COURSE: GRAPHS / TRAVEL GRAPHS
INSTRUCTOR: MODOU S. MBYE (MR.)
TRAVEL GRAPHS
A travel graph tells you about a journey. It shows the relationship between distance and
time for the journey. Travel graphs are sometimes called journey graphs or distance-time
graphs or story graphs.
Time is usually on the horizontal axis.
Distance is usually on the vertical axis.
Distance
Time
Reading Travel Graphs
The simplest travel graph shows a journey made at steady or uniform speed. It is a straight
line graph. The slope or gradient of the graph line represents the speed. To find the average
speed, we use the formula:
Average speed = Total Distance
Total Time Taken
You can find the distance and time for a journey (or part of the journey) from the travel
graph. Then you can work out the average speed.
Different parts of a journey are often done at different speeds. The steeper the slope of the
line, the faster the speed is. Sometimes the journey has a stop in it. The graph shows the stop
too. The distance does not change but the time does, so the line is parallel to the time axis.
Remember to include any stops in the total time taken for the journey.
QUESTION ONE (1)
The diagram shows the graph of the journey of a bus between town A and Town D. stopping
at villages B and C on the way., the bus leaves town A at 09:00 hours. B, C and D are
respectively 40, 100 and160 kilometers from A. The broken line shows the graph of a second
bus which makes the journey from D to A without stopping.
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Distance from A (km)
D 160
120
C
80
B 40
09:00
0
10:00
11:00
12:00
13:00
Time (hours)
a)
b)
c)
d)
e)
How long did the first bus stops on the journey?
What was the average speed of the first bus between B and C?
At what time did the first bus reach D?
How far from A did the two buses meet?
What was the average speed of the second bus on the journey?
QUESTION TWO (2)
A motorist drives from town P to town Q, a distance of 110km in 2 hours. He stays for 1
hour in town Q. he then drives back to town P taking 2 ½ hours.
a) On graph paper, choose a scale of 2cm to one hour on the horizontal axis and 2cm to
20km on the vertical axis. Draw a graph of the motorist’s journey.
b) Calculate his average speed:
i. Oh his journey to town Q;
ii. On his return journey;
iii. For the whole journey including the time he was stopped.
QUESTION THREE (3)
a) Taking values of x from O to 4 and using a scale of 2cm to 1 unit on the x-axis and
2cm to 2 units on the y-axis.
Draw the graphs of the equations y = 3x + 2 and y = 5x
b) From the graph, find the coordinates of x and y at the point of intersection of the two
lines.
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QUESTION FOUR (4)
a). Copy and complete the table of values for the relations
y1 = 2x – 4 and y2 = 5 – x.
x
y1 = 2x – 4
y2 = 5 – x
-4
-12
9
-3
-2
-1
-6
6
8
0
1
-2
2
3
2
a) i) Using a scale of 2cm to 1 unit on the x- axis and 2 units on the y – axis, draw two
perpendicular aces OX and OY on a graph sheet.
ii) On the same graph sheet, draw the graphs of the relations y1 = 2x – 4 and y2 = 5 – x
b) Find the coordinates of the point where y1 and y2 meet.
QUESTION FOUR (4)
A passenger Bus A left Banjul for Kartong via Brufut. An hour later another Bus left
Kartong for Banjul traveling by the same route. The travel graphs of the two buses are shown
below:
Kartong 80
BUS A
DISTANCE IN KM
70
60
50
Brufut
40
30
20
BUS B
10
Banjul
0 900
1000
1300
1100
1200
TIME
Using the graph, find:
a. The time when Bus B left Kartong for Banjul.
b. The time when the two Buses passed by each other.
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c. The speed of Bus A between Banjul to Brufut.
d. How far from Kartong was bus B at 10:30?
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COURSE: BEARING & CONSTRUCTIONS
INSTRUCTOR: VINCENT ODION (MR.)
BEARING
QUESTION ONE (1)
a). A ship sails from S, southwards to town R at a distance of 5m. What is the bearing of R
from S?
b). The bearing of A from B is 1200. Find the bearing of B from A.
QUESTION TWO (2)
The bearing of X from y is 0700 and the bearing of Z from y is 1600, where X, Y and Z are
three points in the plane. If Y is equidistant from X and Z,
a) Find the bearing of Z and X
b) What type of triangle is XY Z?
QUESTION THREE (3)
Mr. Njie walked from point w north ward covering a distance of 40mm and turned left at
point x west ward at an angle of 450.
What is the bearing of y from x?
QUESTION FOUR (4)
The bearing of N from M is 1400 where MNOP are four points in the plane of an equidistant
of 8m.
a) From N to O South wards
b) From O to P North West and parallel to line MN above.
c) From P to M North wards
i. Find the bearing of M from N
ii. Find the bearing of O from N
iii. Find the bearing of P from O
iv. Find the bearing of M from P
v. What type of quadrilateral is MNOP?
