BUKASA HIGH SCHOOL

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KIRINYA
MATHEMATICS DEPARTMENT
0779699579/0706500525
456/1 MATHEMATICS
O’LEVEL MATHEMATICS SEMINAR HELD AT BUKASA HIGH SCHOOL
FOR NEW LOWER SECONDARY CURRICULUM
ON SATURDAY 28TH JUNE 2025
FORMAT OF THE PAPER:
SECTION A: This will comprise of two items and they are compulsory.
SECTION B: This comprises of part 1 and part II each having two questions and a learner answers one question from each part.

ITEM Area of Topics covered

construct
SECTION A: Compulsory

Item Numbers • Number bases

1
• Working with integers

• Fractions, percentages and decimals

• Numerical concept 1 and 2

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 • Ratios and proportions
Item Patterns • Sequences and patterns
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2 and
algebra • Equations of lines and curves

• Algebra 1 and 2

• Mapping and relations

• Inequalities and regions

• Rectangular cartesian plane

• Simultaneous equations

• Linear programming

• Loci
SECTION B

PART I (choose one question)

Item Data and • Data collection and presentation

3 and probability
4 • Graphs

• Set theory

• Matrices

•
Probability
PART II (choose one question)

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 Item Geometry • Geometric construction
5 and and
• Bearings
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6 measures
• General and angle properties of geometric figures

• Reflection

• Business arithmetic

• Time and tables

• Similarities and enlargement

• Circle properties

• Rotation

• Length and area properties of two-dimensional geometrical figures

• Nets, areas and volumes of solids

• Trigonometry 1 and 2

• Matrix transformation
• Lines and planes in 3 dimensions

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 THEME: NUMBERS By the end of the first week of the term, two-thirds of the student
body had settled their fees. In the subsequent week, an additional 100
ITEM 1
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students paid their fees, resulting in a significant increase in the

Your father upon his retirement deposited sh. 10,500,000 in a joint proportion of students who had paid their fees, reaching three-
family account to support you and your two brothers, Moses and quarters of the total student population.
Joshua. You being the eldest will receive 40% while Moses and TASK
Joshua share the rest in a 3:4 ratio respectively.
a) Guide the school management to ascertain the land area,
You withdraw at different intervals: you every 8 days, Moses every 12 ensuring a surd form solution with rationalized numerals and
days and Joshua every 15 days. The last time you all met at the bank simplified radicals.
was 5th May. As per the family tradition, when you meet at the bank b) Ascertain the precise number of students in the school.
on the same day, you visit your father together. Now there are two c) Guide the school administration on whether the construction can
concerns, Moses has an exam on 6th September and Joshua has a trip be undertaken, with supporting reasons.
on 4th August. They wonder if they will have a common date, allowing ITEM 3
you to visit your father together again.
You and your friend Samantha started a partnership business and
TASK agreed to deal in poultry feeds. You contributed Sh. 750,000 while
1
(a) Compute the amount each of you will receive. Samantha contributed Sh. 900,000. You both agreed to save of the
5
(b) Determine the next date all the three of you will meet at the bank. gross profit in a joint account and further use 20% of the remainder to
(c) Will that date be suitable for all of you to visit your father? buy more requirements for business and there after share the net profit
ITEM 2 in the ratio of your initial contributions.
At the end of the 1st year of operation, the business made gross profits
At the beginning of term two, your school faced a pressing issue: - a amounting to Sh. 3,160,000 and Samantha deposited your share on
shortage of dormitory space for students. To address this sleeping
your account but you were not sure if it was the actual amount you
accommodation deficit, the school's management developed a plan to were supposed to get.
construct new hostels with a rectangular design, where the length is √3
You sent Samantha to withdraw some money from your phone but
times the width, resulting in a perimeter of (14+6√3) units. To fund
you had forgotten the pin however you remembered that you had
this project, the administration required a 97% collection rate of
saved it as 31 in decimal base which is supposed to be converted to
school fees from students
base three to get the actual pin.

