O-LevelAdditional MathematicsDifferentiation and integration4037/12 · May/June 2020 · Q912 Marks

A particle P moves in a straight line such that its displacement, xm, from a fixed point O at time ts is given by x = 10 sin2t-5. Find the speed of P when t = π.

(a)(i)A particle P moves in a straight line such that its displacement, xm, from a fixed point O at time ts is given by x = 10 sin2t-5. Find the speed of P when t = π.[1]
(a)(ii)Find the value of t for which P is first at rest.[2]
(a)(iii)Find the acceleration of P when it is first at rest.[2]
(b)(i)The diagram shows the velocity-time graph for a particle Q travelling in a straight line with velocity vms⁻¹ at time ts. The particle accelerates at 3.5ms⁻² for the first 10s of its motion and then travels at constant velocity, Vms⁻¹, for 10s. The particle then decelerates at a constant rate and comes to rest. The distance travelled during the interval 20 ≤ t ≤ 25 is 112.5m. Find the value of V.[1]
(b)(ii)Find the velocity of Q when t = 25.[3]
(b)(iii)Find the value of t when Q comes to rest.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[1] v = 20cos 2t when t = π, ν = 20
[1] Equating their (i) to zero, must be a cosine and attempt to solve
[1] t = π/4
[1] Attempt to differentiate their v, dep on previous M mark, and use their value for (ii)
[1] -40
[1] 35
[1] Use of area under appropriate part of the graph
[1]
[1] x = 10
[1] Using a ratio method or otherwise, find extra time to stop = 2s or equivalent
[1] t' = 2
[1] 27

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