UGANDA NATIONAL EXAMINATIONS BOARD

UGANDA CERTIFICATE OF EDUCATION

MATHEMATICS
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2 �����
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Section A will comprise of two items and section B comprises of Part 1 and Part 2 each having two
questions and a learner answers on question from that part.
ITEM AREA OF CONSTRUCT TOPICS COVERED
SECTION A : Compulsory
Item One Numbers Number Bases
Working with Integers
Fractions, percentages and
decimals
Numerrical Concept 1 and 2
Ratios and Proportions

Item Two Patrens and Algebra Sequences and Patterns

Equation of line and curves
Algebra 1 and 2
Mapping and relation
Inequalites and regions
Equations of straight line
Rectangular Cartesion plane
Simultaneous Equations
Linear programming
Loci

SECTION B
Part 1 (Choose one question)
Item 3 Data and Probability Data Collection and
Item 4 presentation
Graphs
Set theory
Matrices
Probability
Part 2 (Choose one question)
Item 5 Geometry and Measures Geometrical Construction
Item 6 Bearings
General angle properties
Reflection
Business Arithemetics
Time and timetables
Similarities and enlargement
Circles
Rotation

Kyewalabye Trevor William

 Sample Question Paper
SECTION A
(Numbers, Patterns and Algebra)
Item 1 Fractions, percentages and decimals
Item 2 Integers , Fractions, percentages and decimals
Item 3 Linear Programming
Item 4 Linear Programming
Item 5 Linear Programming
Item 6 Number Bases, Ratios and Proportions
Item 7 Algebra 1 &2 , Simutaneuos Equations
Item 8 Algebra 1 &2 , Simutaneuos Equations
SECTION B
PART 1
Item 9 Data Collection and Presentation
Item 10 Data Collection and Presentation
Item 11 Probability
Item 12 Graphs
Item 13 Graphs
PART II
Item 14 Geometrical Construction
Item 15 Geometrical Construction
Item 16 Bearings
Item 17 Bearings
Item 18 Business Arithemetics
Item 19 Business Arithemetics
Item 20 Geometrical Construction
SECTION A
ITEM 1
The new farm manager who started work at Sugar Corporation of Uganda
Limited's Lugazi farm on Monday was unable to access the farm store,
which held critical records, due to password–protected locks. He reached
out to the farm owner, who replied with a message revealing that the
password was a two-digit base eleven numeral. The pin had the
characteristic that the total of its digits equaled 17, with the first digit
being 4 greater than the second. He gained entry to the store and obtained
access to the records, revealing that the farm had a sizable amount of land
available for crop cultivation. He intended to allocate 14% of the land for
corn, 30% for wheat, and the remainder for soya beans, with a specific
focus on planting corn on 42 – acres. Meanwhile, he discovered that his
12 farm workers could cultivate 15 acres every 4 days.
TASK
a) Educate the new manager on the pin’s decimal representation.
b) Ascertain the farm’s total land area and the remaining acres suitable
for soybean growth.
c) Help the farm manager understand how long it will take workers to get
the land ready for planting.

ITEM 2
At the beginning of term two, Kumasi secondary school's administration
faced a pressing issue: - a shortage of dormitory space for students. To
address this sleeping accommodation deficit, the school's management
developed a plan to construct new hostels with a rectangular design,
where the length is √3 times the width, resulting in a perimeter of
(14+6√3) units. To fund this project, the administration required a 97%
collection rate of school fees from students.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 students paid their fees, resulting in a
significant increase in the proportion of students who had paid their fees,
reaching three-quarters of the total student population.
TASK
a) Guide the school management to ascertain the land area, ensuring a
surd form solution with rationalized numerals and simplified radicals.
b) Ascertain the precise number of students in the school.
c) Guide the school administration on whether the construction can be
undertaken, with supporting reasons.

