O-LevelAdditional MathematicsLogarithmic and exponential functions4037/12 · May/June 2019 · Q36 Marks
The number, B, of a certain type of bacteria at time t days can be described by B = 200e²ᵗ +800e⁻²ᵗ.
- (i)Find the value of B when t = 0.[1]
- (ii)At the instant when dB/dt = 1200, show that e⁴ᵗ - 3e²ᵗ - 4 = 0.[3]
- (iii)Using the substitution u = e²ᵗ, or otherwise, solve e⁴ᵗ - 3e²ᵗ - 4 = 0.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.
- [B1] 1000
- [B1] dB/dt = 400e²ᵗ-1600e⁻²ᵗ
- [M1] 3=e²ᵗ-4e⁻²ᵗ oe For equating an equation of the form ae²ᵗ + be⁻²ᵗ to 1200 and dividing by 400
- [A1] e⁴ᵗ - 3e²ᵗ - 4 = 0
- [M1] (e²ᵗ+1)(e²ᵗ-4) = 0 For attempt to factorise and solve, dealing with exponential correctly, to obtain e²ᵗ = ...
- [A1] t = ln 2, 1/2 ln 4 or awrt 0.693 only isw if appropriate
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