O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2019 · Q119 Marks
The velocity, v ms⁻¹, of a particle travelling in a straight line, t seconds after passing through a fixed point O, is given by v = 4/(t+1)³.
- (i)Explain why the direction of motion of the particle never changes.[1]
- (ii)Showing all your working, find the acceleration of the particle when t = 5.[3]
- (iii)Find an expression for the displacement of the particle from O after t seconds.[3]
- (iv)Find the distance travelled by the particle in the fourth second.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.
- [B1] (i) v ≠ 0 or v > 0 oe
- [B2] (ii) Differentiates: -12(t+1)⁻⁴ oe, isw (B1 for k(t+1)⁻⁴ with k ≠ −12)
- [B1] -4/108 oe or -0.00926
- [B3] (iii) Integrates: [s =]-2(t+1)⁻² + 2 oe, isw (B2 for -2(t+1)⁻² [+c] or B1 for k(t+1)⁻² [+c] with k ≠ –2)
- [M1] (iv) Finds their s from (iii) when t = 4 or when t = 3 or finds [their(-2(t+1)^-2 [+2])]^4_3 = their(-2(4+1)^-2 -(-2(3+1)^-2))
- [A1] 9/200 or 0.045
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