MATHEMATICS (121)

PAPER ONE

NATURAL NUMBERS

1. Otego had 3469 bags of maize, each weighing 90 kg. He sold 2654
of them.

a.) How many kilogram of maize was he left with?

b.) If he added 468 more bags of maize, how many bags did he end
up with?

2. All prime numbers less than ten are arranged in descending order
to form a number
a.) Write down the number formed
b.) What is the total value of the second digit?

FACTORS
1. Express the numbers 1470 and 7056, each as a product of its
14702
prime factors. Hence evaluate: Leaving the
√7056
answer in prime factor form.
2. Express 196 as a product of its prime factors hence evaluate;
√ 196

GREATEST COMMON DIVISOR(GCD)

 1. a) Find the greatest common divisor of the term. 144x 3y2 and
81xy4
b) Hence factorize completely this expression 144x3y2-81xy4
2. The GCD of two numbers is 7and their LCM is 140. if one of the
numbers is 20, find the other number.
3. The LCM of three numbers is 7920 and their GCD is 12. Two of
the numbers are 48 and 264. Using factor notation find the third
number if one of its factors is 9.

LEAST COMMON MULTIPLE(LCM)

1. Find the L.C.M of x2 + x, x2 – 1 and x2 – x.

2. Find the least number of sweets that can be packed into polythene bags
which contain either 9 or 15 or 20 or 24 sweets with none left over.

3. A number n is such that when it is divided by 27, 30, or 45, the

remainder is always 3. Find the smallest value of n.
4. A piece of land is to be divided into 20 acres or 24 acres or 28 acres for
farming and Leave 7 acres for grazing. Determine the smallest size of
such land.
5. When a certain number x is divided by 30, 45 or 54, there is always a
remainder of 21. Find the least value of the number x .
6. A number m is such that when it is divided by 30, 36, and 45, the
remainder is always 7. Find the smallest possible value of m.

INTERGERS
 1. The sum of two numbers exceeds their product by one. Their
difference is equal to their product less five. Find the two numbers.

2. Evaluate −12÷ (−3 ) × 4−(−15) a.

−5 ×6 ÷ 2+(−5)
−8 ÷ 2+12 × 9−4 ×6
3. Evaluate; 56÷ 7 × 2
4. Evaluate without using mathematical tables or the calculator
1.9 ×0.032
20× 0.0038

FRACTIONS
1. Evaluate without using a
calculator.
3 5 2
(1 ÷ )×
7 8 3
3 5 4 1
+1 ÷ of 2
4 7 7 3

2. A two digit number is such that the sum of the ones and the tens
digit is ten. If the digits are reversed, the new number formed
exceeds the original number by 54. Find the number.

3.
3
(
3 1 1 1
) 2
Evaluate 8 of {7 5 − 3 1 4 +3 3 × 2 5 }

4. ¨ into fraction
Convert the recurring decimal l2.18
2 1
5. Simplify (0.00243) 5 ×(0.0009) 2 without using tables or calculator.

6. Evaluate without using tables or calculators

6 14 20
of ÷ 80 ×−
7 3 3
−2× 5+(14 ÷7)×3
7. Mr. Saidi keeps turkeys and chickens. The number of turkeys
exceeds the number of chickens by 6. During an outbreak of a
disease, ¼ of the chicken and 1/3 of the turkeys died. If he lost
total of 30 birds, how many birds did he have altogether?

8 ÷2+12 × 9−4 ×6
9. Work out 56 ÷ 7 × 2

−4 of (−4+−5÷ 15 )+−3−4 ÷ 2
10. Evaluate
84 ÷−7+3−−5

11. Write the recurring decimal 0.3̇ as Fraction

5 1 5
of (4 −3 )
6 3 6
12. Evaluate 5 3 15 21 without using a calculator.
× + ÷
12 25 9 3

13. Without using tables or calculators evaluate.

35÷ 5+2 ×−3

−9+ 14 ÷7 +4
Without using tables or calculator, evaluate the following.

−8+ (−13 ) ×3−(−5)

−1+(−6)÷ 2× 2

15. ¨ as a single fraction.

Express 1.93̇+ 0.25

1 1 1 1 2
of 3 + 1 (2 − )
2 2 2 2 3
16. Simplify 3 1
of 2 ÷ 1/2
4 2

17. Evaluate:
 1
of 4
2 2 1
÷ −1
5 9 10
1 1 3
− of
8 6 8

18. Without using a calculator or table, work out the following

leaving the answer as a mixed number in its simplest form:-

3 2 3 1
+1 ÷ of 2
4 7 7 3
9 3 2
( − )×
7 8 3

19. Work out the following, giving the answer as a mixed number in
2 1 4 1
÷ of −1
5 2 9 10
its simplest form. 1 1 3
− ×
8 16 8

20.
3 3 1 1 1
( ) 2
Evaluate; 8 of {7 5 − 3 1 4 +3 3 × 2 5

4 25 2
1 of ÷ 1 × 24
5 18 3
21. Without using a calculator, evaluate: 1 1 5
leaving
2 − of 12 ÷
3 4 3

the answer as a fraction in its simplest form.

22. There was a fund-raising in Matisse high school. One seventh of

the money that was raised was used to construct a teacher’s
house and two thirds of the remaining money was used to
construct classrooms. If shs.300,000 remained, how much
money was raised

DECIMALS
 1.) Without using logarithm tables or a calculator evaluate.
384.16 ×0.0625
96.04
2.) Evaluate without using mathematical table

0.0128
1000
200

SQUARE AND SQURE ROOTS

1.) Evaluate without using tables or calculators

√
3 0.125 × √ 64
0.064 ×√ 629

2.) Evaluate using reciprocals, square and square root tables only.

√¿¿¿

3.) Using a calculator, evaluate √¿¿¿

(Show your working at each stage)

4.) Use tables of reciprocals and square roots to evaluate

√ 2
−
1.06
0.5893 846.3

5.) Use tables to find;

a) i) 4.9782
ii) The reciprocal of 31.65
Hence evaluate to 4.S.F the value of
1
4.9782 – 31.65
 6) Without using mathematical tables or calculator, evaluate:
√ 153 × 1.8
0.68× 0.32
giving

your answer in standard form

ALGEBRAIC EXPRESSION

2 X−Z
1. Given that y= X + 3 Z express x in

terms of y and z
2. Simplify the expression
X−1 2 X +1
− . Hence solve the
X 3X
equation

X−1 2 X +1 2
− =
X 3X 3

3. Factorize a2 – b2
Hence find the exact value of 25572 - 25472
2 2
p −2 pq+ p
4. Simplify 3 2 2 3
p − pq + p q−q

5. Given that y = 2x – z, express x in terms of y and z.

6. Four farmers took their goats to a market. Mohammed had
two more goats as Koech had 3 times as many goats as
Mohammed, whereas Odupoy had 10 goats less than both
Mohammed and Koech.

(i) Write a simplified algebraic expression with one variable,

representing the total number of goats.
 (ii) Three butchers bought all the goats and shared them
equally. If each butcher got 17 goats, how many did
odupoy sell to the butchers?

1 5
7. Solve the equation 4 x = 6 x −7

a b
8. Simplify 2(a+ b) + 2(a−b)

9. Three years ago, Juma was three times as old. As Ali in two
years time, the sum of their ages will be 62. Determine their
ages.

RATES, RATIO, PROPORTION AND PERCENTAGE

1. Akinyi bought and beans from a wholesaler. She then mixed the
maize and beans in the ratio 4:3 she bought the maize at Ksh 21
per kg and the beans 42 per kg. If she was to make a profit of
30%. What should be the selling price of 1 kg of the mixture?
2. Water flows from a tap at the rate of 27 cm 3 per second into a
rectangular container of length 60 cm, breadth 30 cm and height
40 cm. If at 6.00 PM the container was halffull, what will be the
height of water at 6.04 pm?
3. Two businessmen jointly bought a minibus which could ferry 25
paying passengers when full. The fare between two towns A and
B was Ksh 80 per passenger for one way. The minibus made
three round trips between the two towns daily. The cost of fuel
was ksh 1500 per day. The driver and the conductor were paid
daily allowances of Ksh 200 and Ksh 150 respectively. A further
 Ksh 4000 per day was set aside for maintenance, insurance and
loan repayment.

(a) (i) How much money was collected from the passengers
that day?

(ii) How much was the net profit?

(b) On another day, the minibus was 80% full on the average
for the three round trips, how much did each businessman
get if the day’s profit was shared in the ratio 2:3?

4. Wainaina has two dairy farms, A and B. Farm A produces milk

with 3¼ percent fat and farm B produces milk with 4¼ percent
fat.
(a) Determine
(i) The total mass of milk fat in 50 kg of milk from
farm A and 30 kg of milk from farm B

(ii) The percentage of fat in a mixture of 50kg of milk A

and 30kg of milk from B

(b) The range of values of mass of milk from farm B that must
be used in a 50kg mixture so that the mixture may have at
least 4 percent fat.