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 TASK from each double room and sh 60,000 monthly from each single room.
a) Determine the actual amount of money that was to be He wants less than 6 double rooms. He also wants to construct at most
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deposited on your account. twice as many single rooms as double rooms. It will cost him sh12
b) What is the actual pin in order to make the withdraw from your million to construct each double room and sh10 million to construct
phone? each single room.
THEME: PATTERNS AND ALGEBRA TASK:
(a) Write down mathematical statements which relate the number
ITEM 4
of double rooms and single rooms.
Henry is planning for a small birthday party for his son and wishes to (b) With the aid of a cartesian plane, advise Mr. Orute on how to
spend sh. 98,000 to buy 5 kg of meat and 7 kg of rice. However, some maximize monthly rent income.
guests that he had invited communicated that they would not make it
ITEM 6
for the party. Therefore, he had to reduce both quantities by 2 kg
hence leading to a reduction in his expenditure by sh. 32,000. When Ichuli is starting a baking business, selling cakes and cookies. To
Henry went to the market, there was an 80% increase in the price of estimate profits, he consulted a friend in the industry. The friend
each kilogram of meat and an offer given to him on rice as 25% shared data from their own experience: Initial phase: 40 cakes, 30
discount on each kilogram of rice. cookies, total profit UGX 29,000; Later phase: 50 cakes, 20 cookies,
total profit UGX 31,000.
Tasks:
Ichuli aims to start by producing at least 120 items (cakes and cookies
a) Help Henry to estimate the initial price of meat and the rice combined). Since cakes sell more, he wants to make at most 80 cakes
per kilogram. and at most 60 cookies. He needs to determine the optimal number of
b) What amount does he pay in order for the party to be a cakes and cookies to produce initially.
success?
TASK;
ITEM 5
a) What are the expected earnings from each cake and cookie,
Mr. Orute is retired teacher from a government school. The
based on his friend's experience?
government has been deducting NSSF saving for the years he has
b) (i) What mathematical inequalities are making decision-making
taught in this government school and he is now having 120 million on
hard for Ichuli?
his account. He wants to use his savings to construct single and double
(ii) Use the inequalities to help him decide on the highest
rooms for rentals from which he plans to collect a minimum of sh
number of cakes and cookies he can start with.
540,000 monthly. He intends to charge rent of sh 90,000 monthly
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 studies only, and 28 students offer both mathematics and religious
education. The number of students offering both computer studies and
DATA AND PROBABILITY
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mathematics only is twice the number of students offering all the three
ITEM 7 subjects. Students offering religious studies only are half the number
of students offering all the three subjects. The administration is
Solomon sells yoghurt in schools and the yoghurt is of three flavours; missing data on students who offer none of the three subjects, those
chocolate, vanilla and straw berry. He wants to introduce the same who offer religious studies, and those who offer mathematics only.
business in your school. He knows that for business to succeed, the The are also planning to create a new stream either for students
probability that the learners will take at least two of the flavours offering mathematics only, or for those offering computer studies only
should be greater than 0.5. or religious education; depending on which group is larger, to help
He is uncertain of the business success in your school. The study you them focus more during revision,
have carried out in your school indicates that 158 learners like TASK
chocolate flavours, 130 like vanilla, and 188 like straw berry. a) Determine the number of students who offer;
40 learners like chocolate and vanilla, 80 like straw berry and vanilla, (i) All the three subjects
88 like chocolate and straw berry and 10 like none of the three (ii) Religious education
flavours. (iii) Do not offer any of the three subjects
The number of learners who like all the three flavours is equal to the b) Which stream will the school administration need to create?
number of those who like chocolate and vanilla only. Give reason
TASK:
Using the data from your study, do you advice Solomon to introduce
the business in your school? Give a reason for your advice.

ITEM 8

The school administration is conducting a survey to find missing

information and make decisions on the available data. In a class of
200 students, 128 students offer mathematics, some offer religious
education and 92 offer computer studies. Two students offer computer
studies and religious education only, 30 students offer computer

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ITEM 9

A parent intends to make shopping of scholastic materials for his children who are going back to school for a new term by the names of Jane, Mary
and Darin. They budgeted as below basing on the list of requirements that they were given by their class teachers. - Jane: 6 exercise, 3 pencils, 2
Graph books, 3 pens
- Mary: 3 pencils, 1 Graph book, 6 exercise books, 3 pens
- Darien: 2 Graph books, 4 exercise books, 3 pencils and 5 pens
At that time the prices were 1 Graph book shs2000, 1 pencil shs100, 1 exercise book shs1,500 and 1 pen shs500. On reaching school they found out
that the canteen manager had increased prices of the
items by 10 % and also, they found out that the school administration had decided that on each item listed, they should increase the number for each
by 2 since the school administration had decided that the students extend by two weeks when the term ends in order to compensate for the time
students had lost the previous term.
Before leaving their home, they were given by their father shs 200, shs 800 so that they can finish the clearing process at school and then
afterwards they share equally the remaining money to be used as their pocket money.
TASK
a) Assuming they were to buy the items before going to school, using matrices help the father to determine how much he would give to each child.
b) By using matrices determine how much each child paid to the canteen attendant in order to acquire the items. Help the children determine how
much each share shared as pocket money after buying the items from the school canteen

