Pure Mathematics Year 1 (AS) Unit Test 8: Exponentials and Logarithms
1 The graph of passes through the points (2, 400) and (5, 50).
a Find the values of the constants a and b. (5 marks)
b Given that , for some constant k > 0, show that where
log means log to any valid base. (4 marks)
2
Find the value of x showing detailed reasoning. (6 marks)
3 a Sketch the graph of stating the coordinates of any points where the graph
crosses the coordinate axes. (2 marks)
b i Describe fully the transformation which transforms the graph
to the graph (1 marks)
ii Describe the transformation which transforms the graph
to the graph (1 marks)
4 Solve algebraically, showing each step of your working, the equation
(5 marks)
5 a Sketch the graph for x > −a labelling any asymptotes and
points of intersection with the x-axis or y-axis. Leave your answers in terms of a where necessary.
(6 marks)
b For x > −a, describe, with a reason, the relationship between the graphs of
and (2 marks)
6 The population, P, of bacteria in an experiement can be modelled by the formula
, where t is the time in hours after the experiment began.
a Use the model to estimate the population of bacteria 7 hours after the experiment began.
(2 marks)
b Interpret the meaning of the constant 100 in the model. (1 mark)
c How many whole hours after the experiment began does the population of bacteria first exceed
1 million, according to the model? (3 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
1 The graph of passes through the points (2, 400) and (5, 50).
a Find the values of the constants a and b. (5 marks)
b Given that , for some constant k > 0, show that where
log means log to any valid base. (4 marks)
2
Find the value of x showing detailed reasoning. (6 marks)
3 a Sketch the graph of stating the coordinates of any points where the graph
crosses the coordinate axes. (2 marks)
b i Describe fully the transformation which transforms the graph
to the graph (1 marks)
ii Describe the transformation which transforms the graph
to the graph (1 marks)
4 Solve algebraically, showing each step of your working, the equation
(5 marks)
5 a Sketch the graph for x > −a labelling any asymptotes and
points of intersection with the x-axis or y-axis. Leave your answers in terms of a where necessary.
(6 marks)
b For x > −a, describe, with a reason, the relationship between the graphs of
and (2 marks)
6 The population, P, of bacteria in an experiement can be modelled by the formula
, where t is the time in hours after the experiment began.
a Use the model to estimate the population of bacteria 7 hours after the experiment began.
(2 marks)
b Interpret the meaning of the constant 100 in the model. (1 mark)
c How many whole hours after the experiment began does the population of bacteria first exceed
1 million, according to the model? (3 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
