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Math Studies
Review Booklet
1) The table below shows the relative frequencies of the ages of the students at Ingham
High School.
(a)
Age
(in years)
Relative
frequency
13
0.11
14
0.30
15
0.23
16
0.21
17
0.15
Total
1
If a student is randomly selected from this school, find the probability that
(i)
the student is 15 years old;
(ii)
the student is 16 years of age or older.
There are 1200 students at Ingham High School.
(b)
Calculate the number of 15 year old students.
2)
In each of the Venn diagrams, shade the region indicated.
(a)
AB
A
(b)
B
The complement of (A  B)
A
(c)
(d)
B
The complement of (A  B)
A
B
A
B
A  (B  C)
C
3) Let
= {positive integers less than 15};
X= {multiples of 2};
Y = {multiples of 3}.
(a)
Show, in a Venn diagram, the relationship between the sets , X and Y.
(1)
(b)
List the elements of:
(i)
XY
(1)
(ii)
X  Y.
(2)
(c)
Find the number of elements in the complement of (X  Y).
(2)
(Total 6 marks)
4)
A group of 25 females were asked how many children they each had. The results are
shown in the histogram below.
Number of Children per Female
10
9
8
7
6
5
Frequency
4
3
2
1
0
(a)
0
1
2
Number of Children
3
4
Show that the mean number of children per female is 1.4.
(2)
(b)
Show clearly that the standard deviation for this data is approximately 1.06.
(3)
(c)
Another group of 25 females was surveyed and it was found that the mean
number of children per female was 2.4 and the standard deviation was 2. Use
the results from parts (a) and (b) to describe the differences between the
number of children the two groups of females have.
(2)
5)
The universal set U is defined as the set of positive integers less than 10. The subsets A and
B are defined as:
A = {integers that are multiples of 3}
B = {integers that are factors of 30}
(a)
(b)
List the elements of
(i)
A;
(ii)
B.
Place the elements of A and B in the appropriate region in the Venn diagram
below.
U
A
B
6)
The sets
A, B and C are subsets of U. They are defined as follows:
U = {positive integers less than 16}
A = {prime numbers}
B = {factors of 36}
C = {multiples of 4}
(a)
List the elements (if any) of the following:
(i)
A;
(ii)
B;
(iii)
C;
(iv)
A  B  C.
(4)
(b)
(i)
Draw a Venn diagram showing the relationship between the sets U, A, B
and C.
(ii)
Write the elements of sets U, A, B and C in the appropriate places on the
Venn diagram.
(4)
(c)
From the Venn diagram, list the elements of each of the following
(i)
A  (B  C);
(ii)
(A  B);
(iii)
(A  B)  C.
(3)
7)
A committee U has three sub-committees: research R, finance F and purchasing P. No
member belongs to both finance and purchasing sub-committees. Some members
belong to both research and purchasing committees. All members of the finance subcommittee also belong to the research sub-committee.
Draw a Venn diagram, showing the relationship between the sets U, R, F and P.
8) 100 students were asked which television channel (MTV, CNN or BBC) they had
watched the previous evening. The results are shown in the Venn diagram below.
U
MTV
CNN
35
19
23
6
5
2
3
7
BBC
From the information in the Venn diagram, write down the number of students who
watched
(a)
both MTV and BBC;
(b)
MTV or BBC;
(c)
CNN and BBC but not MTV;
(d)
MTV or CNN but not BBC.
(Total 4 marks)
9) Calculate 3.7 × 16.22 – 500, writing your answer
(a)
correct to two decimal places;
(b)
(i)
correct to three significant figures;
(ii)
in the form a × 10k, where 1 ≤ a < 10, k 
10) Let
= {x : 1 ≤ x < 17, x 
.
}.
P , Q and R are the subsets of
such that
P = {multiples of four};
Q = {factors of 36};
R = {square numbers}.
(a)
List the elements of
(i)
(ii)
P  Q  R.
(2)
(b)
Describe in words the set P  Q.
(1)
(c)
(i)
Draw a Venn diagram to show the relationship between sets P, Q and R.
(2)
(ii)
Write the elements of
in the appropriate places on the Venn diagram.
(3
11) Let m = 6.0 ×103 and n = 2.4 ×10–5.
Express each of the following in the form a ×10k, where 1 ≤ a < 10 and k 
(a)
mn;
(b)
m
.
n
.