CONSTRUCTIONS
QUESTION FIVE (5)
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(a) Draw a line of 100mm which inclines at 600 and bisect it.
(b) Construct an angle of 750 and bisect it.
QUESTION SIX (6)
ABCD is rhombus.
DAB = 600
AB = AD = 6cm
Diagonal BD = 6.5cm
(i) Complete the rhombus
(ii) Measure diagonal AC
(iii) What type of triangle is AOB?
QUESTION SEVEN (7)
Construct a quadrilateral ABCD.
GIVEN DATA:
 Diagonal AC = 125mm
 Diagonal BD = 52mm
 Both diagonals intersect at O distance of 25mm from A.
What is the name of the quadrilateral?
QUESTION EIGHT (8)
Construct a quadrilateral EFGH in which EF = 9.5cm,
EFG = 900,
FEH = 450 .
Draw a line to join distance G and H. Measure line GH if EH = 6cm and FG = 4.5cm
Which quadrilateral is the figure EFGH?
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COURSE:
ELEMENTARY STATISTICS / PROBABILITY
INSTRUCTOR: ABDOULIE CONTA (MR.)
QUESTION ONE (1)
A survey conducted on 270 students in Basori at Basori Lower Basic School to find out their
favorite soft drinks. The result was as follows:
SOFT DRINKS
Fanta
Muscatella
Sprite
Coca-Cola
Vimto
NO. OF
STUDENTS
45
30
60
120
15
a)
Draw a bar chart to illustrate the above information.
Draw a pie chart to show the information.
What fraction of the students prefer sprite?
If a student is chosen at random from the students, what is the probability that he
prefer coca-cola?
QUESTION TWO (2)
The frequency table below is the mark distribution of a class of 35 pupils in an English
Examination.
MARKS
1
FREQUENCY 2
2
4
3
5
4
6
5
10
6
1
7
2
8
3
9
1
10
1
(a) Find the mean mark for the class.
(b) What is the median mark?
(c.) What is the modal mark?
What is the probability that a pupil chosen from the class obtained
(i) A mark less than 5?
(i) A mark greater than or equal to 7?
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QUESTION THREE (3)
A box contains 15 red balls, 5 green balls and 10 blue balls.
(a) How many balls are in the box?
(b) What is the probability of picking
(i) Green ball?
(ii) Red Ball?
(iii) Blue ball?
(iv) Red ball or Blue ball?
(v) Red ball and Blue ball?
QUESTION FOUR (4)
Two dice are thrown once.
(a) What is the sample space or possible out come?
(b) If a number is chosen at random, what is the probability that
(i) It is a total of 8
(ii) It is a total of less than 6
(iii) The total is an even number.
QUESTION FIVE (5)
A number is chosen at random from the integers 5 to 25 inclusive. Find the probability that
the number is a
(i) Multiple of 3
(ii) Multiple of 5
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COURSE:
SETS & FORMS /
INSTRUCTOR: ISSA DUMBUYA (MR.)
QUESTION ONE (1)
a), If P = {1, 2, 3, 4} write down all the subsets of P which have exactly two elements.
A = {prime numbers less than 15}
B = {even numbers less than 15}
C = {x : 3 ≤ x < 12, where x is an integer}
List the elements of
i.
A∩ C
ii.
B∩C
iii.
( A U B) ‘∩ C
QUESTION TWO (2)
U is a universal set of all prime numbers.
P and Q are subsets of U given by
P = { x : 3 (x + 1) ≥ 2 (x + 1)} and
Q = { x : 7 < x < 31} where x is an integer.
List the elements of P ∩ Q.
QUESTION THREE (3)
During one year in a village, ⅝ of the children had malaria, ½ had children pox and ⅛ had
neither. What fraction of the children had both chicken pox and malaria?
Show the information in the venn diagram.
QUESTION FOUR (4)
If D(∩) is the set of factors of ∩, find
i.
ii.
D(48) ∩ D (64)
D(12) ∩ D (16)
Listing the members we have
D(48) = { 1, 2, 3, 4, 6, 8, 12, 16, 24, 48}
D(64) = { 1, 2, 4, 8, 16, 32, 64}
D(12) = { 1, 2, 3, 4, 6, 12}
D(16) = { 1, 2, 4, 8, 16}
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COURSE: SOLID MENSURATION
INSTRUCTOR: LAMIN DAMPHA (MR.)
Surface Areas and Volumes.
CUBE
A cube has six faces of equal dimension. This means that it has length, breadth (width) and
height all of which are equal to each other.
