Chapter 10
Chapter 11
Chapter 12
Applications in Trigonometry
10A
p.2
10B
p.15
10C
p.26
Coordinate Geometry of Straight Lines
11A
p.40
11B
p.48
11C
p.59
11D
p.73
11E
p.83
Introduction to Probability
12A
p.89
12B
p.98
12C
p.111
12D
p.118
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1
F3B: Chapter 10A
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Consolidation Exercise
Maths Corner Exercise
10A Level 1
Maths Corner Exercise
10A Level 2
Maths Corner Exercise
10A Level 3
Maths Corner Exercise
10A Multiple Choice
E-Class Multiple Choice
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Book 3B
Lesson Worksheet 10A
(Refer to §10.1)
[In this worksheet, unless otherwise stated, give the answers correct to 3 significant figures if
necessary.]
10.1A
Gradient of an Inclined Plane
Refer to the figure.
If AB represents an inclined plane, then
vertical distance
BC
gradient of AB =
=
horizontal distance
AC
e.g. Gradient of the inclined plane XY in the figure
YZ
=
XZ
5m
=
10 m
1
= (or 1 : 2) Gradients are usually expressed in the
2
1
form of
n
Y
5m
Z
(b)
0.4 m
12 m
(c)
400 m
60 000 m
Example 1
Instant Drill 1
A
B
C
○→ Ex 10A 1
gradient = 1 : 10
1
6
X
or 1 : n, where n is an integer.
1. In each of the following, find the gradient of the inclined road AB.
Vertical
Horizontal
Gradient of AB
distance AC
distance BC
(in the form of 1 : n)
(
) cm
=1:(
)
5 cm
20 cm
(a)
(
) cm
gradient =
10 m
vertical distance
=hm
vertical distance
= 20 m
horizontal
distance
= 70
May walks up along
an inclined
road
asmshown.
Find the value of h.
(
)
Sol Gradient of the road =
(
)
=
horizontal
A car travels
downdistance
along =and m
inclined road as
shown. Find the value of d.
20 m
Sol Gradient of the road =
dm
1
20
=
6
d
d = 20 × 6
= 120
4
2. Kary walks along an inclined road with gradient 1 : 7. If the horizontal distance she walks is
17.5 m, find the vertical distance she walks.
gradient =
Let d m be the vertical distance she walks.
(
)
Gradient of the road =
(
)
(
)
(
)
=
(
)
(
)
=
(
)
(
)
?
_____ m
○→ Ex 10A 2, 3
3. When an ant crawls 85 cm up along a straight stick, the vertical distance
that it crawls is 13 cm.
(a) Find the horizontal distance that the ant crawls.
A
1
(b) Find the gradient of the stick in the form of , where n is correct to
n
the nearest integer.
(a) Let AC = x cm. In △ABC,
x2 + (
)2 = (
)2
=
85 cm
B
13 cm
C
Use Pythagoras’ theorem to find x.
Recall:
c
b
a 2 + b 2 = c2
a
1
be the gradient of the stick.
n
BC
Gradient of the stick =
AC
(
)
1
=
n
(
)
=
(b) Let
○→ Ex 10A 4, 5
10.1BGradient and Inclination
Refer to the figure.
(a) The angle θ between the inclined plane AB and
the horizontal (AC) is called the inclination of AB.
BC
(b) Gradient of AB =
= tan θ
AC
(inclination)
Y
e.g. Gradient of the inclined plane XY in the figure
= tan 38°
= 0.781, cor. to. 3 sig. fig.
X
5
38°
horizontal
Z
4. In each of the following, find the gradient of the inclined plane AB.
(a)
(b)
(c)
B
A
A
24°
horizontal
C
11°
B
horizontal
C
B
A
50°
horizontal
C
In each of the following, find the gradient of an uphill road with the given inclination. [Nos. 5–6]
(Express the answers in the form of 1 : n, where n is correct to the nearest integer.)
5. 7°
6. 4.4°
1

Let 1 : n  i.e.  be the gradient of the road.
n

Gradient of the road = tan (
)
1
= tan (
n
)
=
∴ The gradient of the road is
○→ Ex 10A 6
.
Example 2
The gradient of a road is
Instant Drill 2
Find the inclination of a path with gradient
1 : 18.
Sol
1
. Find its
4
inclination.
Sol Let θ be the inclination of the road.
1
tan θ =
4
θ = 14.0°, cor. to 3 sig. fig.
∴ The inclination of the road is 14.0°.
7.
In each of the following, find the
inclination θ of an inclined road with the
given gradient.
5
(a) 3.7
(b)
8
8.
B
42 m
A
Find the inclination
inclined road
75ofmthehorizontal
AB in the figure.
