MASTERY OF SENIOR SECONDARY O’ LEVEL MATHEMATICS
ECz PrACTICE quEsTIONs
1. Shade the region (P ∩ R) ∪ Q on the Venn diagram in the Venn
diagram in the answer space
(2002 p1)
2. In a class of 40 pupils, 16 like Physics, 17 like Mathematics, 24 like
Chemistry and the rest like other subjects not stated. 5 like all the
three subjects, 4 like mathematics and Physics only, 6 like Physics
and Chemistry only while 3 like Mathematics and Chemistry only.
(2003 p2)
(i)
Illustrate this information on a clearly labeled Venn diagram
(ii)
Hence or otherwise find
(a) The number of pupils who like one subject only
(b) The number of pupils who do not like any of the three
subjects
3. Use the set notation to describe the shaded region in the Venn
diagram (2004 p1)
4. A survey was carried out on 100 pupils at Mwimbo High School to
determine their participation in sports. The results are shown in the
Venn diagram below
(2004 p2)
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A HIDDEN MATHEMATICAL GENIUS WITHIN REVEALED
(a) Given that 42 pupils play Volleyball, find the value of a and b
(b) If a pupil is selected at random from the group, what is the
probability that the pupil
(i)
Does not play football
(ii)
Play volleyball or football but not both
(c) What is the probability that the pupil either participate in only one
of these sports or is not involved at all?
(d) What is the probability that a pupil who plays volleyball also
plays basketball but does not play football?
5. At the show grounds, there were several traditional dances taking
place. Among these traditional dances were Siyomboka, Ngoma and
Kalela. 8 people watched all three dances, 34 watched Siyomboka
and Ngoma but not Kalela, 42 watched Ngoma and Kalela but not
Siyomboka, 24 watched Kalela and Siyomboka but not Ngoma, 84
watched Siyomboka only, y watched Ngoma only, (y - 2) watched
Kalela only and 32 did not watch any of these dances.
(2005 p2)
(i)
Illustrate the information on Venn diagram to show the
number of people in each region
(ii)
Given that 500 people went to the show grounds, calculate
(a) The value of y
(b) The number of people that did not watch Ngoma
6. It is given that n(E) = 50 and that A and B are two sets for which
n(A ∪ B)’ = 24, n(A) = 18 and n(B) = 14.
(2006 p1)
(a) Illustrate the information on a Venn diagram
(b) Find n(A ∩ B’)
7. Pupils at a city school were questioned about the type of transport
they use when going to school. Their responses were as shown below
(2006 p2)
(a) Find the value
of x
(b) Find
the
number
of
pupils
who
cycle only
(ii)
Given also that 43
(i)
Given that there
pupils use vehicles
were 80 pupils who
only, find the total
walk and 75 pupils
number of pupils at
who cycle
this school.
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MASTERY OF SENIOR SECONDARY O’ LEVEL MATHEMATICS
8. Given that n(A) = 18, n(B) = 14, n(A ∩ B) = 6 and n(A ∪ B)’ = 2.
Find n(A ∪ B)
(2007 p1)
9. A survey was conducted on 60 women concerning types of simcards
used in their cell phones for the past 2 years. Their responses are
given in the diagram below
(2007 p2)
(a) Given that 23 women have used Cellz simcards, find the values
of a and b
(b) How many women have used only two different simcards
(c) If a woman is selected at random, what is the probability that
(i)
She has no cell phone
(ii)
She has used only one type of simcard
simca
(d) How many women did not use MTN and Cellz simcards?
(e) How many women used either Celtel or MTN simcards but not
Cellz?
10. Use set notation to describe the shaded region in the diagram below
(2008 p1)
11. A survey carried out among school leavers in a certain town,
involving three institutions, showed that 118 applied to the
University of Zambia (UNZA), 98 applied to the Copperbelt
University (CBU) and 94 applied to the National Resources
Development College (NRDC). To increase the chances of selection,
42 applied
ied to UNZA and CBU, 24 to CBU and NRDC, 34 to UNZA
and NRDC and 8 applied to all the three institutions. (2008 p2)
(i)
Show this information on a Venn diagram
(ii)
Calculate the total number of school leavers who took part in
the survey
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A HIDDEN MATHEMATICAL GENIUS WITHIN REVEALED
12. In a particular year, 150 candidates applied for training as teachers of
Mathematics, Science and Home Economics at COSETCO. The
Venn diagram below illustrates the number of candidates for each
subject.
