
- [M1] [v=] 19t + 5/2 e^(8-2t) (+c) oe
- [A1] FT (their 10) – 78.5 [v=] 19t + 5/2 e^(8-2t) - 137/2 oe
- [M1] dep on previous M1 FT their 137/2 provided not 0 or algebraic [s=] 19t²/2 + 5/2 (-1/2) e^(8-2t) - 137/2 t (+d) oe
- [A1] [s=] 19t²/2 - 5/4 e^(8-2t) - 137/2 t + 509/4 oe, cao
- [A1] s = 392 or 392.2[49...]
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