5. In the year 2001, the price of a sofa set in a shop was Ksh 12,000
(a) Calculate the amount of money received from the sales of
240 sofa sets that year.
(b) (i) In the year 2002 the price of each sofa set increased
by 25% while the number of sets sold decreased by 10%. Calculate
the percentage increase in the amount received from the sales.

(ii) If the end of year 2002, the price of each sofa set
changed in the ratio 16: 15, calculate the price of
each sofa set in the year 2003.

(c) The number of sofa sets sold in the year 2003 was P% less
than the number sold in the year 2001.

(d) Calculate the value of P, given that the amounts received

from sales if the two years were equal.

6. A solution whose volume is 80 liters is made up of 40% of water

and 60% of alcohol. When x liters of water is added, the
percentage of alcohol drops to 40%.

(a) Find the value of x

(b) Thirty liters of water is added to the new solution.

Calculate the percentage of alcohol in the resulting
solution

(c) If 5 liters of the solution in (b) above is added to 2 liters of

the original solution, calculate in the simplest form, the
ratio of water to that of alcohol in the resulting solution.

7. Three business partners, Asha, Nangila and Cherop contributed

Ksh 60,000, Ksh 85,000 and Ksh 105, 000 respectively. They
agreed to put 25% of the profit back into business each year.
They also agreed to put aside 40% of the remaining profit to
 cater for taxes and insurance. The rest of the profit would then be
shared among the partners in the ratio of their contributions. At
the end of the first year, the business realized a gross profit of
Ksh 225, 000.
(a) Calculate the amount of money Cherop received more than
Asha at the end of the first year.

(b) Nangila further invested Ksh 25,000 into the business at

the beginning of the second year. Given that the gross
profit at the end of the second year increased in the ratio
10:9, calculate Nangila’s share of the profit at the end of
the second year.

8. Kipketer can cultivate a piece of land in 7 hrs while Wanjiku can

do the same work in 5 hours. Find the time they would take to
cultivate the piece of land when working together.
9. Mogaka and Ondiso working together can do a piece of work in
6 days. Mogaka working alone takes 5 days longer than Onduso.
How many days does it take Onduso to do the work alone.
10. A certain amount of money was shared among 3 children in the
ratio 7:5:3 the largest share was
Ksh 91. Find the
(a) Total amount of money

(b) Difference in the money received as the largest share and

the smallest share.

LENGTH
1.) Two coils, which are made by winding copper wire of different
gauges and length, have the same mass. The first coil is made by
winding 270 metres of wire with cross sectional diameter 2.8mm
while the second coil is made by winding a certain length of wire
with cross-sectional diameter
2.1mm. Find the length of wire in the second coil.

2. The figure below represents a model of a hut with HG = GF =

10cm and FB = 6cm. The four slanting edges of the roof are
each 12cm long.

Calculate

i Length DF.
ii Angle VHF
iii The length of the projection of line VH on the plane EFGH.
iv The height of the model hut.
v The length VH.
vi The angle DF makes with the plane ABCD.
3. A square floor is fitted with rectangular tiles of perimeters 220
cm. each row (tile lengthwise) carries 20 less tiles than each
column (tiles breadth wise). If the length of the floor is 9.6 m.

Calculate:

a. The dimensions of the tiles

b. The number of tiles needed

c. The cost of fitting the tiles, if tiles are sold in dozens at sh.
1500 per dozen and the labor cost is sh. 3000

AREA

1.) Calculate the area of the shaded region below, given that AC is
an arc of a circle center B. AB=BC=14cm CD=8cm and angle
ABD = 750

A
14 cm

B 750

14 cm

C
8 cm
D
2.) The scale of a map is 1:50000. A lake on the map is 6.16cm 2.
Find the actual area of the lake in hectares.
 3.) The figure below is a rhombus ABCD of sides 4cm. BD is an arc
of circle center C. Given that ∠ABC = 1380. Find the area of
shaded region.

4.) The figure below sows the shape of Kamau’s farm with
dimensions shown in meters
140
m

100m

20 m

80 m

Find the area of Kamau’s farm in hectares

5.) In the figure below AB and AC are tangents to the circle centre
O at B and C respectively, the angle AOC = 600
A

O
C

B
 Calculate

(a) The length of AC

6.) The figure below shows the floor of a hall. A part of this floor is
in the shape of a rectangle of length 20m and width 16m and the
rest is a segment of a circle of radius 12m. Use the figure to
Find:-

(a) The size of angle COD

(b) The area of figure DABCO

(c) Area of sector ODC

(d) Area of the floor of the house.

7.) The circle below whose area is 18.05cm2 circumscribes a triangle

ABC where AB = 6.3cm, BC = 5.7cm and AC = 4.8cm. Find the
area of the shaded part
8.) In the figure below, PQRS is a rectangle in which PS=10k cm
and PQ = 6k cm. M and N are midpoints of QR and RS
respectively

(a) Find the are of the shaded region

(b) Given that the area of the triangle MNR = 30 cm 2. find the
dimensions of the rectangle
(c) Calculate the sizes of angles Ѳ and ꞵ giving your answer to 2
decimal places
9.) The figure below shows two circles each of radius 10.5 cm with
centers A and B. the circles touch each other at T
 Given that angle XAD =angle YBC = 1600 and lines XY, ATB and
DC are parallel, calculate the area of:

d) The minor sector AXTD

e) Figure AXYBCD
f) The shaded region
10.) The floor of a room is in the shape of a rectangle 10.5 m long by
6 m wide. Square tiles of length 30 cm are to be fitted onto the
floor.

(a) Calculate the number of tiles needed for the floor.

(b) A dealer wishes to buy enough tiles for fifteen such rooms.
The tiles are packed in cartons each containing 20 tiles. The
cost of each carton is Kshs. 800. Calculate;

(i) The total cost of the tiles.

(ii) If in addition, the dealer spends Kshs. 2,000 and Kshs. 600
on transport and subsistence respectively, at what price
should he sell each carton in order to make a profit of 12.5%
(Give your answer to the nearest Ksh.)
The figure below is a circle of radius 5cm. Points A, B and C are the vertices of the
triangle ABC in which ∠ABC = 60o and ∠ACB=50o which is in the circle.
Calculate the area of ABC.

60o

50o
A C

11.) Mr.Wanyama has a plot that is in a triangular form. The plot

measures 170m, 190m and 210m, but the altitudes of the plot as
well as the angles are not known. Find the area
of the plot in hectares

12.) Three sirens wail at intervals of thirty minutes, fifty minutes and
thirty five minutes. If they wail together at 7.18a.m on Monday,
what time and day will they next wail together?

13.) A farmer decides to put two-thirds of his farm under crops. Of

this, he put a quarter under maize and four-fifths of the
remainder under beans. The rest is planted with carrots. If
0.9acres are under carrots, find the total area of the farm
 VOLUME AND CAPACITY

1. ) All the water is poured into a cylindrical container of circular

radius 12cm. If the cylinder has height 45cm, calculate the

surface area of the cylinder which is not in contact with water.

2. The British government hired two planes to airlift football fans to

South Africa for the World cup tournament. Each plane took 10
½ hours to reach the destination. Boeing 747 has carrying
capacity of 300 people and consumes fuel at 120 liters per
minute. It makes 5 trips at full capacity. Boeing 740 has carrying
capacity of 140 people and consumes fuel at 200 liters per
minute. It makes 8 trips at full capacity. If the government
sponsored the fans one way at the cost of 800 dollars per fan,
calculate:
(a) The total number of fans airlifted to South Africa.

(b) The total cost of fuel used if one litre costs 0.3 dollars.

(c) The total collection in dollars made by each plane.

(d) The net profit made by each plane.

3. A rectangular water tank measures 2.6m by 4.8m at the base and

has water to a height of 3.2m. Find the volume of water in litres

that is in the tank.

4. Three litres of water (density1g/cm3) is added to twelve litres of

alcohol (density 0.8g/cm3. What is the density of the mixture?

 5. A rectangular tank whose internal dimensions are 2.2m by 1.4m

by 1.7m is three fifth full of milk.

(a) Calculate the volume of milk in litres

(b) The milk is packed in small packets in the shape of a right

pyramid with an equilateral base triangle of sides 10cm.
The vertical height of each packet is 13.6cm. Full packets
obtained are sold at shs.30 per packet. Calculate:

(i) The volume in cm3 of each packet to the nearest

whole number.

(ii) The number of full packets of milk.

(iii) The amount of money realized from the sale of

milk.

6. An 890kg culvert is made of a hollow cylindrical material with

outer radius of 76cm and an inner radius of 64cm. It crosses a

road of width 3m, determine the density of the material ssused in
its construction in Kg/m3 correct to 1 decimal place.

MASSS WEIGHT AND DENSITY

1.) A squared brass plate is 2mm thick and has a mass of 1.05kg. The
density of brass is 8.4g/cm. Calculate the length of the plate in
centimeters.
2.) A sphere has a surface area 18cm 2. Find its density if the sphere has
a mass of 100g.