ITEM 3
Mr. Alex, a seasoned bus driver at the New Taxi Park, was hired to
transport 377 students for a field trip from Mukono to Entebbe. When the
geography teacher inquired about the best route, Mr. Alex expertly
outlined two alternative routes, ensuring a smooth and informed journey
from Mukono to Entebbe. The shorter route takes 2 hours and 26 minutes
to complete, with the driver maintaining an average speed of 54km/h for
the first x kilometers and 37.5km/h for the last y kilometers. The longer
route, which is 5km longer than the shorter route, takes the driver 2 hours
and 12 minutes to complete at an average speed of 60km/h. The driver
also charges a fare of Shs 1000 per student per kilometer.
TASK
a) Create two mathematical models involving x and y which can be used
to help in the analysis of the two routes.
b) Help Mr. Alex by revealing the hidden values of x and y from the
equations in (a) above.
c) Guide the geography teacher on the best route, highlighting the
benefits of your proposed option.
ITEM 4
Garden City Shopping Mall in Kampala is experiencing a massive turnout
of customers, leading to a critical shortage of parking spaces. To address
this challenge, the mall's management has planned an innovative solution:
a triangular rooftop parking design, exclusively designed for compact
cars, each spanning an area of 0.2 square units, to cater for the growing
demand of space. The parking area is bounded by the following
constraints; x+ y< 3 , x− y+ 3 ≥ 0, and y + 1 ≥ 0. To capitalize on the high
demand for parking, the mall plans to impose a parking fee of Shs 2500
per vehicle, with calculations indicating that on peak days, the cars will
occupy a total area of 12.4 square units, yielding a significant income
source.
TASK
a. Create a graphical representation on paper to help Garden City
management accurately visualize and understand the layout and
boundaries of the new parking design.
b. Assist the management in identifying the accurate coordinates and
size of the new parking area in square units.
c. Will the new design accommodate the expected maximum number of
vehicles? If so, determine the maximum number of cars that can be
accommodated in the new parking area at full capacity.
d. Forecast the mall’s daily highest revenue on a peak day.

Item 5
James is starting a baking business, selling cakes and cookies. To
estimate profits, he consulted a friend in the industry. The friend shared
data from their own experience: Initial phase: 40 cakes, 30 cookies, total
profit UGX 29,000; Later phase: 50 cakes, 20 cookies, total profit UGX
31,000. James aims to start by producing at least 120 items (cakes and
cookies combined). Since cakes sell more, he wants to make at most 80
cakes and at most 60 cookies. He needs to determine the optimal
number of cakes and cookies to produce initially.
TASK;
a) What are the expected earnings from each cake and cookie, based on
his friend's experience?
b) (i) What mathematical inequalities are making decision-making hard
for James?
(ii) Use the inequalities to help him decide on the highest number of
cakes and cookies he can start with.

ITEM 6
Following the passing away of your cousin's father, the family convened
a month later to settle his estate and read his will, to which you were
invited. However, a surprise revelation emerged when the youngest child
revealed a mysterious envelope left by their father, containing a secretive
message. The envelope contained a piece of paper with the number 31,
which needed to be converted to a ternary numeral system (base three) to
unlock the secret code to gain access to his office strongbox. But, to
their dismay, none of the family members recalled how to perform the
conversion, and they turned to you for assistance. the secret code to gain
access to his office strongbox. Upon opening the safe, they discovered a
staggering 349 million shillings and a will outlining the distribution of
the wealth. The will stipulated that the wife was to receive 40% of the
total, the eldest son was to receive one-third of the remaining amount, and
the two younger children were to share the balance in a 2:3 ratio
according to their birth order. To ensure a fair and impartial distribution,
your expertise was sought once again to calculate the exact share for each
beneficiary, preventing any potential disputes or biases.
TASK;
a) Show how you helped your cousins unlock the safe, showing each step
clearly
b) Show, with step-by-step calculations, how you helped the family
allocate the funds among its
members, and share your thoughts on the distribution's fairness.

Item 7
A man intends to plant trees on the two sides of the road which leads to
his land.On one side of te road he is to plant a tree after 5m yet on the
other side he is to plant a tree every after 6m, at the start of the road, two
trees are to be planted directly opposite each other.In the first phase of
the planting trees, he will plant trees, until another pair of treess is again
directly opposite.His land has an area of 500�� .He plans to use 25% of
the land to plant maize,one fifth of the land for beans and 205 �� for
growing ground nuts.
Task:
a) Help the man to determine how many tree seedlings he needs to buy
to just plant his first phase.
b) Determine in �� the size of the land to be used for growing beans.
c) Express the area to be used for growing ground nuts in standard form.
d) Do you think he partitioned the entire land propertly? Give a reason.