(2009 p2)
(i)
Given that 70 candidates applied for Science, find the value of
a and b
(ii)
How many candidates applied for two different subjects only
(iii)
How many candidates did not apply for Science or
Mathematics?
(iv)
How many candidates applied for either Science or Home
Economics but not Mathematics?
(v)
If a candidate is selected at random, what is the probability
that the candidate applied for one subject only?
13. At one college, a group of 25 students were asked which cell phones
service providers they subscribed to. The results are shown in the
Venn diagram below.
(2010 p2)
(i)
(ii)
Calculate the value of x
Given that G = {Glo}, R = {Rodgers} and D = {Du}. Find
(a) ∩(G ∩ R)
(b) ∩(D ∪ G’)
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MASTERY OF SENIOR SECONDARY O’ LEVEL MATHEMATICS
14. Shade A’∩(B ∪ C) in the Venn diagram
(2011 p1)
15. In a certain month, a survey was carried out on 250 high school
pupils to find out the number of pupils that bought Oranges (O),
Mangoes (M) and Lemons (L). Their responses were as shown in
Venn diagram below.
(2011 p2)
(i)
(ii)
Find the value of x
How many pupils bought Mangoes and Lemons but not
Oranges?
(iii)
How many pupils bought one type of fruit only?
16. A class of 41 girls takes History (H), Commerce (C) and Geography
(G) as optional subjects. The Venn diagram below shows their choice
distribution.
(2012 p2)
(i)
(ii)
Calculate the value x
Find
(a) ∩(H ∪ G)
(b) ∩(G’ ∩ H’)
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A HIDDEN MATHEMATICAL GENIUS WITHIN REVEALED
17. The diagram below shows three sets A, B and C
(2013 p2)
Given that ∩(A ∪ 𝐵 ∪ C) = 50, find
(i)
The value of x
(ii)
∩(A ∪ B)
(iii)
∩(B ∪ C)’
(iv)
∩(A’ ∩ C’)
18. The Venn diagram below illustrates the number of pupils who take
Mathematics (M), Physics (P) and Chemistry (C) at Imishila
Secondary School.
(2014 p1)
(a) Write an expression, in terms of x, for the total number of pupils
who take Physics.
(b) Given that 22 pupils take Physics and 20 pupils take
Mathematics, find the value of x and y.
19. Tokhozani sports club offers Squash (S), Badminton (B) and Tennis
(T). The Venn diagram below shows choices of the 73 members of
the club.
(2014 p2)
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MASTERY OF SENIOR SECONDARY O’ LEVEL MATHEMATICS
(i)
(ii)
Calculate the value of y
Find the number
mber of members who played Squash or Tennis
but not Badminton
(iii)
How many members played two different sports only?
(iv)
Find the number of members who played one sport only
20. The Venn diagram below shows the number of students who took
Business studies (B), Human Resources
R
(H) and Community
Development (C) at Mafundisho College. 100 students took these
three courses.
(2015 p2)
(a) Find
(i)
The value of x,
(ii)
The number of students who took Human Resources,
(iii)
∩(B ∩ C) ∩ H’
(iv)
∩(B ∪ C) ∩ H’
(b) If a student is chosen at random, what is the probability that the
student took
(i)
One course
(ii)
At least two courses?
21. At Gender Technical Secondary School, a group of 70 girls take
optional subjects as illustrated in the Venn diagram
diagra
below.
…....(2015 p2 gce )
(i)
(ii)
Calculate the value of x
Find the number of girls who take
(a) History only
(b) Commerce
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A HIDDEN MATHEMATICAL GENIUS WITHIN REVEALED
22. Using the Venn diagram in the answer space, shade the region
represented by B’ ∩ (A ∩ C)
(2016 p1 gce)
23. The Venn diagram below shows the results of a survey conducted at
Mayuka Clinic on patients who were attended on a particular day. Set
W represents those who complained of body weakness, set H those
who complained of Headache and set F those who complained of
Fever.