3.) Nyahururu Municipal Council is to construct a floor of an open

wholesale market whose area is 800m2. The floor is to be covered
with a slab of uniform thickness of 200mm. In order to make the
slab, sand, cement and ballast are to be mixed such that their masses
are in the ratio 3:2:3. The mass of dry slab of volume 1m 3 is
2000kg. Calculate

(a) (i) The volume of the slab

(ii) The mass of the dry slab.

(iii) The mass of cement to be used.

(b) If one bag of the cement is 50kg, find the number of bags to be
purchased.

(c) If a lorry carries 10 tonnes of ballast, calculate the number of lorries

of ballast to be purchased.
4.) A sphere has a surface area of 18.0cm2. Find its density if the sphere
has a mass of 100 grammes.

5.) A piece of metal has a volume of 20 cm 3 and a mass of 300g.

Calculate the density of the metal in kg/m3.
6.) 2.5 liters of water density 1g/cm 3 is added to 8 liters of alcohol
density 0.8g/cm3. Calculate the density of the mixture

TIME
1.) A van travelled from Kitale to Kisumu a distance of 160km. The
average speed of the van for the first 100km was 40km/h and the
remaining part of the journey its average speed was 30km/h.
Calculate the average speed for the whole journey.
2.) A watch which looses a half-minute every hour was set to read
the correct time at 0545h on Monday. Determine the time, in the
12 hour system, the watch will show on the following Friday at
1945h.
3.) The timetable below shows the departure and arrival time for a
bus plying between two towns M and R, 300km apart
0710982617

Town Arrival Departure

M 0830h
N 1000h 1020h
P 1310h 1340h
Q 1510h 1520h
R 1600h
(a) How long does the bus take to travel from town M to R?

(b) What is the average speed for the whole journey?

LINEAR EQUATIONS

1. A cloth dealer sold 3 shirts and 2 trousers for Ksh 840 and 4
shirts and 5 trousers for Ksh 1680 find the cost of 1 shirt and
the cost of 1 trouser
2. Solve the simultaneous equations

2x – y = 3
 x2 – xy = -4

3. The cost of 5 skirts and blouses is Ksh 1750. Mueni bought

three of the skirts and one of the blouses for Ksh 850. Find
the cost of each item.
4. Akinyi bought three cups and four spoons for Ksh 324.
Wanjiru bought five cups and Fatuma bought two spoons of
the same type as those bought by Akinyi, Wanjiku paid Ksh
228 more than Fatuma. Find the price of each cup and each
spoon.
5. Mary has 21 coins whose total value is Ksh 72. There are
twice as many five shillings coins as there are ten shilling
coins. The rest one shillings coins. Find the number of ten
shillings
coins that Mary has.
6. The mass of 6 similar art books and 4 similar biology books is
7.2 kg. The mass of 2 such art books and 3 such biology
books is 3.4 kg. Find the mass of one art book and the mass
of one biology book.
7. Karani bought 4 pencils and 6 biros – pens for Ksh 66 and
Tachora bought 2 pencils and 5 biro pens for Ksh 51.
(a) Find the price of each item

(b) Musoma spent Ksh 228 to buy the same type of pencils
and biro – pens if the number of biro pens he bought were
4 more than the number of pencils, find the number of
pencils bought.

8. Solve the simultaneous equations below

 2x – 3y = 5

-x + 2y = -3

9. The length of a room is 4 metres longer than its width. Find

the length of the room if its area is 32m2.
10. Hadija and Kagendo bought the same types of pens and
exercise books from the same types of pens and exercise
books from the same shop. Hadija bought 2 pens and 3
exercise books for Ksh 78. Kagendo bought 3 pens and 4
exercise books for Ksh 108. Calculate the cost of each item

11. In fourteen years time, a mother will be twice as old as her

son. Four years ago, the sum of their ages was 30 years. Find
how old the mother was, when the son was born.
12. Three years ago Juma was three times as old as Ali. In two
years time the sum of their ages will be 62. Determine their
ages.
13. Two pairs of trousers and three shirts costs a total of Ksh 390.
Five such pairs of trousers and two shirts cost a total of Ksh
810. Find the price of a pair of trousers and a shirt.
14. A shopkeeper sells two- types of pangas type x and type y.
Twelve x pangas and five type y pangas cost Kshs 1260,
while nine type x pangas and fifteen type y pangas cost 1620.
Mugala bought eighteen type y pangas. How much did he pay
for them?
 COMMERCIAL ARITHMETIC

1. The cash prize of a television set is Kshs 25000. A

customer paid a deposit of Kshs 3750. He repaid the
amount owing in 24 equal monthly installments. If he was
charged simple interest at the rate of 40% p.a how much
was each installment?
2. Mr Ngeny borrowed Kshs 560,000 from a bank to buy a
piece of land. He was required to repay the loan with
simple interest for a period of 48 months. The repayment
amounted to Kshs 21,000 per month.
Calculate

(a) The interest paid to the bank

(b) The rate per annum of the simple interest

3. A car dealer charges 5% commission for selling a car. He

received a commission of Ksh 17,500 for selling car. How
much money did the owner receive from the sale of his
car?
4. A company saleslady sold goods worth Ksh 240,000 from
this sale she earned a commission of Ksh 4,000
(a) Calculate the rate of commission

(b) If she sold good whose total marked price was Ksh
360,000 and allowed a discount of 2% calculate the
amount of commission she received.
5. A business woman bought two bags of maize at the same
price per bag. She discovered that one bag was of high
quality and the other of low quality. On the high quality
bag she made a profit by selling at Ksh 1,040, whereas on
the low quality bag she made a loss by selling at Ksh 880.
If the profit was three times the loss, calculate the buying
price per bag.
6. A salesman gets a commission of 2. 4 % on sales up to
Ksh 100,000. He gets an additional commission of 1.5%
on sales above this. Calculate the commission he gets on
sales worth Ksh 280,000.
7. Three people Koris, Wangare and Hassan contributed
money to start a business. Korir contributed a quarter of
the total amount and Wangare two fifths of the remainder.
Hassan’s contribution was one and a half times that of Koris.
They borrowed the rest of the money from the bank which
was Ksh 60,000 less than Hassan’s contribution. Find the
total amount required to start the business.

8. A Kenyan tourist left Germany for Kenya through

Switzerland. While in Switzerland he bought a watch
worth 52 deutsche Marks. Find the value of the watch in:
(a) Swiss Francs.

(b) Kenya Shillings

Use the exchange rates below:

1 Swiss Franc = 1.28 Deutsche Marks.

 1 Swiss Franc = 45.21 Kenya Shillings

9. A salesman earns a basic salary of Ksh 9000 per month. In

addition he is also paid a commission of 5% for sales
above Ksh 15000. In a certain month he sold goods worth
Ksh 120, 000 at a discount of 2½ %. Calculate his total
earnings that month
10. In this question, mathematical table should not be used
A Kenyan bank buys and sells foreign currencies as shown
below
Buying Selling
(In Kenya shillings) (In Kenya
Shillings)
1 Hong Kong dollar 9.74 9.77
1 South African rand 12.03 12.11
A tourists arrived in Kenya with 105 000 Hong Kong dollars
and changed the whole amount to Kenyan shillings. While in
Kenya, she pent Kshs 403 897 and changed the balance to
South African rand before leaving for South Africa. Calculate
the amount, in South African rand that she received.

11. A Kenyan businessman bought goods from Japan worth 2,

950 000 Japanese yen. On arrival in Kenya custom duty of
20% was charged on the value of the goods. If the
exchange rates were as follows

1 US dollar = 118 Japanese Yen

1 US dollar = 76 Kenya shillings

 Calculate the duty paid in Kenya shillings

12. Two businessmen jointly bought a minibus which could

ferry 25 paying passengers when full. The fare between
two towns A and B was Ksh 80 per passenger for one
way. The minibus made three round trips between the two
towns daily. The cost of fuel was Ksh 1500 per day. The
driver and the conductor were paid daily allowances of
Ksh 200 and Ksh 150 respectively.
A further Ksh 4000 per day was set aside for maintenance.

(a) One day the minibus was full on every trip.

(i) How much money was collected from the

passengers that day?

(ii) How much was the net profit?

(b) On another day, the minibus was 80% on the average for
the three round trips. How much did each business get if
the days profit was shared in the ratio 2:3?

13. A traveler had sterling pounds 918 with which he bought

Kenya shillings at the rate of Ksh 84 per sterling pound.
He did not spend the money as intended. Later, he used the
Kenyan shillings to buy sterling pound at the rate of Ksh
85 per sterling pound. Calculate the amount of money in
sterling pounds lost in the whole transaction.
14. A commercial bank buys and sells Japanese Yen in Kenya
shillings at the rates shown below
Buying 0.5024
 Selling 0.5446

A Japanese tourist at the end of his tour of Kenya was left

with Ksh 30000 which he converted to Japanese Yen through
the commercial bank. How many Japanese Yen did he get?