Item 8
A mathematician gave your friend a carpenter a task of making a
rectangular ground floor of a rabbit house. The length of the house is to
be (� + �)� and width to be y m .Its perimeter should be 25 m and its
area suld be 25�� .The mathematician adds that he needs the work to be
finished in only one day but he has ever contracted 3men working at the
same rate and they only managed to work on 5 �� . To be given this
contract your friend is required to mke a clear diagram showing the
numeiacl sizes of the length and width but fails to do so and comes to
yu for help.
Tasks;
a. Determine the length and width of the floor to be occupied by the
house.
b. Make a sketch of the floor your friend can present to the
mathematician to get the contract.
c. Determine the number of workers who are needed to complete the
house if they all work at the same rate as the group the man has ever
used.
SECTION B
PART 1
ITEM 9
To enhance the yields of Rice, Beans, Sugarcane, and Peas in Iganga
district, the Ministry of Agriculture's Farmer Training and Capacity
Building program conducted a survey, yielding the following findings;
Among the 80 rice farmers surveyed, 45 also grow beans, 60 cultivate
sugarcane, and 5 focus solely on peas and rice. Additionally, 5 farmers
dedicate their land solely to rice. The number of farmers who grow beans,
sugarcane, peas, and rice is equal to those who grow peas,
sugarcane, and rice. Moreover, the farmers who cultivate rice, and
sugarcane only are equal in number to those who grow rice, peas, and
beans, and are 5 fewer than those who grow all four crops.
The ministry plans to provide support to these farmers as follows:
• A farmer who cultivates all four crops (beans, sugarcane, peas, and rice) will
receive a package consisting of 4 tractors and a cash grant of UGX 3,000,000.
• One who plants only three crops will receive 3 tractors and UGX 2,000,000.
• 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.
The ministry needs to calculate the total cost of tractors for farmers,
based on the number of tractors needed for each group, with each tractor
costing UGX 68,000,000.
TASK
a) Assist the ministry in determining:
(i) The total number of farmers cultivating all four crops
(ii) The number of farmers growing only three crops
(iii) The chance of selecting a farmer who grows only two crops in
Iganga district.
(iv) The likelihood of selecting a farmer who does not grow Peas
b) Set the total funding required for the ministry's farmer support
initiative.
ITEM 10
To address concerns about battery durability, Uganda Batteries Limited
(UBL), a trusted manufacturer since 1967, conducted a thorough test on a
random sample of 50 batteries. Their experts carefully selected and
examined these batteries, yielding the following results (rounded to the
nearest minute):
423 369 387 411 393 394 405 369 372 410
371 377 389 409 392 408 409 396 431 391
431 401 363 391 405 382 396 381 438 422
400 381 399 415 428 422 397 399 401 398
396 372 410 419 386 390 362 373 391 402
The director has decided to withdraw batteries with a life equal to or less
than the average lifespan of the tested samples and has directed the
experts to manufacture only batteries that achieve at least 99 % of the
median life of the 50 tested batteries.
TASK
a) (i) Organize the data into intervals of 10 using a statistical table and
analyze the trends to recommend the most effective battery replacement
strategy to the director
(ii) Elaborate on the reasoning that led to your conclusion in a) i)
b) (i) Develop a graphical display to illustrate the data, allowing the
director and their team to estimate the median, visualize and analyze the
information
(ii) Identify the target battery lifespan for manufacturing, as
recommended by the director.
(iii) Analyze the graph and explain the situation, backing your argument
with data and logical reasoning.
c) Aid the manager in recognizing the chance of selecting a battery with a
lifespan greater than or equal to the median value.

