LUBIRI SECONDARY SCHOOL

SENIOR FIVE E.O.T ASSISEMENT EXAMINATION

MATHEMATICS PAPER ONE

TIME: 3 HOURS
ANSWER ITEM ONE IN SECTION A (40 MARK) AND

ANY FOUR ITEMS IN SECTION B (60 MARKS)

SECTION A

ITEM ONE

a. You are given three points 𝑃(−2 − 3), 𝑄(2 0)𝑎𝑛𝑑 𝑅(8 − 8). Illustrate that 𝑃𝑄̂ 𝑅 = 900
and generate the area of ∆𝑃𝑄𝑅.
b. It’s known that 𝑓(𝑥) ≡ 3 − 7𝑥 + 5𝑥 2 − 𝑥 3 , explain mathematically the fact that 3 − 𝑥
gives a reminder zero and identify the other two factors which give zero reminder.
c. Identify the range of values of 𝑥 for which 4𝑥 2 − 12𝑥 + 5 < 0
d. Trigonometry is a branch of mathematics which uses identities. Use knowledge of this
branch of mathematics to explain the fact that 𝑠𝑒𝑐 2 𝐴 + 𝑐𝑜𝑠𝑒𝑐 2 𝐴 = 4𝑐𝑜𝑠𝑒𝑐 2 2𝐴.
e. James is a farmer dealing in carrots He monitors production depending on number of
sacks got due climatic factors basing on the equation 𝑄 = 8𝑐𝑜𝑠𝑥 − 15𝑠𝑖𝑛𝑥 , where Q is
number of sacks and x is in degrees.(00 ≤ 𝑥 ≤ 600 ) Help James to know the price of
𝑥
each sack when the Quantity Q is zero, given that 𝑃 = 3600 × 100,000 𝑠ℎ𝑠.
f. Th distance covered by a car depends on the velocity V of the car according to the
equation 𝐷 = 3(22𝑣 ) + 2(2𝑣 ) . Find the velocity V when the distance is 1 unit.
g. In how many ways can seven boys and three girls sit on a bench is girls must not sit
together and two of the girls are identical twins.
𝛼 𝛽 𝛼 𝛽
h. Evaluate the value of the sum 𝛽 𝑎𝑛𝑑 and the value of the product of 𝛽 𝑎𝑛𝑑 ,by
𝛼 𝛼
applying the equation 2𝑥 2 − 3𝑥 + 2 = 0

SECTION B
ITEM TWO

A community has been contributing towards the construction of a new place of worship.
Their plan is to have face with an isosceles triangular roof shape. The corners are assumed to be
at (5 , −3) (−6, 1) 𝑎𝑛𝑑 (1,8). A construction company has been contracted to develop the
building and you are one of the engineers in charge of the task. The challenge is to identify
which points should form the horizontal base and which point to form tip. Further more a big
cross is to be placed in the middle of the shape as an identification symbol of the faith.

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 The community also requested that the front compound should be in a triangular shape
with total perimeter of 4200m, one side to be of length 1400m and total area of compound to be
2100√15𝑚2

Task: As the head of the team of engineers help the company to a accomplish the request of the
community by identifying the horizontal corners, tip and how to place the cross. Also identify the
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other lengths of the compound and make sure the cosine of the largest angle formed is 4.

ITEM THREE.

A chemical company has come up with a new jelly product on the market developed
11𝑥+12
using the formula 2𝑥 3 +𝑥 2−15𝑥−18 from three different components. The feedback from the market
shows that the product has been rejected due to fact that affects the skin which has caused losses
to the company. Their chemist believes that the percentages of the three components used where
not balanced so there is need to separate the formula into different units to identify the
percentages used of each component. The chemist is challenged with this task and the company
has contacted a mathematic consultancy company to solve the task.

Task; As an expert in this area in the mathematic company identify the units from the formula so
that the chemist can balance the percentages.

ITEM FOUR.

One side of a container at KUKU sea port has a rectangular face marked as PKMN with
PK being on a horizontal plane having a length of 8 meters and MN being 6 meters. The
container is to be lifted by a crane attached at point N and turned through an angle of 300 at
point P. The driver of the crane is concerned about vertical and horizontal distance of the point N
from point P before he continues to lift the container due to risks that might occur.
At the same port there are two observation tours in a straight line one kilometer apart.
Peter and Tom are in charge of them to observe incoming ships with containers. Tom is due east
of Peter. Peter observes a ship on baring of 1670 and Tom observe the same ship on bearing of
2050 .
Task: You are tasked with generating the distance Vertical and horizontal distance of point N
from point P so that the driver of the crane is aware of the risks involved in lifting he
container.
Also help Peter and Tom Know how far the ship is from the port so that their get set to off
load the containers

ITEM FIVE.
A road from Kampala towards Entebbe goes via a hill is expressed using the equation
𝑦 = 𝑥 − 9𝑥 2 + 20𝑥 − 8. Due to heavy trucks which use the road a company has been
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contracted to find the inclination of the road at a point 1600m horizontally and 4000m vertically
where trucks find it hard to ascend the hill so that adjustments are made to ease movement of the
trucks

An alternative road is to be constructed at a lower point on the same hill 1100m

Task: As a young engineer you are tasked with find the inclination of the road at that point
(1600m, 4000m) and the angle of inclination. Also form the expression of the alternative road in
terms of x and y at the point (1100m, 3000m)

ITEM SIX.

In mathematics students study a branch dealing with factorial and arrangements. Mary
has arranged a birth day party and invited twelve friend’s young boys and girls from her form
schools. She has one round table to fit the friends

A couple is organizing a party to thank friends who have stood with them during the
period of organizing their wedding successfully but they want to use invitation cards with six-
digit figures. They have decided to use these six digits 5,1,6,4,6,6 to form the figure but the value
of the figure formed should not be less than 400,000. This will determine the number of cards to
be printed hence eliminating n invited visitors. The cost of print a card is 6500 shillings
Task: Help Mary identify the different ways in which she can arrange the friends around the
table in order without fighting for chairs.
A friend advised her that among the invited friends there are two who cannot sit next to each
other, help Mary make adjustments in the different ways of siting the friends on the table.

Also help the couple to know how many visitors they can invite and the total cost needed to
print the cards