Name:…………………………………………………………………………………………
COMBINATION…………………………………….Signature…………………………
DARA CHRISTIAN HIGH SCHOOL-LIRA CITY
S 5 END OF TERM II EXAMINATION 2025
PRINCIPAL MATHEMATICS P425
TIME: 2 HOURS :30 MINUTES HAVE A VISION
INSTRUCTIONS: This paper consist of Two-Sections Section A (pure
mathematics) and Section B (Applied mathematics).
Answer only THREE items from section A and two items from section B from
which item 5 is compulsory
Answer FIVE ITEMS in total,
SECTION A: PURE MATHEMATICS
Answer ANY THREE items in this SECTION
Item 1
Malaria disease is the leading cause of death among children in Uganda, to help
bring down the mortality rate among the children, Lira regional referral hospital
through the ministry of health, have embarked on the vaccination of children
aged two years and below. Recently, the hospital receives a consignment of this
vaccines. Each child requires from 𝟐𝟎𝟎𝐦𝐠 to 𝟒𝟎𝟎𝐦𝐠 of the drug per dose to be
effective and safe.
A new automatic dispenser is being calibrated by the technician to dispense the
vaccines. However, as the technician was explaining to the community how it
works, he noted that the dispenser sensor fluctuates, therefore he had modeled
an expression to help determine the amount of medicine dispensed per dose as,
𝐐(𝐱) = 𝐱 𝟐 + 𝟏𝟎𝐱 + 𝟏𝟖𝟎.
Where;
𝐐(𝐱) - the amount of medicine in milligram (mg) dispensed depending on the
sensor adjustment level 𝐱, a non negative number
TASK.
1
As a mathematics student who attended this training, help the technician to;
a) Form an inequality representing the safe dosage condition
b) Solve the inequality to find the range of adjustment level 𝐱 for which the
dosage remains within the safe limit, so that he can advise the community
effectively.
Item 2
A team of students is working on a digital irrigation system for a school project.
They encountered different mathematical models in form of rational functions
to represent pressure, flow rate and energy loss in the water pipes. To analyze
these models, they need to break each expression into simpler fractions which
are easier to use in computations. The team encountered the following
expressions but they couldn't express them into simpler forms:
The function for water pressure along a straight pipe section is modeled as
2𝑥 2 −5𝑥+7
(𝑥−2)(𝑥−1)2
4𝑥 2 +3𝑥+1
The expression for the flow rate of water in a narrow pipe is given as (𝑥 2 +1)2
𝑥 4 −𝑥 3 +𝑥 2 +1
The function representing energy loss around a bend in the pipe is 𝑥 3 +𝑥
Task:
As a student of mathematics, help the team of students on how they can
express the given functions into simpler fractions so as to analyze the models.
Item 3
An engineer is installing a solar panel on a building. The owner of the building
insisted that to tap maximum solar energy, the panel should be slanted at an
angle of 700 ≤ 𝒙 ≤ 𝟗𝟎𝟎 to the eastern direction. To ensure this, the engineer
used the equation 𝒄𝒐𝒔𝟐𝒙 = 𝟒𝒄𝒐𝒔𝟐 𝒙 − 𝟐𝒔𝒊𝒏𝟐 𝒙 for 𝟎𝟎 ≤ 𝒙 ≤ 𝟑𝟔𝟎𝟎 . But the owner of
the building was not satisfied with the model equation by the engineer and
suggested to the engineer to use the identity 𝒔𝒊𝒏(𝒙 + 𝜶) = 𝑷𝒔𝒊𝒏(𝒙 − 𝜶), where
𝒙 and 𝜶 are angles
Task
As a student of mathematics:
(a) Help the owner of the building to prove that if
𝑷+𝟏
𝒔𝒊𝒏(𝒙 + 𝜶) = 𝑷𝒔𝒊𝒏(𝒙 − 𝜶), then 𝒕𝒂𝒏𝒙 = (𝑷−𝟏) 𝒕𝒂𝒏𝜶 .Hence solve the
equation 𝐬𝐢𝐧(𝒙 + 𝟐𝟎𝟎 ) = 𝟐𝐬𝐢𝐧(𝒙 − 𝟐𝟎𝟎 ) for
𝟎𝟎 ≤ 𝒙 ≤ 𝟑𝟔𝟎𝟎 .
(b) By solving the equations above for both engineer and the owner of the
building, whose equation should be used during the installation of the
panel and at what angle?
2
Item 4
At your school, the agriculture club of the school is planning to sell their
poultry (one month old birds) to their customers. This is the third sales the
club is making since they embarked in this business, from the previous market
experience, they observe that;
As they reduce the selling price, more customers buy birds, increasing their
total revenue. Their revenue function R(x) in thousands shillings,from selling
birds depends on the price (𝑥), in thousand shillings per bird and is modeled
by the polynomial function 𝐑(𝒙) = −𝟐𝒙𝟐 + 𝟐𝟎𝒙
Where; x −the price per bird in thousand shillings.
R(x) −the total revenue in thousand shillings.
TASK:
As a student of mathematics student of s.5, help the club to;
a) Know what type of polynomial function is 𝐑(𝒙) = −𝟐𝒙𝟐 + 𝟐𝟎𝒙
b) Calculate the total revenue when x = 5, x = 10, x = 15
c) What price per bird will maximize the total revenue.
d) What is the maximum total revenue they can earn?
SECTION B: APPLIED MATHEMATICS
Answer item 5 and ONLY one other item in part II,
PART I: COMPULSORY
ITEM 5
One Saturday you were having good time driving on a good road with your dad,
he drives at a constant speed of 15m/s for 300 seconds and then he received a
call about your uncle waiting at home so he accelerated the car uniformly to a
speed of 25m/s over a period of 20 seconds. This speed is maintained for 300
seconds before the car is brought to rest in 30 seconds at home. He was not
supposed to exceed average speed limit of 21m/s for overall journey otherwise
he is to pay a fine of shillings 20,000 to UNRA
Task
As a mathematics student, use graphical analysis to represent the journey
described above and use it to find:
(i) the acceleration
(ii) the total distance travelled in the time described,
(iii) whether he is supposed to pay the fine
(iv) How much was this journey in terms of fuel, if 1 metre = ugx sh 5
3
PART II:
ANSWER ONLY ONE ITEM IN THIS PART
ITEM 6
The "Hope for All" NGO is assisting a local health clinic in lira city, Northern
Uganda, with their annual covid-19 vaccination drive. They want to understand
the reach and effectiveness of their outreach efforts. They collected data from a
sample of 40 residents who were present for vaccinations on a particular day
and they wanted to do some data analysis and close if the most likely age of
people vaccinated is more than 54 years
Age 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99
No of residents 4 6 3 5 7 8 5 2
Task
(a) Assist in-charge in determining the average age and standard deviation of
the age of the members who got vaccinated from COVID-19
(b) Present a graphical analysis to determine whether they closed at the right
time
Item 7
Your brother is a director of one nursey school in the city. Eight teachers were
short listed for both written and oral interviews. You were invited to be present
at the board room, their scores are given below
Written(x) 55 54 35 62 87 53 71 50
Oral (x) 57 60 47 65 83 56 74 43
Task, As a student of mathematics,
(a) Represent their scores graphically and comment on the relationship
between the two assessments.
(b) If the successful candidate should score a minimum of 70 in written
interview, what should be the minimum score for the oral interview
(c) Use the data above to test significance at 5%
End