ITEM 11
The recently concluded Uganda Secondary Schools Sports Association
(USSSA) tournament was largely dominated by schools from the western
region, prompting head teachers of participating schools to request
detailed reports from their games’ teachers on transportation and prize
monies received during the football competition
The four schools that dominated include; Fort Porto SS, Tororo SS,
Nyakasura HS, and Kyogera HS. Due to limited funds, the four schools
decided to use two buses; Fort Portal ss and Tororo ss used the Tausi bus
which charges UGX. 24,000 per Km while Kyogera HS and Nyakasura
HS used Global coaches that charge UGX. 28,000 per Km. On the
tournament day, Tausi Bus embarked on its journey from Mbarara to
Kampala at 4:30 a.m., cruising at a steady 80 Km/hr and arriving in
Kampala at 9:00 a.m. Simultaneously, Global Bus set off from Sanga
town, 50 Km from Mbarara, at 4:30 a.m. and traveled at a constant 50
Km/hr for 3 hours and 30 minutes before pausing for 30 minutes. It then
resumed its journey at a steady 67.5 Km/hr until it reached Kampala, with
the bus fare being equally distributed among the participating schools
that used the bus.
Upon arrival in Kampala, the four schools competed in a two-round
football tournament.
1ST round
Win Draw Loss
Fort Portal 1 3 2
Tororo 2 2 2
Nyakasura 3 2 1
Kyogera 0 2 4
2 round
nd

Win Draw Loss

Fort Portal 1 2 3
Tororo 2 1 3
Nyakasura 2 3 1
Kyogera 1 4 1
The tournament followed a standard points system: three points for a
victory, one point for a draw, and zero points for a defeat. Additionally,
the four teams shared a prize pot of UGX 24,000,000, allocated
proportionally to their points tally
TASK
a) (i) Assist the games teachers in plotting the buses' routes on a graph,
enabling a more helpful evaluation of their journey.
(ii) Provide the games teachers with the information needed to determine
each school's transportation expenditure.
(iii) Ascertain the first bus to arrive in Kampala and the time gap between
its arrival and the subsequent bus.
b) Ascertain the winning and last teams and amount given to each team
that participated in the tournament.
ITEM 12
In a certain town, there is a section of the road where many accidents occur and the
residents believe it is due to over speeding so they have requested the authorities to
build humps along that section, the chairperson of the roads committee has decided to
do some research so a checkpoint has been put at that section to measure the speed of
50 vehicles passing that point. They will put humps if the research shows that the
percentage of vehicles passing that point at a speed greater than the speed
limit is greater than those who abide by the speed limit. The road sign shows a speed
limit of 55km/hr for that section. The results for the 50 vehicles sampled are shown in
the table below.
Speed(km/hr) 20- 30- 40- 50- 60- 70- 80- 90-
30 40 50 60 70 80 90 100
Number of 5 8 7 9 6 5 4 6
Vehicles
TASK;
(a) Assist the chairperson in determining the average speed at which vehicles pass that
point.
(b) Present a graphical analysis to guide the committee's choice of implementing
traffic calming measures.

Item 13
Maria, a surveyor, embarks on a journey from Mukono to a Kampala
construction site to perform crucial soil testing, as the soil's sand content
plays a critical role in ensuring foundation stability and preventing
potential settlement or foundation failure due to excessive sand. Maria's
journey begins with a 30 Km stretch on a bearing of 080° to Kalagi,
followed by a 330° turn and a 40 Km drive to Gayaza. Finally, she heads
on a bearing of 30° to reach the construction site in Kampala which is on
a bearing of 020° from her starting point in Mukono. Upon arrival, she
collects soil samples at various depths and records the sand content
percentage in the table below:
Soil depth 35 65 55 25 45 75 20 90 51 60
(x)

Percentage 86 70 84 92 79 68 96 58 86 77
of sand (y)

Maria needs to create an appropriate graph to visualize the relationship between depth
and sand content and calculate the total cost of surveying materials; including 50
meters of measuring tape at UGX. 10,000, 20 soil sampling bags at UGX. 5,000 each,
and fuel at UGX. 6,000 per Km that she traveled. She will submit the calculations to
apply for funding from her company.
TASK
a) Help Maria draw a precise diagram illustrating her journey, including
bearings and distances.
b) (i) Develop a scatter plot to illustrate the relationship between depth
and sand content, aiding Maria in her data analysis.
(ii) Describe the relationship between soil depth and sand percentage,
including any trends or patterns you observe.
(iii) Plot a line of best fit through the scatter diagram data, and use it to:
- Predict the sand percentage at a depth of 31 cm
- Estimate the depth at which the sand percentage is 54%.
c) Assist Maria in preparing a budget proposal to fund her project
activities, including her return trip to Mukono via the same route.

