O-LevelAdditional MathematicsDifferentiation and integration4037/12 · May/June 2018 · Q47 Marks
A particle P moves so that its displacement, x metres from a fixed point O, at time t seconds, is given by x = ln (5t + 3).
- (i)Find the value of t when the displacement of P is 3m.[2]
- (ii)Find the velocity of P when t = 0.[2]
- (iii)Explain why, after passing through O, the velocity of P is never negative.[1]
- (iv)Find the acceleration of P when t = 0.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.
- B1 3 = ln (5t + 3) e³ = 5t +3 or better
- B1 t=3.42
- M1 dx/dt = 5/(5t+3) for k₁/(5t+3)
- A1 all correct When t = 0, dx/dt = 5/3, 1.67 or better
- B1 dep on M1 in (ii) If t>0 each term in k₁/(5t+3) >0 so never negative oe FT on their k₁/(5t+3) provided k₁ > 0
- M1 d²x/dt² = k₂/(5t+3)²
- A1 all correct d²x/dt² = -25/(5t+3)² When t = 0, d²x/dt² = -25/9 or -2.78
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