15. In the month of January, an insurance salesman earned

Ksh 6750 which was commission of 4.5% of the premiums
paid to the company.
(a) Calculate the premium paid to the company.

(b) In February the rate of commission was reduced by 66 2/3%

and the premiums reduced by 10% calculate the amount
earned by the salesman in the month of February

16. Akinyi, Bundi, Cura and Diba invested some money in a

business in the ratio of 7:9:10:14 respectively. The
business realized a profit of Ksh 46800. They shared 12%
of the profit equally and the remainder in the ratio of their
contributions. Calculate the total amount of money
received by Diba.

17. A telephone bill includes Ksh 4320 for a local calls Ksh
3260 for trank calls and rental charge Ksh 2080. A value
added tax (V.A.T) is then charged at 15%, Find the total
bill.
18. During a certain period. The exchange rates were as
follows
1 sterling pound = Ksh 102.0

1 sterling pound = 1.7 us dollar

 1 U.S dollar = Ksh 60.6

A school management intended to import textbooks worth

Kshs 500,000 from UK. It changed the money to sterling
pounds. Later the management found out that the books the
sterling pounds to dollars. Unfortunately a financial crisis
arose and the money had to be converted to Kenya shillings.
Calculate the total amount of money the management ended
up with.

19. A fruiterer bought 144 pineapples at Ksh 100 for

every six pineapples. She sold some of them at Ksh 72 for
every three and the rest at Ksh 60 for every two. If she
made a 65% profit, calculate the number of pineapples
sold at Ksh 72 for every three.

COORDINATES AND GRAPHS

1.) Copy and complete the table and hence draw the corresponding
graph.
Y= 4x + 3

x -2 -1 0 1 2
y

2.) Draw the graph of the following:

a.) Y + 2x =5
b.) y/2 + 2x =5
 ANGLES AND PLANE FIGURES

In the figure below, lines AB and LM are parallel.

A B
y z

830
x
L 1300 M
Find the values of the angles marked x, y and z

GEOMETRIC CONSTRUCTIONS

1. Using a ruler and a pair of compasses only,

a) Construct a triangle ABC in which AB = 9cm, AC = 6cm and angle

BAC = 37½0

a) Drop a perpendicular from C to meet AB at D. Measure CD

and hence find the area of the triangle ABC
b) Point E divides BC in the ratio 2:3. Using a ruler and Set
Square only, determine point E.

Measure AE.

2Using ruler and pair of compasses only for constructions in this question.

a Construct triangle ABC such that AB=AC=5.4cm and angle

ABC=300. Measure BC
 b On the diagram above, a point P is always on the same side of
BC as A. Draw the locus of P such that angle BAC is twice
angle BPC
c Drop a perpendicular from A to meet BC at D. Measure AD
d Determine the locus Q on the same side of BC as A such that the
area of triangle BQC = 9.4cm2
3 (a) Without using a protractor or set square, construct a triangle ABC in
which AB = 4cm, BC = 6cm and ∠ABC = 67½0. Take AB as the
base.

a. Measure AC.

b. Draw a triangle A1BC1 which is indirectly congruent to triangle ABC.

4 Construct triangle ABC in which AB = 4.4 cm, BC = 6.4 cm and

AC = 7.4 cm. Construct an escribed circle opposite angle ACB

a Measure the radius of the circle

b Measure the acute angle subtended at the centre of the circle by

AB

c A point P moves such that it is always outside the circle but

within triangle AOB, where O is the centre of the escribed
circle. Show by shading the region within which P lies.
5 (a) Using a ruler and a pair of compasses only, construct a
parallelogram PQRS in which PQ = 8cm, QR = 6cm and ∠PQR =
1500

(b) Drop a perpendicular from S to meet PQ at B.

Measure SB and hence calculate the area of the parallelogram.
 (c) Mark a point A on BS produced such that the area of triangle
APQ is equal to three quarters the area of the parallelogram
(d) Determine the height of the triangle.

6 Using a ruler and a pair of compasses only, construct triangle ABC in

which AB = 6cm, BC = 8cm and angle ABC = 45 o. Drop a
perpendicular from A to BC at M. Measure AM and AC
8. a) Using a ruler and a pair of compasses only to construct a
trapezium ABCD such that AB=12cm , ∠DAB=600, ∠ABC=750 and AD
=7cm

b)From the point D drop a perpendicular to the line AB to meet the

line at E. measure DE hence calculate the area of the trapezium.

9. Using a pair of compasses and ruler only;

(a) Construct triangle ABC such that AB = 8cm, BC = 6cm and
angle ABC = 300.
(b) Measure the length of AC
(c) Draw a circle that touches the vertices A,B and C.
(d) Measure the radius of the circle Using a set square, ruler and
pair of compases divide the given line into 5 equal portions.
10. Using a ruler and a pair of compasses only, draw a
parallelogram ABCD, such that angle DAB = 75 0. Length AB =
6.0cm and BC = 4.0cm from point D, drop a perpendicular to
meet line AB at N

a) Measure length DN

b) Find the area of the parallelogram

 11. Using a ruler and a pair of compasses only, draw a
parallelogram ABCD in which AB = 6cm, BC= 4cm and angle
BAD = 60o. By construction, determine the perpendicular distance
between the lines AB and CD

12. Without using a protractor, draw a triangle ABC where

∠CAB = 30o, AC = 3.5cm and AB = 6cm. measure BC
13. (a) Using a ruler and a pair of compass only, construct a triangle
ABC in which
∠ABC =37.5o, BC =7cm and BA = 14cm
(b) Drop a perpendicular from A to BC produced and measure its height
(c) Use your height in (b) to find the area of the triangle ABC
(d) Use construction to find the radius of an inscribed circle of triangle
ABC
14. In this question use a pair of compasses and a ruler only
a) Construct triangle PQR such that PQ = 6 cm, QR = 8 cm and
<PQR = 135°
b) Construct the height of triangle PQR in (a) above, taking QR as the
base
15. On the line AC shown below, point B lies above the line such
that ∠BAC = 52.5o and AB = 4.2cm. (Use a ruler and a pair
of compasses for this question)
(a) Construct ∠BAC and mark point B

(b) Drop a perpendicular from B to meet the line AC at point F .

Measure BF
 SCALE DRAWING

1. A point B is on a bearing of 0800 from a port A and at a

distance of 95 km. A submarine is stationed at a port D, which is on
a bearing of 2000 from AM and a distance of 124 km from B. A ship
leaves B and moves directly southwards to an island P, which is on
a bearing of 140 from A. The submarine at D on realizing that the
ship was heading fro the island P, decides to head straight for the
island to intercept the shipUsing a scale 0f 1 cm to represent 10 km,
make a scale drawing showing the relative positions of A, B, D, P.
Hence find
(i) The distance from A to D

(ii) The bearing of the submarine from the ship was setting off
from B

(iii) The bearing of the island P from D

(iv) The distance the submarine had to cover to reach the

island P

2. Four towns R, T, K and G are such that T is 84 km

directly to the north R, and K is on a bearing of 2950 from R at a
distance of 60 km. G is on a bearing of 3400 from K and a
distance of 30 km. Using a scale of 1 cm to represent 10 km,
make an accurate scale drawing to show the relative positions
of the town. Find
(a) The distance and the bearing of T from K
 (b) The distance and the bearing G from T
(c) The bearing of R from G

3. Two aeroplanes, S and T leave airports A at the same time. S

flies on a bearing of 060 at 750 km/h while T flies on a bearing
of 2100 at 900km/h.

(a) Using a suitable scale, draw a diagram to show the

positions of the aeroplane after two hours.

(b) Use your diagram to determine

(i) The actual distance between the two aeroplanes

(ii) The bearing of T from S

(iii) The bearing of S from T

4. A point A is directly below a window. Another point B is 15 m

from A and at the same horizontal level. From B angle of
elevation of the top of the bottom of the window is 300 and the
angle of elevation of the top of the window is 350. Calculate the
vertical distance.

(a) From A to the bottom of the window

(b) From the bottom to top of the window
4. Find by calculation the sum of all the interior angles in the figure
ABCDEFGHI below
 5. Shopping centers X, Y and Z are such that Y is 12 km south
of X and Z is 15 km from X. Z is on a bearing of 330 0 from Y.
Find the bearing of Z from X.

6. The figure below shows a triangle ABC

a) Using a ruler and a pair of compasses, determine a point D on

the line BC such that BD:DC = 1:2.
b) Find the area of triangle ABD, given that AB = AC.

7. A boat at point x is 200 m to the south of point Y. The boat sails

X to another point Z. Point Z is 200m on a bearing of 310 0 from
X, Y and Z are on the same horizontal plane.
(a) Calculate the bearing and the distance
of Z from Y
(b) W is the point on the path of the boat
nearest to Y. Calculate the distance WY
 (c) A vertical tower stands at point Y. The angle of point X from
the top of the tower is 6 0 calculate the angle of elevation of
the top of the tower from W.
8. The figure below shows a quadrilateral ABCD in which AB = 8
cm, DC = 12 cm, ∠BAD = 450, ∠CBD = 900 and ∠BCD = 300.

