C3 : EXPONENTIAL FUNCTIONS (Year 13 Mathematics)
Name: …………………………….
Score:
Percentage:
Grade:
Target grade:
1) a) Solve the equation 2e x  5 , giving your answer as an exact natural logarithm.
b) (i)
By substituting y  e x , show that the equation 2e x  5e  x  7 can be written as
(ii)
[2]
[2]
2 y2  7 y  5  0
x
x
Hence solve the equation 2e  5e  7 , giving your answers as exact values of x.
[3]
[AQA June 2005]
2) A function f is defined by f ( x)  2e3 x  1 for all real values of x.
a) Find the range of f.
1  x 1
b) Show that f 1 ( x)  ln 

3  2 
[2]
[3]
[AQA June 2005]
3) A particular species of orchid is being studied. The population p at time t years after the study
started is assumed to be
2800ae0.2t
p
, where a is a constant.
1  ae0.2t
Given that there were 300 orchids when the study started,
(a) show that a = 0.12
[3]
(b) use the equation with a = 0.12 to predict the number of years before the population of orchids
reaches 1850.
[4]
336
(c) Show that p 
[1]
0.12  e 0.2t
(d) Hence show that the population cannot exceed 2800.
[2]
[Edexcel June 2005]
4) On 1 January 1900, a sculpture was valued at £80. When the sculpture was sold on 1 January
1956, its value was £5000.
The value, £V, of the sculpture, is modelled by the formula V  Ak t , where t is the time in years
since 1 January 1900 and A and k are constants.
a) Write down the value of A.
[1]
b) Show that k ≈ 1.07664.
[3]
c) Use this model to
(i)
show that the value of the sculpture on 1 January 2006 will be greater than £200 000.
[2]
(ii)
find the year in which the value of the sculpture will first exceed £800 000.
[3]
The topics that I need to study further are …