(2016 p2
gce)
(i)
Use the information on the Venn diagram to find the number
of people who complained of
(a) Body weakness only
(b) Body weakness and fever only
(c) Headache
(ii)
Calculate the percentage of those who complained of all the
three ailments.
24. Of the 50 villagers who can tune in to Kambani Radio station, 29
listen to news, 25 listen to sports, 22 listen to music, 11 listen to both
news and sports, 9 listen to both sports and music, 12 listen to both
news and music, 4 listen to all the three programs and 2 do not listen
to any programme
(2016 p2)
(i)
Draw a Venn diagram to illustrate this information
(ii)
How many villagers
(a) Listen to music only
(b) Listen to one type of programme only
(c) Listen to two types of prgramme only
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MASTERY OF SENIOR SECONDARY O’ LEVEL MATHEMATICS
25. Shade B’ ∩ (A ∩ C) in the Venn diagram in the answer space
26. The Venn diagram below shows tourists attractions visited by
students in a certain week (2017 p2 gce)
(i)
(ii)
Find the value of y, if 7 students visited Mambwe falls only
How many students visited
(a) Victoria falls but not Gonya falls
(b) Two tourist attractions only
(c) One tourist attraction only
27. A survey carried out at Kamulima Farming block showed that 44
farmers planted maize, 32 planted sweet potatoes, 37 planted cassava,
14 planted both maize and sweet potatoes, 24 planted both sweet
potatoes and cassava, 20 planted both maize and cassava, 9 planted
all the three crops and 6 did not plant any of the three crops.
(2017 p2)
(i)
Illustrate this information on a Venn diagram
(ii)
How many farmers
(a) Were at this farming block
(b) Planted maize only
(c) Planted two different crops
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A HIDDEN MATHEMATICAL GENIUS WITHIN REVEALED
28. Use set notation to describe the shaded region in the Venn diagram
below
(2018 p1 gce)
29. Use set notation to describe the shaded region in the diagram below
(2018 p1)
30. The diagram below shows how learners in a Grade 12 class at
Twaenda School travel to school. The learners use either buses (B),
cars (C) or Walk to school.
(2018 p2 gce)
(i)
(ii)
If 22 learners walk to school, find the value of x
How many learners use
(a) Only one mode of transport
(b) Two different modes of transport
~ 38 ~
MASTERY OF SENIOR SECONDARY O’ LEVEL MATHEMATICS
31. At Sambilileni College, 20 students study at least one of the three
subjects, Mathematics (M), Chemistry (C) and Physics (P). All those
who study chemistry also study mathematics. 3 students study all the
three subjects, 4 students study mathematics only, 8 students study
chemistry and 14 students study mathematics.
(2018 p2)
(i)
Draw a Venn diagram to illustrate this information
(ii)
How many students study
(a) Physics only
(b) Two subjects only
(c) Mathematics and Physics but not chemistry
32. The Venn diagram below shows the optional subjects that all the
grade 10 learners at Kusambilila Secondary School took, in a
particular year.
(2019 p2)
(i)
(ii)
(iii)
Given that 12 learners took music, find the value of x
How many learners were in grade 10 this particular year?
Find the number of learners who took
(a) One optional subject only
(b) Two optional subjects only
33. Use set notation to describe the shaded region in the diagram below
(2020 p1 gce)
34. The Venn diagram below shows the number of students in each of
the three courses at a University.
(2020 p2 gce)
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A HIDDEN MATHEMATICAL GENIUS WITHIN REVEALED
(i)
Given that there were 25 students altogether, find the value of
x
(ii)
How many students studied
(a) Mathematics and chemistry only,
(b) One course only,
(c) Chemistry and Physics but not Mathematics
35. Of the 115 students who attended an end of year party, 74 took Fanta,
93 took sprite, 87 took coke, 61 took Fanta and sprite, 71 took sprite
and coke, 60 took Fanta and coke and 50 took all the three drinks.
(i)
Illustrate this information on a Venn diagram
(ii)
How many students took
(a) None of the drinks
(b) Fanta and sprite but not coke
(c) At least two different drinks
36. At a certain secondary school, all the learners in Grade twelve take at
least one of the following optional subjects: French (F), Home
Economics (HE) and Geography (G). 20 learners take French only,
25 learners take Home Economics only and 22 learners take
Geography only. Furthermore, 9 learners take French and Geography,
12 learners take French and Home Economics, 15 learners take
Geography and Home Economics and 4 learners take all the three
subjects. (2021 p1)
(i)
Illustrate this information in a Venn diagram.