Find:

a) The length of BD
b) The size of the angle A D B
9. In the figure below, ABCDE is a regular pentagon and ABF is an
equilateral triangle

Find the size of

a) ∠ ADE
b) ∠ AEF
c) ∠ DAF
10. In this question use a pair of compasses and a ruler only
 (a) construct triangle ABC such that AB = 6 cm, BC = 8cm
and ∠ABC 1350
(b) Construct the height of triangle ABC in a) above taking
BC as the base
11. The size of an interior angle of a regular polygon is 3x 0 while its
exterior angle is (x- 20)0. Find the number of sides of the
polygon

12. Points L and M are equidistant from another point K. The

bearing of L from K is 3300. The bearing of M from K is 220 0.
Calculate the bearing of M from L

13. Four points B,C,Q and D lie on the same plane point B is the 42
km due south- west of town Q. Point C is 50 km on a bearing of
5600 from Q. Point D is equidistant from B, Q and C.

(a) Using the scale 1 cm represents 10 km, construct a

diagram showing the position of B, C, Q and D

(b) Determine the

(i) Distance between B and C

(ii) Bearing D from B

14. Two aeroplanes P and Q, leave an airport at the same time flies
on a bearing of 2400 at 900km/hr while Q flies due East at 750
km/hr

(a) Using a scale of 1v cm drawing to show the positions of

the aeroplanes after 40 minutes.
 (b) Use the scale drawing to find the distance between the two
aeroplane after 40 minutes

(c) Determine the bearing of

(i) P from Q ans 2540

(ii) Q from P ans 740

15. In the figure below, ABCDE is a regular pentagon and M is the

midpoint of AB. DM intersects EB at N. (T7)

Find the size of

(a) ∠BAE

(b) ∠BED

(c) ∠BNM

16. Use a ruler and compasses in this question. Draw a parallelogram

ABCD in which AB = 8cm, BC = 6 cm and BAD = 75. By
construction, determine the perpendicular distance between AB
and CD.
17. The interior angles of the hexagon are 2x 0, ½ x0, x + 400, 1100,
1300 and 1600. Find the value of the smallest angle.

18. The size of an interior angle of a regular polygon is 156 0. Find

the number of sides of the polygon.

COMMON SOLIDS

1. The figure below represents a square based solid with a path

marked on it.

Sketch and label the net of the solid.

2. The figure below represents below represents a prism of length 7

cm

AB = AE = CD = 2 cm and BC – ED = 1 cm
 Draw the net of the prism

3. The diagram below represents a right pyramid on a square base

of side 3 cm. The slant of the pyramid is 4 cm.

Draw a net of the pyramid

(a) On the net drawn, measure the height of a triangular face from the top
ofthe Pyramid (a) Draw a regular pentagon of side 4 cm

(b) On the diagram drawn, construct a circle which touches all the sides of
the pentagon

8(a) Sketch the net of the prism shown below

 (b) Find the surface area of the solid

INDICES AND LOGARITHMS

1. Use logarithms to evaluate

2.
√
3 36.15 × 0.02573
1938
Find the value of x which satisfies the equation.
16x2 = 84x-3

(1934) × √ 0.00324
2
3. Use logarithms to evaluate
436
4. Use logarithms to evaluate
55.9 ÷ (02621 x 0.01177) 1/5

5. Simplify 2x x 52x÷2-x
6. Use logarithms to evaluate
(3.256 x 0.0536)1/3
7. Solve for x in the equation
32(x-3) ÷8 (x-4) = 64 ÷2x
2X X
81 ×27 =729
8. Solve for x in the equations
9X
9. Use reciprocal and square tables to evaluate to 4 significant figures,
1 2
the expression: 24.55 + 4.346

10. Find the value of x in the following equation

49(x +1) + 7(2x) = 350

2
(0.07284 )
11. Use logarithms to evaluate 3
√ 0.06195
12. Find the value of m in the following equation
(1/27m)x (81)-1 = 243
13. Given that P = 3y express the equation 3(2y-1) + 2 x 3 (y-1) = 1 in terms
of P hence or otherwise find the value of y in the equation 3 (2y – 1) +
2 x 3 (y-1) = 1
14. Use logarithms to evaluate 55.9÷(0.2621 x 0.01177)1/5

GRADIENT AND EQUATIONS OF STRAIGHT LINES

1. The coordinates of the points P and Q are (1, -2) and (4, 10)
respectively.
A point T divides the line PQ in the ratio 2: 1
(a) Determine the coordinates of T
(b) i. Find the gradient of a line perpendicular to PQ
ii. Hence determine the equation of the line perpendicular PQ
and passing through T
(iii) If the line meets the y- axis at R, calculate the distance
TR, to three significant figures
2. A line L1 passes though point (1, 2) and has a gradient of 5. Another
line L2, is perpendicular to L1 and meets it at a point where x = 4.
Find the equation for L2 in the form of y = mx + c
3. P (5, -4) and Q (-1, 2) are points on a straight line. Find the equation
of the perpendicular bisector of PQ: giving the answer in the form
y = mx+c.
4. On the diagram below, the line whose equation is 7y – 3x + 30 = 0
passes though the points A and B. Point A on the x-axis while point
B is equidistant from x and y axes.
 Calculate the co-ordinates of the points A and B
5. A line with gradient of -3 passes through the points (3. k) and (k.8).
Find the value of k and hence express the equation of the line in the
form a ax + ab = c, where a, b, and c are constants.
6. Find the equation of a straight line which is equidistant from the
points (2, 3) and (6, 1), expressing it in the form ax + by = c where
a, b and c are constants.
7. The equation of a line -3/5x + 3y = 6. Find the:
(a) Gradient of the line

(b) Equation of a line passing through point (1, 2) and perpendicular

to the given line b
8. Find the equation of the perpendicular to the line x + 2y = 4 and
passes through point (2,1)
9. Find the equation of the line which passes through the points P
(3,7) and Q (6,1)
10. Find the equation of the line whose x- intercepts is -2 and y-
intercepts is 5
11. Find the gradient and y- intercept of the line whose equation is 4x –
3y – 9 = 0

ROTATION

1. A translation maps a point (1, 2) onto) (-2, 2). What would be the
coordinates of the object whose image is (-3, -3) under the same
translation?
2. Use binomial expression to evaluate (0.96) 5 correct to 4
significant figures
3. In the figure below triangle ABO represents a part of a
school badge. The badge has as symmetry of order 4 about O.
Complete the figures to show the badge.

O
C
 4. A point (-5, 4) is mapped onto (-1, -1) by a translation. Find the
image of (-4, 5) under the same translation.
5. A triangle is formed by the coordinates A (2, 1) B (4, 1) and C (1,
6). It is rotated clockwise through 900 about the origin. Find the
coordinates of this image.
6. The diagram on the grid provided below shows a trapezium ABCD

On the same grid

(a) (i) Draw the image A’B’C’D of ABCD

under a rotation of 900 clockwise about the

origin .

(ii) Draw the image of A”B”C”D” of A’B’C’D’ under a reflection in

line y = x. State coordinates of A”B”C”D”.
 (b) A”B”C”D” is the image of A”B”C”D under the reflection
in the line x=0.
Draw the image A”B” C”D” and state its coordinates.
(c) Describe a single transformation that maps A” B”C”D
onto ABCD.
7. on the Cartesian plane below, triangle PQR has vertices P(2, 3), Q (
1,2) and R ( 4,1) while triangles P” q “ R” has vertices P” (-2, 3),
Q” ( -1,2) and R” ( -4, 1)

(a) Describe fully a single transformation which maps triangle PQR

onto triangle P”Q”R”
(b) On the same plane, draw triangle P’Q’R’, the image of triangle
PQR, under reflection in line y = x
(c) Describe fully a single transformation which maps triangle
P’Q’R’ onto triangle P”Q”R
(d) Draw triangle P”Q”R” such that it can be mapped onto triangle
PQR by a positive quarter turn about (0, 0)
(e) State all pairs of triangle that are oppositely congruent