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ITEM 10 and Capacity Building program conducted a survey, yielding

the following findings; Among the 80 rice farmers surveyed,
To enhance the yields of Rice, Beans, Sugarcane, and Peas in 45 also grow beans, 60 cultivate sugarcane, and 5 focus solely
Iganga district, the Ministry of Agriculture's Farmer Training
on peas and rice. Additionally, 5 farmers dedicate their land The ministry needs to calculate the total cost of tractors for
solely to rice. The number of farmers who grow beans, farmers, based on the number of tractors needed for each
sugarcane, peas, and rice is equal to those who grow peas, group, with each tractor costing UGX 68,000,000.
sugarcane, and rice. Moreover, the farmers who cultivate rice,
TASK
and sugarcane only are equal in number to those who grow
rice, peas, and beans, and are 5 fewer than those who grow all a) Assist the ministry in determining:
four crops. (i) The total number of farmers cultivating all four
The ministry plans to provide support to these farmers as crops
follows: (ii) The number of farmers growing only three crops
(iii) The chance of selecting a farmer who grows only
• A farmer who cultivates all four crops (beans, sugarcane, two crops in Iganga district
peas, and rice) will receive a package consisting of 4 (iv) The likelihood of selecting a farmer who does not
tractors and a cash grant of UGX 3,000,000. grow Peas
• One who plants only three crops will receive 3 tractors b) Set the total funding required for the ministry's farmer
and UGX 2,000,000. support initiative.
• A farmer who grows two crops only will receive 2
tractors and UGX 1,500,000.
• For a single crop will receive 1 tractor and UGX
1,000,000.
This support aims to motivate farmers to diversify their crops
and boost their productivity.

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 ITEM 11 most effective battery replacement strategy to the
director
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To address concerns about bulb durability, a LED bulb factory

(ii) Elaborate on the reasoning that led to your
in Uganda, conducted a thorough test on a random sample of
conclusion in a) i)
50 bulbs. Their experts carefully selected and examined these
b) (i) Develop a graphical display to illustrate the data,
bulbs, yielding the following results (rounded to the nearest
allowing the director and their team to estimate the
minute):
median, visualize and analyze the information
423 369 387 411 393 (ii) Identify the target battery lifespan for
manufacturing, as recommended by the director.
394 405 369 372 410
(iii) Analyze the graph and explain the situation,
371 377 389 409 392 backing your argument with data and logical
408 409 396 431 391 reasoning.
c) Aid the manager in recognizing the chance of selecting a
431 401 363 391 405 battery with a lifespan greater than or equal to the median
382 396 381 438 422 value.
400 381 399 415 428 THEME: GEOMETRY AND MEASURES
422 397 399 401 398 ITEM 12:
396 372 410 419 386 Four tourists plan to visit sites A, B, and C in Uganda. Their
390 362 373 391 402 journey begins in Entebbe, from where they will travel 400 km
on a bearing of 060° to reach site A, then 600 km due east of
The director has decided to withdraw batteries with a life equal
site A to site B. From site B, they will travel 700 km on a
to or less than the average lifespan of the tested samples and
bearing of 240° to site C. Finally, they will fly directly from
has directed the experts to manufacture only batteries that
site C back to Entebbe. They’ve budgeted UGX 12,000,000 for
achieve at least 99% of the median life of the 50 tested
transport, and the private plane costs $35 per 100 km per
batteries.
person. According to the latest forex rates at a local bureau, 1
TASK US dollar equals 3,700 Ugandan shillings. The tourists wonder
if their budget will cover the trip. Additionally, a weather
a) (i) Organize the data into intervals of 10 using a warning advises flights from site C to Entebbe to use a bearing
statistical table and analyze the trends to recommend the

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 below 320° due to unfavourable conditions beyond that b) Identify the type of triangle represented by the
direction. Pilots have been told to choose a bearing below 320° triangular garden and give a reason for your answer.
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for safety. c) Determine the amount of money needed to buy the