(ii)
How many Grade twelve learners
(a) are at this school,
(b) take Home Economics and Geography but not French,
(c) take two optional subjects only?
~ 40 ~
MASTERY OF SENIOR SECONDARY O’ LEVEL MATHEMATICS
Solutions
1.
5.
6.
2.
3.
4.
(a) 1 + 5 + 10 = 16pupils
(b) 6pupils
A n(B U C)’
.
(a) a + 2 + 10 + 12 = 42
a + 24 = 42
a = 42 – 24
a = 18
b + 15 + 27 + 42 + 11 = 100
b + 95 = 100
b=5
60 3
(b) P(not ftball) =
=
100 5
(ii) P(V or F
27
 0 . 27
100
only)
7.
=
8.
65
13

100
20
10
1
(d) P(V and B only) =

100 10
(c) P(one or not) =
.
~ 41 ~
(a) 222 + 2y = 500
2y = 500 – 222
2y = 278
y = 139
(b) 84 + 24 + 137 + 32 = 277people
.
(a)
(b) n(A n B’) = 12
.
(a) x + 16 + 32 + 21 = 80
x + 69 = 80
x = 80 – 69
x = 11
(b) 75 – (16 + 32 + 11)
75 – 59 = 16 pupils
(c) 80 + 16 + 11 + 43 = 150pupils
n(AUB) = n(A) + n(B) – n(AnB)
= 18 + 14 – 6
= 32 – 6
= 26
A HIDDEN MATHEMATICAL GENIUS WITHIN REVEALED
9.
(a) a + 4 + 2 + 3 = 23
a + 9 = 23
a = 23 – 9
a = 14
23 + 14 + 10 + 8 + b = 60
55 + b = 60
b = 60 – 55
b=5
(b) 3 + 4 + 5 = 12women
(c) (i)
P(no cellphone)
8
2

60 15
(ii)
(iii)
(iv)
(v)
=
13. .
(i)
(ii) P(one type of simcard) =
19
30
(ii)
(d) 14 + 4 + 14 + 5 + 10 + 8 =
55women
(e) 14 + 5 + 10 = 29women
10. A n(BUC)
11. .
70 + 40 + b + 35 = 150
b = 150 – 145
b=5
10 + 30 + 5 = 45candidates
35candidates
15 + 30 + 35 =
80candidates
P(one subject only) =
90
3

150 5
10 + 3 + x + 2 = 25
15 + x = 25
X = 25 – 15
x = 10
(a) n(G n R) = 3
(b) n(D U G’) = 2 + 10 = 12
14.
(i)
(ii)
12. .
(i)
15. .
(i)
118 + 40 + 16 + 44 =
218school leavers
a + 10 + 15 + 30 = 70
a + 55 = 70
a = 70 – 55
a = 15
(ii)
(iii)
~ 42 ~
22 + 23 + 16 + 51 + 13 + 20 +
60 + x = 250
205 + x = 250
x = 250 – 205
x = 45
13pupils
22 + 16 + 20 = 58pupils
MASTERY OF SENIOR SECONDARY O’ LEVEL MATHEMATICS
16.
(i)
(ii)
17. .
(i)
(ii)
2x + x + 1 + x + x + 15 = 41
5x + 16 = 41
5x = 41 – 16
5x = 25
X=5
.
(a) n(HUG) = 10 + 5 + 20 =
35
(b) n(G’ n H’) = 5 + 1 = 6
(ii)
(iii)
(iv)
20. .
(a) .
(i)
10 + x + 3 + 3x + 5 = 50
4x + 18 = 50
4x = 50 – 18
4x = 32
x=8
n(AUB) = 10 + 8 + 3 + 24 =
45
n(B U C)’ = 10 + 4 = 14
n(A’ n C’) = 3 + 4 = 7
(ii)
(iii)
(iv)
(b) .
(i)
(iii)
(iv)
18. .