THE PYTHAGORA’S THEOREM

 1. Given sin (90 - a) = ½ , find without using trigonometric tables
the value of cos a

θ
2. If tan = ,find without using tables or calculator, the
tanθ−cosθ
value of cosθ+ sinθ
3. At point A, David observed the top of a tall building at an angle of
o
30 . After walking for 100meters towards the foot of the building he
stopped at point B where he observed it again at an angle of 60o. Find
the height of the building
4. Find the value of θ, given that ½ sinθ = 0.35 for 0o ≤ θ ≤ 360o
5. A man walks from point A towards the foot of a tall building
240m away. After covering 180m, he observes that the angle of
elevation of the top of the building is 45 o. Determine the angle of
elevation
of the top of the building from A
6.Solve for x in 2 Cos2x0 = 0.6000 00≤ x ≤ 3600.
7. Wangechi whose eye level is 182cm tall observed the angle of
elevation to the top of her house to be 32º from her eye
level at point A. she walks 20m towards the house on a
straight line to a point B at which point she observes the
angle of elevation to the top of the building to the 40º.
Calculate, correct to 2 decimal places the ;
a) distance of A from the house
b) The height of the house
8. Given that cos A = 5/13 and angle A is acute, find the value of:-
2 tan A + 3 sin A
9. Given that tan 5° = 3 + 5, without using tables or a calculator,
determine tan 25°, leaving your answer in the form a + b c
10. If tan θ =8/15, find the value of Sinθ - Cosθ without using a
calculator or table
Cosθ + Sinθ

AREA OF A TRIANGLE
1. The sides of a
triangle are in the ratio 3:5:6. If its perimeter is 56 cm, use the
Heroes formula to find its area

2. The figure below is a triangle XYZ. ZY = 13.4cm, XY = 5cm

and angle xyz = 57.7o

Calculate
i. Length XZ
ii. Angle XZY.
iii. If a perpendicular is dropped from point X to cut ZY at M, Find the
ratio MY: ZM.
iv. Find the area of triangle XYZ. (2 mks)

AREA OF QUADRILATERALS

∠PSR = 400 and PS = 10cm. Calculate the area of the

1. PQRS is a trapezium in which PQ is parallel to SR, PQ = 6cm, SR =
12cm,
trapezium.
 P > Q

40

S > R
2. A regular octagon has an area of 101.8 cm2. calculate the length of one
side of the octagon
3. Find the area of a regular polygon of length 10 cm and side n,
given that the sum of interior angles of n : n –1 is in the ratio
4:3.
4. Calculate the area of the quadrilateral ABCD shown:-

D
14cm
12cm
C

6cm
A B
18cm

AREA PART OF A CIRCLE

1. The figure below shows a circle of radius 9cm and centre O.

Chord AB is 7cm long. Calculate the area of the shaded region.

A B
 2. The figure below shows two intersecting circles with centres P
and Q of radius 8cm and 10cm respectively. Length AB = 12cm

P ө ß

B
Calculate:

∠APB
∠AQB
a)
b)
c) Area of the shaded region

3.

5cm 5cm

A B

The diagram above represents a circle centre o of radius 5cm.

The minor arc AB subtends an angle of 120 0 at the centre. Find
the area of the shaded part.
4. The figure below shows a regular pentagon inscribed in a circle
of radius 12cm, centre O.
 Calculate the area of the shaded part.
5. Two circles of radii 13cm and 16cm intersect such that they share a
common chord of length 20cm. Calculate the area of the shaded part
22
π=
7

6. Find the perimeter of the figure below, given AB,BC and AC are
diameters.

7. The figure below shows two intersecting circles. The radius of a

circle A is 12cm and that of circle B is 8 cm.
 If the angle MBN = 72o, calculate
i. The size of the angle MAN
ii. The length of MN
iii. The area of the shaded region.
8.

In the diagram above, two circles, centres A and C and radii 7cm
and 24cm respectively intersect at B and D. AC = 25cm.

a) Show that angel ABC = 900

b) Calculate

i) the size of obtuse angel BAD

ii) the area of the shaded part

9. The ends of the roof of a workshop are segments of a circle of

radius 10m. The roof is 20m long. The angle at the centre of the
circle is 120o as shown in the figure below:
 1200
10cm

(a) Calculate :-
(i) The area of one end of the roof
(b) The area of the curved surface of the roof What would be the cost to
the nearest shilling of covering the two ends and the curved surface
with galvanized iron sheets costing shs.310 per square meter.
10. The diagram below, not drawn to scale, is a regular pentagon
circumscribed in a circle of radius 10cm at center O

Find;
(a) The side of the pentagon

(b) The area of the shaded region

11. Triangle PQR is inscribed in he circle PQ= 7.8cm, PR = 6.6cm
and QR = 5.9cm. Find:
 Q
P 7.8cm

6.6cm
5.9cm

R
(a) The radius of the circle, correct to one decimal place
(b) The angles of the triangle
(c) The area of shaded region

SURFACE AREA OF SOLIDS

1. A swimming pool water

surface measures 10m long and 8m wide. A path of uniform width is made
all round the swimming pool. The total area of the water surface and the path
is 168m2
(a) Find the width of the path

(b) The path is to be covered with square concrete slabs. Each

corner of the path is covered with a slab whose side is equal to
the width of the path. The rest of the path is covered with
slabs of side 50cm. The cost of making each corner slab is sh
600 while the cost of making each smaller slab is sh.50.
Calculate

(i) The number of the smaller slabs used

(ii) The total cost of the slabs used to cover the whole path

1. The figure below shows a solid regular tetra pack of sides 4cm.

(a) Draw a labelled net of the solid.

 (b) Find the surface area of the solid.

2. The diagram shows a right glass prism ABCDEF with

dimensions as shown.
E

F
5.2cm D

A 5.2cm C

7.4 cm 14.7cm

B
Calculate:
(a) the perimeter of the prism

(b) The total surface area of the prism

(c) The volume of the prism
(d) The angle between the planes AFED and BCEF
3. The base of a rectangular tank is 3.2m by 2.8m. Its height is 2.4m. It
contains water to a depth of 1.8m. Calculate the surface area inside the tank
that is not in contact with water. Draw the net of the solid below and

calculate surface area of its faces

6.
G

4cm F
8cm D
B
A
5cm
The figure above is a triangular prism of uniform cross-section in
which AF = 4cm, AB = 5cm and BC = 8cm.
(a) If angle BAF = 300, calculate the surface area of the prism.
(b) Draw a clearly labeled net of the prisms.
7. Mrs. Dawati decided to open a confectionary shop at corner
Baridi. She decorated its entrance with 10 models of cone ice
cream, five on each side of the door. The model has the
following shape and dimensions. Using π= 3.142 and
calculations to 4 d.p.

(a) Calculate the surface area of the conical part.

(b) Calculate the surface area of the top surface.

(c) Find total surface area of one model.

(d) If painting 5cm2 cost ksh 12.65, find the total cost of painting the
models (answer to 1 s.f).
8. A right pyramid of height 10cm stands on a square base ABCD
of side 6 cm.
(a) Draw the net of the pyramid in the space provided below.
(b) Calculate:-

(i) The perpendicular distance from the vertex to the side AB.

(ii) The total surface area of the pyramid.

c) Calculated the volume of the pyramid.
9. The figure below shows a solid object consisting of three parts. A
conical part of radius 2 cm and slant height 3.5 cm a cylindrical part of
height 4 cm. A hemispherical part of radius 3 cm . the
cylinder lies at the centre of the hemisphere. ( π = 3.142 )

Calculate to four significant figures:

I.The surface area of the solid
II. The volume of the solid
10. A lampshade is in the form of a frustrum of a cone. Its bottom
and top diameters are 12cm and 8cm respectively. Its height is
6cm.Find;
(a) The area of the curved surface of the lampshade
(b) The material used for making the lampshade is sold at Kshs.800 per
square metres. Find the cost of ten lampshades if a lampshade is sold
at twice the cost of the material
11. A cylindrical piece of wood of radius 4.2cm and length 150cm is
cut lengthwise into two equal pieces. Calculate the surface area
of one piece
12. The base of an open rectangular tank is 3.2m by 2.8m. Its height
is 2.4m. It contains water to a depth of 1.8m. Calculate the
surface area inside the tank that is not in contact with water

13. The figure below represents a model of a solid structure in the

shape of frustrum of a cone with ahemisphere top. The diameter of the
hemispherical part is 70cm and is equal to the diameter of thetop of the
frustrum. The frustrum has a base diameter of 28cm and slant height of
60cm.
Calculate:
(a) The area of the hemispherical surface
(b) The slant height of cone from which the frustrum was cut
(c) The surface area of frustrum
(d) The area of the base
(e) The total surface area of the model
14. A room is 6.8m long, 4.2m wide and 3.5m high. The room has two
glass doors each measuring 75cm by 2.5m and a glass window
measuring 400cm by 1.25m. The walls are to be painted except the
window and doors.
a) Find the total area of the four walls
b) Find the area of the walls to be painted
c) Paint A costs Shs.80 per litre and paint B costs Shs.35 per litre. 0.8
litres of A covers an area of 1m2 while 0.5m2 uses 1 litre of paint
B. If two coats of each paint are to be applied. Find the cost of
painting the walls using:
i) Paint A
ii) Paint B
d) If paint A is packed in 400ml tins and paint B in 1.25litres tins,
find the least number of tins of each type of paint that must be
bought.
15. The figure below shows a solid frustrum of pyramid with a square
top of side 8cm and a square base of side 12cm. The slant edge of the
frustrum is 9cm

Calculate:
a) The total surface area of the frustrum

b) The volume of the solid frustrum

c) The angle between the planes BCHG and the base EFGH.