PART II
Item 14
Kasomba is a farm manager at Ngori production ltd. He has a task to buy
fencing material to make a rectangular partition of an area 450m2 for
cabbage production. The material is meant to cover three sides, since the
other is already fenced with live fence.
It has been confirmed a metre require fencing material at a cost of
UGX.2000. Kasomba is given UGX.120,000 which is the exact amount
needed to buy the fencing material for the three sides.
(a) Advise the farmer on how to choose the length and width of the
partition to achieve his objective.
(10 marks)
(b) The manager uses a rectangular piece of card measuring 30 cm by 24
cm to make an open box he would use to pack the produce. The net of an
open box is made by removing a square from each corner of this piece of
card.
Each square that is removed has side x cm. The area of the net is 576 c�2

Task;
(i) Form an equation in x and solve it to find the value of the height of the box.
(ii) Determine the length and the width of the box. (05 marks)
ITEM 15
Uganda Crop Care Limited (UCCL) has secured a contract to supply
liquid fertilizer in Kenya, with a requirement to package it in cylindrical
tanks measuring 15 meters in height and 4 meters in radius. Currently, the
company stores its liquid fertilizer in metallic buckets with dimensions of
10 meters in height, 1 meter in lower radius, and 3 meters in upper radius.
To fulfill the order, UCCL needs to determine the number of buckets
required to fill 100 tanks. Each metallic bucket costs UGX 8000 to
manufacture, and the company sells the fertilizer at UGX 3600 per liter.
The manager at UCCL needs to calculate the number of buckets needed
and evaluate the cost implications.
Task:
a) Ascertain the number of buckets needed to fill all the required tanks.
b) Establish the total cost that will be required to manufacture the
required metallic buckets.
c) Based on calculations, evaluate UCCL’s potential for success.

ITEM 16
Stanbic Bank, a prominent African financial institution, seeks to revamp
its logo to align with its values and appeal to a newer, younger
demographic generation. The current logo, a triangle with coordinates
A(2, 3), B(4, 1), and C(1, 2) on a white rectangular background, is due for
a refresh. The bank's graphic designer has suggested the following design
modifications to enhance the logo;
Keep the original triangle in place, but turn it 90 degrees
counterclockwise around the origin. Then, mirror the resulting triangle
across the horizontal axis. Next, scale up the new triangle by a factor of 3
about the center (-5, -2), creating a logo with four triangles. Paint only the
enlarged triangle with a red-to-white ratio of 3:5, using red paint that
costs UGX 20,000 per square centimeter and white
paint that costs UGX 15,000 per square unit. The bank has set a budget
limit of UGX 205,000 per logo for painting.
TASK
a) (i) Assist the designer in creating a precise layout of the logo,
showcasing the exact placement of the four triangles on the same material.
(ii) Specify the exact vertices of the new triangles.
b) Using data-driven insights, recommend to the bank owners whether to
adjust their allocation for logo painting expenses.

ITEM 17
Your father is one of the organizers of a marathon they want to draw the
map of the route that the participants will during the race.
At their chosen starting point they chose to take a road that turned E300 S
and they moved for 5km where they set up point B which will be used as
a checkpoint, they then turned through 2350 moving a distance of 9km to
point C which will be the finishing point, however on returning to the
office they decided that the finishing point should be put at point A to cut
costs of organizing the two places but they were not sure of the details of
that route from C to A that had to be included on the map.
They want to hire a vehicle that will be used to film the racers, the vehicle
available consumes 2 litres per km and a litre costs UGX.5500, the owner
of the vehicle has asked for UGX.500,000 plus fueling the vehicle for the
total distance to be covered but the vehicle owner plans to buy fuel from
the fuel station where he is given a discount of 5% for every 100,000
worth of fuel he buys since he is a regular
customer.
Task:
(a) Help your father determine the direction from point C to the new
finishing point A that will be
shown on the map to be drawn.
(b) (i) You are required to determine the total cost of hiring the vehicle.
(ii) Do you think the vehicle owner will save some money on fuel if so
how much?