(a) X + 3 + x + 5
2x + 8
(b) 2x + 8 = 22
2x = 22 – 8
2x = 14
X=7
3y + 7 + 3 + 2y = 20
5y = 20 – 10
5y = 10
y=2
19. .
(i)
20 + 13 + 10 + 4 + 5 + y + 2y
= 73
52 + 3y = 73
3y = 73 – 52
(ii)
21. .
(i)
(ii)
~ 43 ~
3y = 21
y=7
20 + 4 + 14 = 38members
4 + 7 + 10 = 21 members
20 + 13 + 14 = 47members
5 + 15 + 10 + 10 + 5 +
15 + x = 100
70 + x = 100
X = 100 – 70
x = 30
10 + 10 + 5 + 15 = 40
students
n(B n C) n H’ = 15
n(B U C) n H’ = 15 + 15
+ 30 = 60
P(one course)
60
3

100 5
P(at least 2)
40
2

100 5
3x – 5 + x + 9 + 2x = 70
6x + 4 = 70
6x = 70 – 4
6x = 66
x = 11
.
(a) 3(11) – 5 = 33 – 5
28girls
(b) 11 + 9 + 22 = 42girls
=
=
A HIDDEN MATHEMATICAL GENIUS WITHIN REVEALED
(c) 21students
27. .
22.
23. .
(i)
(ii)
.
(a) 18people
(b) 13people
(c) 20people
14
 100  25 %
56
(i)
(ii)
.
(a) 44 + 3 + 15 + 2 + 6 =
70farmers
(b) 19farmers
(c) 11 + 5 + 15 =
31farmers
28. C ’ n(A U B)
29. A n(B U C) ’
30. .
(i)
x + 4 + 3 + 7 = 22
x + 14 = 22
x = 22 – 14
x=8
(ii)
.
(a) 14 + 7 + 7 =
28learners
(b) 2 + 4 + 3 = 9learners
31. .
24. .
(i)
(ii)
.
(a) 5villagers
(b) 10 + 9 + 5 =
24villagers
(c) 8 + 7 + 5 = 20villagers
25.
26. .
(i)
(ii)
(i)
(ii)
2y + 1 = 7
2y = 6
y=3
.
(a) 8students
(b) 7students
~ 44 ~
.
(a) 6students
(b) 7students
(c) 2students
MASTERY OF SENIOR SECONDARY O’ LEVEL MATHEMATICS
32. .
(i)
(ii)
(iii)
x – 3 + x + 2 + x – 2 = 12
3x – 3 = 12
3x = 12 + 3
3x = 15
X=5
59 + 2 + 7 + 3 + 43 = 114
learners
.
(a) 59 + 7 + 43 = 109
learners
(b) 2 + 3 = 5 learners
(i)
(ii)
.
(a) 3students
(b) 11students
(c) 11 + 10 + 50 + 21 = 92
students
36. .
33. P ’ n(Q n R)
34. .
(i)
6+5–x+7+4–x+x+3
– x + 2 = 25
27 – 2x = 25
2x = 25 – 27
x=1
(ii)
.
(a) 4students
(b) 15students
35. 2students.
(I)
(II)
~ 45 ~
.
(a) 95 learners
(b) 11 learners
(c) 24 learners
A HIDDEN MATHEMATICAL GENIUS WITHIN REVEALED
3. ALGEbrA
This is the branch of mathematics that uses letters to represent numbers.
ALGEbrAIC TErMINOLOGIEs




Expression – combination of algebraic terms with mathematical
operators (+, -, ×, ÷)
Term – a single product number with two factors, a numerical factor
and a letter factor.
Coefficient – a numerical factor in a term.
Variable – a letter factor in a term.
3.1. sIMPLIfyING ALGEbrAIC ExPrEssIONs
ADDITION AND subTrACTION
Only terms with same variables can be added/subtracted
ExAMPLE 1
Simplify the following algebraic expressions
(c) 2t2 – t + 3t2 + 4t
(a) 5x + 2y – x – 2y
(b) -3x + 2y + x – y
Solutions
(a) 5x – x + 2y – 2y
4x
(b) -3x + 2y + x – y
-2x + y
2
(c) 2t + 3t2 –t + 4t
5t2 + 3t
~ 46 ~