Find the gradient
first.
6
○→ Ex 10A 9
○→ Ex 10A 8
9. The gradient of an inclined path is 0.62.
(a) Find the inclination of the path.
(b) Andy walks 100 m up along the path. Find the horizontal distance he walks.
100 m
inclination (
)
?
Use ‘sin’, ‘cos’ or ‘tan’?
○→ Ex 10A 10, 11
10.1C
Gradient on Map
The figure shows a contour map. If AB represents an inclined
400 m
A
straight path with horizontal distance 3 000 m, we have:
300 m
(i) Since A lies on the contour line with label ‘400 m’,
A is 400 m above the sea level. Similarly, B is 100 m above
200 m
the sea level.
100 m
B
(ii) Vertical distance between A and B = (400 − 100) m
= 300 m
300 m
(iii) Gradient of AB =
Recall:
3 000 m
vertical distance
Gradient =
1
=
horizontal distance
10
7
Example 3
Instant Drill 3
350 m
500 m 400 m
P
M
300 m
250 m
200 m
300 m
200 m
Q
N
The figure shows a contour map, where PQ
represents a straight road. It is given that the
horizontal distance between P and Q is 900 m.
Find
(a) the vertical distance between P and Q,
(b) the gradient of the straight road, and
express the answer as a fraction.
The figure shows a contour map, where MN
represents a straight road. It is given that the
horizontal distance between M and N is
3 600 m. Find
(a) the vertical distance between M and N,
(b) the gradient of the straight road, and
express the answer as a fraction.
Sol (a) Vertical distance between P and Q
= (350 – 250) m
= 100 m
(b) Gradient of the road
100 m
=
900 m
1
=
9
Sol (a) Vertical distance between M and N
= [(
)–(
)] m
=
(b)
10.The figure shows a contour map of a hill. A straight path is built from
1
point A to point B. It is given that the gradient of the path is . Find
5
(a) the vertical distance between A and B,
(b) the horizontal distance between A and B.
650 m
675 m
625 m
A
600 m
B
○→ Ex 10A 13, 14
8
‘Explain Your Answer’ Question
11. PQ and RS are two straight highways. It is given that the gradient of PQ is 1 : 7 and the
inclination of RS is 10°.
The greater the
(a) Find the gradient of RS.
gradient, the
(b) Which highway is steeper, PQ or RS? Explain your answer.
steeper the
(a) Gradient of RS =
(b) Gradient of PQ =
∵ Gradient of PQ ( > / < ) gradient of RS
∴ Highway (PQ / RS) is steeper.
 Level Up Question
12. In the figure, AB and BC represent two straight roads. ADC is a
horizontal line, BD ⊥ AC, BD = 2 m, AC = 30 m and the
inclination of AB is 8°.
A
(a) Find the lengths of AD and DC.
(b) Find the inclination of BC.
9
B
8°
2m
D
30 m
C
New Century Mathematics (2nd Edition) 3B
10
Applications in Trigonometry

Consolidation Exercise
10A
Level 1
1. (a) AB is a straight road. The vertical distance between A and B is 5 m. The horizontal distance
between A and B is 50 m. Find the gradient of the road AB.
(b) Jason travels up along a straight road. When he travels a horizontal distance of 16 m, he
rises 6 m vertically. Find the gradient of the road.
2. In the figure, David runs from P to Q along a road with gradient 1 : 30.
Q
P
1 800 m
If he runs a horizontal distance of 1 800 m, find the vertical distance travelled by him.
3. The figure shows a road AB with gradient
1
.
17
A
3.4 m
B
If a car travels a vertical distance of 3.4 m, find the horizontal distance travelled by the car.
4. In the figure, the length of the lane XY is 25.7 m. The horizontal distance between X and Y is
25.5 m.
Y
25.7 m
X
25.5 m
(a) Find the vertical distance between X and Y.
(b) Express the gradient of the lane in the form of
1
, where n is correct to the nearest integer.
n
5. In each of the following, find the gradient of an uphill road with the given inclination.
(a) 8°
(b) 2.3°
( Express the answers in the form of 1 : n, where n is correct to the nearest integer.)
10
6. The figure shows a road AB with inclination θ. Find the gradient of the road in each of the
following situations.
A
B
θ
(a) tan θ = 0.06
(b) θ = 5.2°
(c) sin θ = 0.4
1
(Express the answers in the form of , where n is correct to the nearest integer.)
n
7. In each of the following, find the inclination of a straight road with the given gradient.
7
(a) 1.5
(b)
(c) 1 : 18
3
(Give the answers correct to the nearest 0.1°.)
