O-LevelAdditional MathematicsDifferentiation and integration4037/11 · May/June 2011 · Q99 Marks

A body moves in a straight line such that, t s after passing through a fixed point O, its displacement from O is s m. The velocity v ms⁻¹ of the body is such that v = 5cos4t.

(i)Write down the velocity of the body as it passes through O.[1]
(ii)Find the value of t when the acceleration of the body is first equal to 10ms⁻².[4]
(iii)Find the value of s when t = 5.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.

[1] B1 for 5.
[1] M1 for attempting to differentiate v = 5cos4t to get acceleration a = -20sin4t.
[1] A1 for a = -20sin4t.
[1] DM1 for setting a=10 and attempting to solve for 4t: 10 = -20sin4t => sin4t = -0.5.
[1] A1 for t = 7π/24 (or 0.916 radians).
[1] M1 for attempting to integrate v = 5cos4t to get displacement s = (5/4)sin4t + c.
[1] A1 for s = (5/4)sin4t + c.
[1] DM1 for using the condition that s=0 when t=0 to find c (c=0). Requires using t in radians for integration.
[1] A1 for s = 1.14 (when t=5, in radians).

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 9 marks.
Source: O-Level Additional Mathematics, May/June 2011 Paper 1 Variant 1, question 9.

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