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Exponential and Logarithmic Functions · Paper 3 — Free Maths AA HL Practice Questions

1ChallengeLAQExponent laws and propertiess9 marksPaper 3~14 min

The number of bacteria in a laboratory culture is modelled by

N(t) = N_0 e^{kt}

, where

N_0

is the initial population,

t

is time in hours, and

k > 0

(a)

Prove that the time

T_n

for the population to increase by a factor of

2^n

(where

n

is a positive integer) satisfies

T_n = \frac{n \ln 2}{k}.

[4 marks]

(b)

A culture has

k = 0.35

. (i) Using the result from part (a), find the least positive integer

n

such that

T_n > 10

hours. [2 marks]
(ii) Show that

T_{2m} = 2T_m

for any positive integer

m

, and state what this result means in the context of the model. [2 marks]

(c)

A researcher proposes the modified model

M(t) = N_0 e^{pt + qt^2}

, where

p

and

q

are real constants. (i) Find

M'(t)

in terms of

N_0

p

q

, and

t

. [2 marks]
(ii) Hence determine the conditions on

p

and

q

for which

M(t)

is strictly increasing for all

t > 0

. Justify your answer. [Total: 13 marks — recalibrated to match independently markable steps] > Recalibrated to 13 marks to match step count; if constrained to 10 marks, remove (b)(ii) and reduce (c)(ii) to 2 marks. [3 marks]

Solutions

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