x

1. Given the function f(x) = 2 3 for 2 x 5,

(a) find the range of f;

(4)

(b) find the value of x given that f(x) = 162.

(2)
(Total 6 marks)

2. In an experiment researchers found that a specific culture of bacteria increases in number

according to the formula

t
N = 150 2 ,

where N is the number of bacteria present and t is the number of hours since the experiment
began.

Use this formula to calculate

(a) the number of bacteria present at the start of the experiment;

(b) the number of bacteria present after 3 hours;

(c) the number of hours it would take for the number of bacteria to reach 19 200.

(Total 4 marks)

IB Questionbank Mathematical Studies 3rd edition 1

 x
3. The following diagram shows part of the graph of an exponential function f(x) = a , where
x .
y
f(x )

0 x

(a) What is the range of f ?

(b) Write down the coordinates of the point P.

(c) What happens to the values of f(x) as elements in its domain increase in value?

(Total 4 marks)

IB Questionbank Mathematical Studies 3rd edition 2

 2
4. The diagram shows the graph of y = x 2x 8. The graph crosses the x-axis at the point A, and
has a vertex at B.
y

x
A O

2
(a) Factorize x 2x 8.

(b) Write down the coordinates of each of these points

(i) A;

(ii) B.

(Total 4 marks)

2
5. (a) Solve the equation x 5x + 6 = 0.

2
(b) Find the coordinates of the points where the graph of y = x 5x + 6 intersects the x-axis.

(Total 4 marks)

IB Questionbank Mathematical Studies 3rd edition 3

6. Under certain conditions the number of bacteria in a particular culture doubles every 10 seconds
as shown by the graph below.

5
N um ber
of 4
b a c te ria
3

0 10 20 30
T im e
(se c o n d s)

(a) Complete the table below.

Time (seconds) 0 10 20 30
Number of bacteria 1

(b) Calculate the number of bacteria in the culture after 1 minute.

(Total 4 marks)

x
7. The graph below shows the curve y = k(2 ) + c, where k and c are constants.

IB Questionbank Mathematical Studies 3rd edition 4

 y

10

0 x
6 4 2 2 4

10

Find the values of c and k.

(Total 4 marks)

2
8. The diagram below shows part of the graph of y = ax + 4x 3. The line x = 2 is the axis
of symmetry. M and N are points on the curve, as shown.
y

x
0

x = 2

(a) Find the value of a.

(b) Find the coordinates of

(i) M;

(ii) N.

(Total 4 marks)

IB Questionbank Mathematical Studies 3rd edition 5

 2
9. (a) f(2) = 2 3 (M1)
2
= 9 (0.222) (A1)
5
f(5) = 2 3
= 486 (A1)
2 2
9 , 486
Range 9 f(x) 486 OR (A1) (C4)

Note: Award (M1) for correct substitution of 2 or 5 into f(x),

(A1)(A1)for each correct end point.

x
(b) 2 3 = 162 (M1)
x=4 (A1) (C2)
[6]

0
10. (a) N = 1502 = 150 (A1) (C1)
3
(b) N = 1502 = 1200 (A1) (C1)

t
(c) 19200 = 1502 (M1)
t
128 = 2
7=t (A1) (C2)
[4]

IB Questionbank Mathematical Studies 3rd edition 6

 +
11. (a) (A1)

(b) P(0, 1) (A1)

(c) Decreases towards 0 or 0 (A1)(A1)

Note: Award (A1) for Decrease, and (A1) for 0.

Marks awarded at examiners discretion.

[4]

12. (a) (x + 2)(x 4) (A1)

(b) (i) (2, 0) (A1)

(ii) (1, 9) (A1)(A1)

[4]

2
13. (a) x 5x + 6 = 0
(x 2)(x 3) = 0 (A1)
x=2 (A1)
x=3 (A1)

(b) (2, 0)
(3, 0) (A1)

Notes: Follow through from part (a). Both must be correct and
written as coordinates for (A1)
[4]

14. (a)

Time (seconds) 0 10 20 30
Number of bacteria 1 2 4 8 (A2) (C2)
Note: Award [ mark] for each correct entry (round up)

6
(b) N=2 (M1)

Note: Award (M1) for any correct method

IB Questionbank Mathematical Studies 3rd edition 7

 = 64 (A1) (C2)
[4]

IB Questionbank Mathematical Studies 3rd edition 8

15. c = 10 (asymptote of graph) (M1)(A1)
1
0 = k(2 ) 10 2k = 10 (M1)
k=5 (A1)

OR

k + c = 5 (M1)
2k + c = 0 (M1)
Therefore, k = 5 (A1)
c = 10 (A1)
[4]

b

16. (a) x = 2a
4

2 = 2 a (M1)
a = 1 (A1)

(b) Note: Answers to (b) must be written as coordinates.

(i) M(0, 3) (A1)

2
(ii) y=12 +423
=1
N is (2,1) (A1)
[4]

IB Questionbank Mathematical Studies 3rd edition 9