Item 18
James, a petroleum engineering master's graduate from Makerere
University, has landed a job at a Ugandan NGO. The organization offers
a comprehensive benefits package, including.
o Housing allowance: Shs. 14,000 per month
o Marriage allowance:
o Medical allowance: Shs. 50,700 per annum
o Transport allowance: Shs. 10,000 per month

However, James must pay an annual insurance premium of Shs. 68,900.

He has five children, with three under 8, one 16-year-old, and a 20-year-
old. The NGO provides a family allowance for four children, as follows:
Shs. 3,400 for each child above 18 years; Shs. 4,200 for each child
between 10- 18 years; Shs. 5,400 for each child below 9 years.
The tax rates for working-class citizens in Uganda are shown in the table
below:

Income (Shs) per annum Tax Rate(%)

1st Shs 80,000 7.5
800001-160000 12.5
160001-240,000 20.0
240,001-320,000 30.0
320,001-400,000 36.5
400,001-480,000 45.0
Above 480,000 52
The accountant revealed to James that his annual income taxes would be
Shs 100,320. James was confused because he didn't understand how his
income was calculated, and he didn't know how to figure out his gross
annual income. He also learned that his annual total tax free – income
would exceed his taxable income by 24%. James aims to constantly set
aside half of his annual net income to purchase a 40 m x 22 m plot in
Kayunga village within the next ten years, taking advantage of the stable
land prices. The land is expected to be priced at UGX 4,000 per square
meter within this time frame.
TASK
a) (i) Help James arrive at his accurate taxable income figure through
careful calculation and logical thinking.
(ii) Assist James in understanding his annual marriage allowance
compensation.
(iii) Support James in figuring out his annual take-home pay.
b) Assist James in determining if he can reach his goal of purchasing the
land within the desired timeframe.

ITEM 19
Moses, an employee at a gift box manufacturing company, has been
tasked with designing a foldable gift box with square faces and a capacity
of 2744 cubic centimeters. He's struggling to create a sketch that will
guide him in arranging the faces of the box to fold and close, as well as
determining the dimensions of each cardboard piece to cut and join to
achieve the desired outcome. He normally receives a monthly salary of
300,000 Ugandan shillings with no additional allowances.
However, the company will start deducting taxes from his pay every
month, and he needs to determine his new take-home pay. The company
will follow the tax brackets shown below:
Monthly Taxable Income (Ugx.) Rate(%)
0-100,000 0
100,000-200000 5
200,000-300,000 10
He needs to deliver the gift box to the customer and charges a delivery
fee based on the amount of fuel used. His motorcycle uses 0.035 liters of
fuel per kilometer, and he travels at an average speed of 20 meters per
second. With fuel priced at 5,000 Ugandan shillings per liter, he wants to
calculate the delivery fee. According to the customer, the journey will
take 45 minutes.
TASK:
a) Help Moses develop a sketch outlining the specifications he needs.
b) What is the new salary Moses will be receiving monthly?
c) How much will Moses charge the customer for delivery

Item 20
A bucket in a shape of a frustrum with an open end of diameter 30cm and
a bottom of 20cm,the bucket which is 42cm deep is used to fill an empty
cylindrical tank of diameter 1.8m and height 1.2m

Three hundred and sixty litres of a homogenous paint is made by mixing three paints
A,B and C. The ratio by amount of point A to point B is 3:2 and that of B to C is 1:2
Paint A costs shs 1800 per litre paint B costs shs 2400 per litre and paint C shs 1275
per litre.

Task:

a) Determine the capacity of the bucket in litres correct to 3dps.

b) The capacity of the tank in litres correct to 2dps.
c) The number of buckets that must be drawn to fill the tank.
d) The amount of each paint in the mixture.
e) The amount of money needed to make 1 litre of the mixture.
f) The percentage profit made by selling the mixture at shs 2210 per litre.

END