O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2014 · Q77 Marks
A particle moving in a straight line passes through a fixed point O. The displacement, x metres, of the particle, t seconds after it passes through O, is given by x = t + 2 sin t.
- (i)Find an expression for the velocity, v ms⁻¹, at time t.[2]
- (ii)When the particle is first at instantaneous rest, find the value of t,[2]
- (iii)When the particle is first at instantaneous rest, find its displacement and acceleration.[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2014). The mark or mark type is shown in brackets.
- [B1] (i) v = 2cost + 1. Mark final answer.
- [M1] (ii) 2cost + 1 = 0. Equate their v to zero (must be a differential) and attempt to solve to find an angle.
- [A1] (ii) t = 2π/3 or 2.09. Awrt.
- [B1] (iii) x = (2π/3) + 2sin(2π/3) = 3.83m. Awrt.
- [B1ft] (iii) a = -2sint. Ft their v (2nd differential).
- [DB1ft] (iii) t=(2π/3) → a = -2sin(2π/3) = -√3 = -1.73 ms⁻². ft using their angle t in correct a awrt.
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