VOLUME OF SOLIDS

1. Metal cube of side

4.4cm was melted and the molten material used to make a sphere. Find
22
to 3 significant figures the radius of the sphere take π= 77
2. Two metal spheres of diameter 2.3cm and 3.86cm are melted.
The molten material is used to cast equal cylindrical slabs of
radius 8mm and length 70mm. If 1/20 of the metal is lost during
casting. Calculate the number of complete slabs casted.

3. The volume of a rectangular tank is 256cm 3. The dimensions are

as in the figure. Find the value of x

¼x
x-8
16cm

4.
R=14c
m

22.5cm

21cm

The diagram represent a solid frustum with base radius 21cm

and top radius 14cm. The frustum is 22.5cm high and is made of
a metal whose density is 3g/cm3 π = 22/7.
a) Calculate
i. the volume of the metal in the frustrum.
ii. the mass of the frustrum in kg.
b) The frustrum is melted down and recast into a solid cube. In the process
20% of the metal is lost. Calculate to 2 decimal places the length of each side
of the cube.
5. The figure below shows a frustrum
2.2 cm

4.8 cm
Find the volume of the frustrum

6. The formula for finding the volume of a sphere is given by

4
V = 3 π r 3 . Given that V = 311 and π =3.142, find r.

7. A right conical frustrum of base radius 7cm and top radius

3.5cm, and height of 6cm is stuck onto a cylinder of base radius
7cm and height 5cm which is further attached to a hemisphere to
form a closed solid as shown below

Find:
(a) The volume of the solid
 (b) The surface area of the solid

8. A lampshade is made by cutting off the top part of a square-

based pyramid VABCD as shown in the figure below. The base
and the top of the lampshade have sides of length 1.8m and 1.2m
respectively. The height of the lampshade is 2m

Calculate
a) The volume of the lampshade
b) The total surface area of the slant surfaces
c) The angle at which the face BCGF makes with the base
ABCD.
9. A solid right pyramid has a rectangular base 10cm by 8cm and
slanting edge 16cm.
calculate:
(a) The vertical height
(b) The total surface area
(c) The volume of the pyramid
10. A solid cylinder of radius 6cm and height 12cm is melted and
cast into spherical balls of radius
3cm. Find the number of balls made
 11. The sides of a rectangular water tank are in the ratio 1: 2:3. If the
volume of the tank is 1024cm3.
Find the dimensions of the tank. (4s.f)
The figure below represents sector OAC and OBD with
12.
radius OA and OB respectively. Given that OB is twice OA and
angle AOC = 60o. Calculate the area of the shaded region in m2,
given that OA = 12cm
B
C

O 600

14. The figure below represents a frustrum of a right pyramid on a square base.
The vertical height of the frustrum is 3 cm. Given that EF = FG = 6 cm and that
AB = BC = 9 cm

Calculate;
 a) The vertical height of the pyramid.
b) The surface area of the frustrum.
c) Volume of the frustrum.
d) The angle which line AE makes with the base ABCD.

15. The diagram below shows a metal solid consisting of a cone

mounted on hemisphere. The height of the cone is 1½ times its
radius;

Given that the volume of the solid is 31.5π cm3, find:

d) The radius of the cone
e) The surface area of the solid
f) How much water will rise if the solid is immersed totally in a
cylindrical container which contains some water, given the radius
of the cylinder is 4cm
g) The density, in kg/m3 of the solid given that the mass of the solid
is 144gm
16. A solid metal sphere of volume 1280 cm3 is melted down and recast
into 20 equal solid cubes. Find the length of the side of each cube.
Calculate the volume of the frustum.
 QUADRATIC EQUATIONS AND EXPRESSIONS

2 2
2 y − xy−x
1. Simplify 2 2
2x −y

2. Solve the following quadratic equation giving your answer to 3

23 1
d.p. x − 2 −¿120=0
x

3. Simplify as simple as possible ¿ ¿

4. The sum of two numbers x and y is 40. Write down an
expression, in terms of x, for the sum of the squares of the two
numbers.Hence determine the minimum value of x2 + y2
5. Mary has 21 coins whose total value is Kshs 72. There are twice
as many five shillings coins as there are ten shillings coins. The
rest one shilling coins. Find the number of ten shilling coins that
Mary has.
6. Four farmers took their goats to the market Mohamed had two
more goats than Ali Koech had 3 times as many goats as
Mohamed. Whereas Odupoy had 10 goats less than both
Mohamed and Koech.
I.) Write a simplified algebraic expression with one variable.
Representing the total number of goats
II.) Three butchers bought all the goats and shared them equally. If
each butcher got 17 goats. How many did Odupoy sell to the
butchers?

LINEAR INEQUALITIES

1. Find the range of x if 2≤ 3 – x <5

2. Find all the integral values of x which satisfy the inequalities:
2(2-x) <4x -9<x + 11
3. Solve the inequality and show the solution
2 x +5
3 – 2x<x ≤ 3
on the number line

4. Solve and write down all the integral values satisfying the
inequality.
X – 9 ≤ - 4 < 3x – 4
5. Show on a number line the range of all integral values of x which
satisfy the following pair of inequalities:
3–x≤1–½x
-½ (x-5) ≤ 7-x
6. Solve the inequalities 4x – 3< 6x – 1 ≤ 3x + 8; hence represent
your solution on a number line
7. Find all the integral values of x which satisfy the inequalities
2(2-x) < 4x -9< x + 11
8. Given that x + y = 8 and x²+ y²=34
9. Find the value of:- a) x²+2xy+y²
b) 2xy
10. Find the inequalities satisfied by the region
labelled R

11. Find all the integral values of x which satisfy the inequality
3(1+ x) < 5x – 11 <x + 45
12. The vertices of the unshaded region in the figure below are
O(0, 0) , B(8, 8) and A (8, 0). Write down the inequalities which satisfy the
unshaded region
 y B(8, 8)

x
14. WriteO(0,
down0)
the inequalitiesA(8,
that satisfy
0) the given region simultaneously. (3mks )

15. Write down the inequalities that define the unshaded region
marked R in the figure below. (3mks)
 16. Write down all the inequalities represented by the regions R.

17. a) On the grid provided draw the graph of y = 4 + 3x – x 2 for

the integral values of x in the interval -2≤X≤5. Use a scale of
2cm to represent 1 unit on the x – axis and 1 cm to represent 1
unit on the y – axis.
b) State the turning point of the graph.
c) Use your graph to solve.
(i) -x2 + 3x + 4 = 0
(ii) 4x = x2
 LINEAR MOTION

1. A bus takes 195 minutes to travel a distance of (2x + 30) km at an

average speed of (x -20) km/h Calculate the actual distance
traveled. Give your answers in kilometers.
2.) The table shows the height metres of an object thrown vertically
upwards varies with the time t seconds. The relationship between s and
t is represented by the equations s = at2 + bt + 10 where b are constants.
t 0 1 2 3 4 5 6 7 8 9 10
s 45.1
I.) Using the information in the table, determine the values of a
and b
II.) Complete the table
(b) (i) Draw a graph to represent the relationship between s and t
(ii) Using the graph determine the velocity of the object when t = 5 seconds
3.) Two Lorries A and B ferry goods between two towns which are
3120 km apart. Lorry A traveled at km/h faster than lorry B and B takes
4 hours more than lorry A to cover the distance.Calculate the speed of
lorry B
4.) A matatus left town A at 7 a.m. and travelled towards a town B at
an average speed of 60 km/h. A second matatus left town B at 8 a.m.
and travelled towards town A at 60 km/h. If the distance between the
two towns is 400 km, find;
I.) The time at which the two matatus met
II.) The distance of the meeting point from town A
5. The figure below is a velocity time graph for a car.
 y