TASKS: pavers.
(a) Find out if the transport budget will be enough for the ITEM 15
whole journey.
(b) Advise them if the bearing they will set from site C to Mugu Moja who is new employee of a company has just learnt
Entebbe will be safe for them to use that direct route. that his monthly income tax is UGX 1,102,000. He wants to
ITEM 14 buy a used car priced at USD 8,900. One option is to pay cash
and get a 5% discount, while the other involves paying a
Your brother wants to design a children’s playground. The deposit of UGX 10,000,000 and covering the balance over six
playground will have a triangular garden and a circular fence months with monthly equal instalments of UGX 4,500,000. He
around the garden. The whole two sides of the rectangular is also not sure of how to arrive at his actual take-home pay
garden will measure 50m and 70m, and the angle between due to PAYE.
them will be 450. Assume the exchange rate is: 1 USD = UGX 3,700
PAYE Taxation Schedule
Your brother also wants to construct a circular fence around Monthly taxable income Tax rate
the garden such that the circular fence perfectly touches the UGX 0 – 235,000 0%
three vertices of the triangular garden. Your brother intends to UGX 235,001 – 335,000 10%
put pavers in the region outside the triangular garden but UGX 335,001 – 410,000 20%
inside the circular fence. Each square meter of pavers costs
UGX 410,001 and above 30%
shs 35, 000.
Task
Your brother needs help in identifying the type of triangle
a) Compare the payment options clearly showing the financial
represented by the triangular garden, coming up with an
difference.
accurate design of the playground as well as the cost of the
b) Estimate the Mugu Moja’s monthly net pay.
pavers.
c) If he is to save 60% of his monthly net income, how many
Task: months would it take in order to purchase the car?
Help your brother to:
a) Construct an accurate design of the children’s
playground.

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 ITEM 16 ITEM 17
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A man took out a loan of 5,000,000 Ugandan shillings from Moses, an employee at a gift box manufacturing company, has
the bank at a compound interest rate of 20% and was supposed been tasked with designing a foldable gift box with square
to repay it within 3 years. However, with the deadline faces and a capacity of 2744 cubic centimeters. He's struggling
approaching, he has yet to raise the necessary amount. to create a sketch that will guide him in arranging the faces of
the box to fold and close, as well as determining the
He decides to sell one of his plots of land for 14,000,000
dimensions of each cardboard piece to cut and join to achieve
Ugandan shillings to raise the funds to pay off the loan.
the desired outcome.
However, the buyer negotiates a 10% discount on the price. If
he accepts the offer and sells the land, the broker will charge a He normally receives a monthly salary of 300,000 Ugandan
5% commission on the sale price. Additionally, the LC1 is shillings with no additional allowances. However, the
requesting 350,000 Ugandan shillings to sign the agreements. company will start deducting taxes from his pay every month,
He is uncertain whether the remaining amount will be and he needs to determine his new take-home pay. The
sufficient to cover the loan he needs to repay. company will follow the tax brackets shown below:
He needs to pick up the child from school by 5:00 pm. Monthly taxable income (Ugx.) Rate
(%)
However, it's currently 4:45 pm, and he's just leaving the
broker's house, which is located a certain distance away from 0–100,000 0
the school (as shown below). Considering he drives at an 100,001–200,000 5
average speed of 50 km/h, he's unsure if he'll arrive on time.
200,001–300,000 10
He needs to deliver the gift box to the customer and charges a
TASK: delivery fee based on the amount of fuel used. His motorcycle
a) (i) What is the outstanding loan amount he needs to settle uses 0.035 liters of fuel per kilometer, and he travels at an
with the bank? average speed of 20 meters per second. With fuel priced at
(ii) Will he be able to pay off the loan if he sells the 5,000 Ugandan shillings per liter, he wants to calculate the
land? delivery fee. According to the customer, the journey will take
b) Will he arrive at the child's school on time? 45 minutes.
TASK:

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 a) Help Moses develop a sketch outlining the specifications c) What is the total amount of income tax Jane will
he needs. deduct from her employees?
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b) What is the new salary Moses will be receiving monthly? d) How much will she pay each of the employees as net
c) How much will Moses charge the customer for delivery? salary?

ITEM 18

Jane has a room that she plans to use as a hair salon. She has a
wall paper of 4.7m2. The wall she plans to cover with the paper
Practice makes mathematics easier.
is 240 cm by 300 cm. She knows the wall paper she has is not
enough. So she needs your advice on how much more she
should buy. A roll of 10 m2 of wall paper is sold at UGX
32,000. She also plans to use three employees and pay each of
her employees a gross monthly salary of UGX 350,000. The
gross monthly salary includes a non- taxable transport
allowance of UGX 90,000. However, before she pays them,
she has to deduct income tax as a requirement by the tax
authority. The tax bands are shown the table below.

Monthly taxable income Tax rate (%)

(shillings)
0-100,000 0
100,001-200,000 5
200,001-300,000 15

Jane is having difficulty in using the tax bands.

Task:
a) How many more square meters of wall paper should
Jane buy?
b) How much will she pay for the extra wall paper?

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