x

1. Let f(x) = log3 + log3 16 – log3 4, for x > 0.

2

(a) Show that f(x) = log3 2x.

(2)

(b) Find the value of f(0.5) and of f(4.5).

(3)

ln ax
The function f can also be written in the form f(x) = .
ln b

(c) (i) Write down the value of a and of b.

(ii) Hence on graph paper, sketch the graph of f, for –5 ≤ x ≤ 5, –5 ≤ y ≤ 5, using a scale of 1 cm to
1 unit on each axis.

(iii) Write down the equation of the asymptote.

(6)

(d) Write down the value of f–1(0).

(1)

The point A lies on the graph of f. At A, x = 4.5.

(e) On your diagram, sketch the graph of f–1, noting clearly the image of point A.
(4)
(Total 16 marks)

2. The functions f (x) and g (x) are defined by f (x) = ex and g (x) = ln (1+ 2x).

(a) Write down f −1(x).

(b) (i) Find ( f ◦ g) (x).

(ii) Find ( f ◦ g)−1 (x).

(Total 6 marks)

3. The population of a city at the end of 1972 was 250 000. The population increases by 1.3% per year.

(a) Write down the population at the end of 1973.

(b) Find the population at the end of 2002.

(Total 6 marks)

4. Let f (x) = loga x, x > 0.

(a) Write down the value of

(i) f (a);

(ii) f (1);

(iii) f (a4 ).
(3)

(b) The diagram below shows part of the graph of f.

 y

1
f

–2 –1 0 1 2 x

–1

–2

On the same diagram, sketch the graph of f−1.

(3)
(Total 6 marks)

5. The area A km2 affected by a forest fire at time t hours is given by A = A0 ekt.
When t = 5, the area affected is 1 km2 and the rate of change of the area is 0.2 km2 h−1.

(a) Show that k = 0.2.

(4)

1
(b) Given that A0 = , find the value of t when 100 km2 are affected.
e
(2)
(Total 6 marks)

6. Solve the following equations.

(a) ln (x + 2) = 3.

(b) 102x = 500.

(Total 6 marks)

7. Find the exact value of x in each of the following equations.

(a) 5x+1 = 625

(b) loga (3x + 5) = 2

(Total 6 marks)

8. A machine was purchased for $10000. Its value V after t years is given by V =100000e−0.3t. The machine
must be replaced at the end of the year in which its value drops below $1500. Determine in how many years
the machine will need to be replaced.
(Total 6 marks)

9. There were 1420 doctors working in a city on 1 January 1994. After n years the number of doctors, D,
working in the city is given by

D = 1420 + 100n.

(a) (i) How many doctors were there working in the city at the start of 2004?

(ii) In what year were there first more than 2000 doctors working in the city?
(3)
 At the beginning of 1994 the city had a population of 1.2 million. After n years, the population, P, of the city
is given by

P = 1 200 000 (1.025)n.

(b) (i) Find the population P at the beginning of 2004.

(ii) Calculate the percentage growth in population between 1 January 1994 and 1 January 2004.

(iii) In what year will the population first become greater than 2 million?
(7)

(c) (i) What was the average number of people per doctor at the beginning of 1994?

(ii) After how many complete years will the number of people per doctor first fall below 600?
(5)
(Total 15 marks)

10. The population p of bacteria at time t is given by p = 100e0.05t.

Calculate

(a) the value of p when t = 0;

(b) the rate of increase of the population when t = 10.

(Total 6 marks)

11. The mass m kg of a radio-active substance at time t hours is given by

m = 4e–0.2t.

(a) Write down the initial mass.

(b) The mass is reduced to 1.5 kg. How long does this take?

(Total 6 marks)

12. $1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum number of
months required for the value of the investment to exceed $3000.

(Total 6 marks)

13. Each year for the past five years the population of a certain country has increased at a steady rate of 2.7% per
annum. The present population is 15.2 million.

(a) What was the population one year ago?

(b) What was the population five years ago?

(Total 4 marks)

14. Michele invested 1500 francs at an annual rate of interest of 5.25 percent,
compounded annually.

(a) Find the value of Michele’s investment after 3 years. Give your answer to the nearest franc.
(3)

(b) How many complete years will it take for Michele’s initial investment to double in value?
(3)

(c) What should the interest rate be if Michele’s initial investment were to double in value in 10 years?
(4)
(Total 10 marks)
15. A group of ten leopards is introduced into a game park. After t years the number of leopards, N, is modelled
by N = 10 e0.4t.

(a) How many leopards are there after 2 years?

(b) How long will it take for the number of leopards to reach 100? Give your answers to an appropriate
degree of accuracy.

Give your answers to an appropriate degree of accuracy.

(Total 4 marks)

16. The diagram shows three graphs.

B
y
A

A is part of the graph of y = x.

B is part of the graph of y = 2x.

C is the reflection of graph B in line A.

Write down

(a) the equation of C in the form y =f (x);

(b) the coordinates of the point where C cuts the x-axis.

(Total 4 marks)

17. Initially a tank contains 10 000 litres of liquid. At the time t = 0 minutes a tap is opened, and liquid then
flows out of the tank. The volume of liquid, V litres, which remains in the tank after t minutes is given by

V = 10 000 (0.933t).

(a) Find the value of V after 5 minutes.

(1)

(b) Find how long, to the nearest second, it takes for half of the initial amount of liquid to flow out of the
tank.
(3)

(c) The tank is regarded as effectively empty when 95% of the liquid has flowed out.
Show that it takes almost three-quarters of an hour for this to happen.
(3)

(d) (i) Find the value of 10 000 – V when t = 0.001 minutes.

(ii) Hence or otherwise, estimate the initial flow rate of the liquid.
Give your answer in litres per minute, correct to two significant figures.
(3)
(Total 10 marks)
18. A population of bacteria is growing at the rate of 2.3% per minute. How long will it take for the size of the
population to double? Give your answer to the nearest minute.
(Total 4 marks)

19. Solve the equation 4x–1 = 2x + 8.

(Total 5 marks)

20. Solve the equations

x
ln =1
y
ln x3 + ln y2 = 5.
(Total 5 marks)

21. Let g(x) = log5│2log3x│. Find the product of the zeros of g.

(Total 5 marks)

22. Write ln (x2 – 1) – 2 ln(x + 1) + ln(x2 + x) as a single logarithm, in its simplest form.
(Total 5 marks)