O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2024 · Q96 Marks

In this question time is measured in seconds.

(a)A particle is moving in a straight line with constant velocity of 6ms⁻¹. At time t = 0, it passes a fixed point A. At time t = 5 it suddenly changes direction and moves with a different constant velocity along the same straight line. It passes the point A again at time t = 15. Sketch the velocity-time graph for the motion.[3]
(b)Another particle is moving in a straight line with constant acceleration. At time t = 0 it passes a fixed point B with velocity -8ms⁻¹. It passes the point B again at time t = 20. Sketch the velocity-time graph for the motion.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[B1] Horizontal line, v = 6 for 0 ≤ t ≤ 5
[B2] Horizontal line, v = −3 for 5 ≤ t ≤ 15 [B1] for horizontal line for 5 < t < 15 with v = k where k < 0 or v = 3 for 5 ≤ t ≤ 15 or v = -3 for t > 5 and at least 7 ≤ t ≤ 13
If 0 scored, award: [SC2] for Horizontal line, v = –6 for 0 ≤ t ≤ 5 and Horizontal line, v = 3 for 5 ≤ t ≤ 15
OR [SC1] for Horizontal line, v = –6 for 0 ≤ t ≤ 5 and Horizontal line for 5 < t < 15 with v = k where k > 0
[B3] Single line with positive gradient passing through (0, -8), (10, 0) and (20, 8) [B2] for a single line with positive gradient • passing through a point indicated as (0, -8) or (20, 8) and the point (10, 0) • or passing through points indicated as (0, -8) and (20, 8) which does not pass through (10, 0)
OR [B1] for a single line with positive gradient • passing through a point indicated as (0, -8) or (20, 8) to or through any point on the t-axis • or passing through (10, 0) but with both endpoints incorrect or unlabelled
If 0 scored, award [SC1] for a single line with negative gradient passing through (0, 8), (10, 0) and (20, -8)

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