Example:
A cube has a length of 6cm, calculate
i)
ii)
iii)
Surface area
Length of a diagonal
Volume
CYLINDER
Example: A measuring cylinder of radius 3 cm contains water to a height of 49 cm. If this
water is poured into a similar cylinder of radius 7 cm, what will be the height of water column?
Example:
A doorway is made up of an arch, ABC which is a quarter of a circle of radius 2 m, two
uprights,
, of height 3 m, and a horizontal base
C
A
2m
2m
3m
D
3m
E
a) Show that the perimeter of the doorway in meters is
b) Find a similar expression for the area of the doorway in terms of
Simplify your
answer as far as possible (
Example:
A cylindrical container has a base radius of 8 cm and its height is 19 cm. calculate:
a) Total surface area
b) Curved surface area
c) Volume
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(Hint: assume that it is opened at the top)
SPHERE / HEMISPHERE
Example:
A solid sphere has a radius of 6 cm and is made of metal of density
calculate the
mass of the sphere in kg.
CONE
Example:
A cone has a base radius of 6 cm and a height of 9 cm. calculate:
a) Volume;
b) Curved surface area
c) Total surface area
Example:
A solid consist of a cone on top of cylinder of the same radius. The length of the solid is 15 cm.
If the height of the cone is equal to the common radius, show that the volume of the solid is
Example:
A rectangular garden 30 m by 20 m consist of a lawn 28 m by 18 m surrounded by a path. The
path is to be made of concrete to a depth of 8cm. calculate the volume of the cement required in
The cement is delivered in bags each holding one cubic meter. Calculate the costs if the cement
cost D600 per bag and the labour charges are D2000. The company states that the cost will be
approximately D70.00 per square meter. Is this correct
Example: The solid below is a cylinder surmounted by a hemisphere bowl.
a) calculate its total surface area
7cm
b) volume (Take
=
)
10cm
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Example:
A solid consist of a cone attached to a hemisphere as shown below
7cm
Calculate the volume of the solid if the diameter of the hemisphere is 3 cm and the overall height
of the object is 7 cm.
CONE FORM FROM A SECTOR
O
O
A,B
A
B
Example:
In the diagram below, O is the center of the circle radius 32 cm. If
calculate,
correct to two decimal places, the area of the
a) Minor sector POQ
b) Shaded portion
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c) Of the sector POQ is used to form the curved surface of a cone with vertex O, calculate
the base radius of the cone, correct to one decimal place
R
O
P
Q
Example:
The figure below shows a frustum of a cone while bottom and top radii are respectively 12 cm
and 8 cm. If the height is 13 cm, calculate its:
8 cm
13 cm
12 cm
a) Curved surface area
b) Total surface area
c) Volume
Example: A frustum of a pyramid consists of a square base of length 10 cm and a top square of 7
cm. The height of the frustum is 6 cm. Calculate:
a) Surface area
b) Volume
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PYRAMID
Example:
1. A pyramid with vertex O stands on a square base ABCD
and
. Calculate :
a)
b)
c)
d)
e)
the height
The volume
The angle between plane OAB and the base
The total surface area of triangular faces
The angle between faces OAB and OCD
Example:
PQRV is a right pyramid with base PQR which is an equilateral triangle of sides 18 cm.
each slopping edge of the pyramid is of length 13 cm. G is the center of the symmetry of
triangle PQR. Given that
= 10.4cm. Calculate:
a) The angle between PR and the base to the nearest degree
b) The height
of pyramid correct to one decimal places
c) The volume of the pyramid in
correct to the nearest integer
O
Example
The model below shows a pyramid EFGHV on a cuboid ABCDEFGH.
. The volume of the model is
Find , correct to the nearest whole number
a) The height of the pyramid
b) The slant height of the pyramid
c) The length of the slant edge VG
d) The angle between the face VFG and the base EFGH.
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.
Exercise:
1.
A pyramid VABCD has a square base of 8 cm. the volume of the pyramid is
.
Find
a) The height of the pyramid
b) The slant height of the pyramid
c) The length of the slopping edge
2. A pyramid VABCD has a square base of side 8 cm and each side of the slopping edge is 9
cm. calculate:
a) The height of the pyramid
b) The angle between the plane VAB and VCD
c) The volume of the pyramid to two significant figures
3. A right pyramid on a base 10 m square is 15 m high.
a) Find the volume of the pyramid
b) If the top 6 m of the pyramid are removed, what is the volume of the remaining frustum?
4. A right circular cylinder of height 14 cm and a radius 6 cm is filled with water. A heavy
circular cone of height 11 cm and a base- radius 8 cm is lowered, with vertex downwards and
axis vertical, into the cylinder until the cone rests on the rim of the cylinder. Find
a) The volume of water which spills over from the cylinder, and
b) The height of the water in the cylinder after the cone has been removed.
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