Velocity (m/s)
80

0 4 20 24 x
Time (seconds)
(a) Find the total distance traveled by the car.
(b) Calculate the deceleration of the car.
6. A bus started from rest and accelerated to a speed of 60km/h as it
passed a billboard. A car moving in the same direction at a speed
of 100km/h passed the billboard 45 minutes later. How far from
the billboard did the car catch up with the bus?
7. Nairobi and Eldoret are each 250km from Nakuru. At 8.15am a
lorry leaves Nakuru for Nairobi. At 9.30am a car leaves Eldoret
for Nairobi along the same route at 100km/h. Both vehicles
arrive at Nairobi at the same time.
(a) Calculate their time of arrival in Nairobi
(b) Find the cars speed relative to that of the lorry.
(c) How far apart are the vehicles at 12.45pm.
8. Two towns P and Q are 400 km apart. A bus left P for Q. It
stopped at Q for one hour and then started the return journey to
P. One hour after the departure of the bus from P, a trailer also
heading for Q left P. The trailer met the returning bus ¾ of the
way from P to Q. They met t hours after the departure of the bus
from P.
(a) Express the average speed of the trailer in terms of t
(b) Find the ration of the speed of the bus so that of the trailer.
9. The athletes in an 800 metres race take 104 seconds and 108
seconds respectively to complete the race. Assuming each athlete
is running at a constant speed. Calculate the distance between
them when the faster athlete is at the finishing line.
10. A and B are towns 360 km apart. An express bus departs form A
at 8 am and maintains an average speed of 90 km/h between A
and B. Another bus starts from B also at 8 am and moves
towards A making four stops at four equally spaced points
between B and A. Each stop is of duration 5 minutes and the
average speed between any two spots is 60 km/h. Calculate
distance between the two buses at 10 am.
11. Two towns A and B are 220 km apart. A bus left town A at 11.
00 am and traveled towards B at 60 km/h. At the same time, a
matatu left town B for town A and traveled at 80 km/h. The
matatu stopped for a total of 45 minutes on the way before
meeting the bus. Calculate the distance covered by the bus before
meeting the matatu.
12. A bus travels from Nairobi to Kakamega and back. The average
speed from Nairobi to Kakamega is 80 km/hr while that from
Kakamega to Nairobi is 50 km/hr, the fuel consumption is 0.35
litres per kilometer and at 80 km/h, the consumption is 0.3 litres
per kilometer .Find
i) Total fuel consumption for the round trip
ii) Average fuel consumption per hour for the round trip.
13. The distance between towns M and N is 280 km. A car and a
lorry travel from M to N. The average speed of the lorry is 20
km/h less than that of the car. The lorry takes 1h 10 min more
than the car to travel from M and N.
(a) If the speed of the lorry is x km/h, find x
(b) The lorry left town M at 8: 15 a.m. The car left town M and
overtook the lorry at 12.15 p.m. Calculate the time the car left
town M.
14. A bus left Mombasa and traveled towards Nairobi at an average
speed of 60 km/hr. after 21/2 hours; a car left Mombasa and
traveled along the same road at an average speed of 100 km/ hr.
If the distance between Mombasa and Nairobi is 500 km,
Determine
(a) (i) The distance of the bus from Nairobi when the car took off
(ii) The distance the car traveled to catch up with the bus
(b) Immediately the car caught up with the bus
(c) The car stopped for 25 minutes. Find the new average
speed at which the car traveled in order to reach Nairobi at
the same time as the bus.
15. A rally car traveled for 2 hours 40 minutes at an average speed of
120 km/h. The car consumes an average of 1 litre of fuel for
every 4 kilometers. A litre of the fuel costs Kshs 59. Calculate
the amount of money spent on fuel
16. A passenger notices that she had forgotten her bag in a bus 12
minutes after the bus had left. To catch up with the bus she
immediately took a taxi which traveled at 95 km/hr. The bus
maintained an average speed of 75 km/ hr. determine
(a) The distance covered by the bus in 12 minutes
(b) The distance covered by the taxi to catch up with the bus
17. The athletes in an 800 metre race take 104 seconds and 108
seconds respectively to complete the race. Assuming each athlete
is running at a constant speed. Calculate the distance between
them when the faster athlete is at the finishing line.
18. Mwangi and Otieno live 40 km apart. Mwangi starts from his
home at 7.30 am and cycles towards Otieno’s house at 16 km/ h
Otieno starts from his home at 8.00 and cycles at 8 km/h towards
Mwangi at what time do they meet?
19. A train moving at an average speed of 72 km/h takes 15 seconds
to completely cross a bridge that is 80m long.
(a) Express 72 km/h in metres per second
 (b) Find the length of the train in metres

STATISTICS (I)

1. The height of 36 students in a class was recorded to the nearest

centimeters as follows.
148 159 163 158 166 155 155 179 158 155 171 172
156 161 160 165 157 165 175 173 172 178 159 168
160 167 147 168 172 157 165 154 170 157 162 173
(a) Make a grouped table with 145.5 as lower class limit and class width of 5.
2. Below is a histogram, draw.

5.0
Frequency density
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
11.5 13.517. 15.5 23.5
Length
Use the histogram above to complete the frequency table below:
Length Frequency
11.5 ≤ x ≤13.5
13.5 ≤ x ≤15.5
15.5 ≤ x ≤ 17.5
17.5 ≤ x ≤23.5
 3.
Kambui spent her salary as follows:

Food 40%
Transport 10%
Education 20%
Clothing 20%
Rent 10%
Draw a pie chart to represent the above information

4. The examination marks in a mathematics test for 60 students

were as follows;-

60 54 34 83 52 74 61 27 65 22
70 71 47 60 63 59 58 46 39 35
69 42 53 74 92 27 39 41 49 54
25 51 71 59 68 73 90 88 93 85
46 82 58 85 61 69 24 40 88 34
30 26 17 15 80 90 65 55 69 89
Class Tally Frequency Upper class
limit
10-29
30-39
40-69
70-74
75-89
90-99
From the table;
 (a) State the modal class
(b) On the grid provided , draw a histogram to represent
the above information
5. The marks scored by 200 from 4 students of a school were
recorded as in the table below.
Marks 41 – 51 – 56 – 66 – 71 –
50 55 65 70 85
Frequency 21 62 55 50 12
a.) On the graph paper provided, draw a histogram to represent this
information.
b.) On the same diagram, construct a frequency polygon.
c.) Use your histogram to estimate the modal mark.
6. The diagram below shows a histogram representing the marks
obtained in a certain test:-
Frequency Density
7

1
0
4.5 9.5 Marks 19.5 39.5 49.5

(a) If the frequency of the first class is 20, prepare a frequency

distribution table for the data
(b) State the modal class
(c) Estimate:
(i) The mean mark
(ii) The median mark
 ANGLE PROPERTIES OF A CIRCLE

1. The figure below shows a circle centre O and a cyclic quadrilateral

ABCD. AC =CD, angle ACD is 80 o and BOD is a straight line. Giving
reasons for your answer, find the size of :-
C

(i) Angle ACB

(ii) Angle AOD
(iii) Angle CAB
(iv) Angle ABC
(v) Angle AXB
2 In the figure below CP= CQ and <CQP =
1600. If ABCD is a cyclic quadrilateral, find < BAD.
 3 In the figure below AOC is a diameter of the circle centre O;
AB = BC and < ACD = 250, EBF is a tangent to the circle at
B.G is a point on the minor arc CD.

(a) Calculate the size of

(i) < BAD

(ii) The Obtuse < BOD

(iii) < BGD

(b) Show the < ABE = < CBF. Give reasons

4 In the figure below PQR is the tangent
 to circle at Q. TS is a diameter and TSR and QUV are straight
lines. QS is parallel to TV. Angles SQR = 40 0 and angle TQV =
550

Find the following angles, giving reasons for each answer

(a) QST
(b) QRS
(c) QVT
(d) UTV
4. In the figure below, QOT is a diameter. QTR = 480, TQR =
760 and SRT = 370
 Calculate
(a) <RST
(b) <SUT
(c) Obtuse <ROT
5. In the figure below, points O and P are centers of intersecting
circles ABD and BCD respectively. Line ABE is a tangent to
circle BCD at B. Angle BCD = 420
 (a) Stating reasons, determine the size of
(i) <CBD
(ii) Reflex <BOD
(b) Show that ∆ ABD is isosceles
6. The diagram below shows a circle ABCDE. The line FEG is a
tangent to the circle at point E. Line DE is parallel to CG, <
DEC = 280 and < AGE = 320
 Calculate:
(a) < AEG
(b) < ABC
7. In the figure below R, T and S are points on a circle centre
OPQ is a tangent to the circle at T. POR is a straight line and
<QPR = 200

Find the size of <RST

 VECTORS
10 −14
1. Given that 4p−3q = and p + 2q = find
5 15
a. (i) pandq
(ii) | p+2q|
b. Show that A (1, -1), B (3, 5) and C (5, 11) are collinear
1 6 −3
1
2. Given the column vectorsa= −2 b =−3 c = 2 and that p= 2a− b+c
3
1 9 3
a. (i) Express p as a column vector
b. (ii) Determine the magnitude of p
3. Given the points P(-6, -3), Q(-2, -1) and R(6, 3) express PQ and
QR as column vectors. Hence show that the points P, Q and R
are collinear.

4. The position vectors of points x and y are x = 2i+ j−3k and y

=3i+2j−2k respectively. Find x y as a column vector
1 −4 −3
5. Given that a~ 2, b~ = 5 , c~ = 2 and P~ = 2a~ + b~ − 3c~ .find p
~

6. The position vectors of A and B are 2 and 8

respectively. Find the coordinates of M 5 -7
which divides AB in the ratio 1:2.
7. The diagram shows the graph of vectors EF, FG and GH .
~ ~ ~
 Find the column vectors;

(a) EH
(b) | EH |
8. OA = 2i− 4k and OB =−2i+ j − k . Find AB
9. Given that p = 2i – j + k and q = i + j +2k, determine

(a.) │p + q│

(b) │ ½ p – 2q │