8. In the figure, a cable car travels along a straight cable. If it travels a horizontal distance of
960 m and a vertical distance of 128 m, find the inclination of the cable, correct to 3 significant
figures.
128 m
960 m
9. In the figure, a cat walks down along a path with gradient 0.2.
gradient = 0.2
(a) What is the inclination of the path?
(b) If the cat walks 15 m along the path, find the horizontal distance the cat walks.
(Give the answers correct to 3 significant figures.)
8
. The inclination of another road QR is 16°. Wendy claims that
25
the road PQ is steeper than the road QR. Do you agree? Explain your answer.
10. The gradient of a road PQ is
11
11. The figure shows a contour map, where XY represents a
straight road of gradient 1 : 12.
(a) Find the vertical distance between X and Y.
(b) Find the horizontal distance between X and Y.
250 m Y
200 m
150 m
100 m
12. The figure shows a contour map. PQ represents a straight
road and the horizontal distance between P and Q is 400 m.
(a) Find the gradient of the road, and express the answer as
a fraction.
(b) Find the inclination of the road.
(Give the answer correct to the nearest degree.)
P
X
450 m
500 m
400 m
Q
350 m
Level 2
13. In the figure, Jenny drives at a speed of 14 m/s from P to Q along a straight road with
1
gradient . She rises 56 m vertically in the whole journey.
3
Q
P
(a) Find the inclination of the road.
(b) Find the time taken in the whole journey.
(Give the answers correct to 3 significant figures.)
14. Fig. A shows a ladder PQ which leans against a vertical wall. The inclination of the ladder is
32° and the foot of the ladder is 1.4 m from the corner O of the wall.
P
X
0.7 m
Q
32°
Y
1.4 m
Fig. A
Fig. B
(a) Find the length of the ladder.
(b) Later, the top P slides down to X which is 0.7 m above O, and the foot Q slides to Y at the
same time, as shown in Fig. B. Find the new inclination of the ladder.
(Give the answers correct to 3 significant figures.)
12
15. In the figure, XY and YZ represent two straight roads, where X and Z are two points on the
horizontal ground. N is a point on XZ such that YN ⊥ XZ. YN = 80 m and XZ = 650 m.
Y
80 m
X
Z
650 m
It is given that the inclination of the road XY is 13°.
(a) Find the length of NZ, correct to 3 significant figures.
(b) Which path, XY or YZ, is less steep? Explain your answer.
16. The figure shows two straight roads PQ and QR of lengths 2 000 m and 960 m respectively. The
gradient of PQ is 0.32. ONR is a horizontal line.
P
2 000 m
Q
96
O
R
N
(a) Find the inclination of PQ. Hence, find the horizontal distance between P and Q.
(b) It is given that the horizontal distance between P and R is 2 800 m. Find the gradient
of QR.
(Give the answers correct to 3 significant figures.)
17. On a contour map of scale 1 : 8 000, AB represents a straight road. The vertical distance
between A and B is 50 m. If the inclination of the road is 9°, find the length of AB on the map
correct to the nearest 0.01 cm.
18. The figure shows a contour map of scale 1 : 30 000. XY
and YZ represent the two parts of a hiking trail. The
length of XY on the map is measured as 3 cm.
(a) Find the inclination of the path XY, correct to
3 significant figures.
(b) If the actual total length of the trail is 1.7 km, which
path, XY or YZ, is steeper? Explain your answer.
450 m
Y
Z
250 m
400 m
350 m
X
300 m
200 m
Scale 1 : 30 000
13
Answer
Consolidation Exercise 10A
1

1. (a) 0.1  or 
 10 
2. 60 m
3. 57.8 m
4. (a) 3.2 m
 3
(b) 0.375  or 
 8
(b)
1
8
(b) 1 : 25
5. (a) 1 : 7
1
1
1
6. (a)
(b)
(c)
17
11
2
7. (a) 56.3°
(b) 66.8°
(c) 3.2°
8. 7.59°
9. (a) 11.3°
(b) 14.7 m
10. yes
11. (a) 150 m
(b) 1 800 m
1
12. (a)
4
(b) 14°
13. (a) 18.4°
(b) 12.6 s
14. (a) 1.65 m
(b) 25.1°
15. (a) 303 m
(b) XY
16. (a) inclination = 17.7°,
horizontal distance = 1 900 m
(b) 0.387
17. 3.95 cm
18. (a) 9.46°
(b) XY
14
F3B: Chapter 10B
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Consolidation Exercise
Maths Corner Exercise
10B Level 1
Maths Corner Exercise
10B Level 2
Maths Corner Exercise
10B Level 3
Maths Corner Exercise
10B Multiple Choice
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15
Teacher’s
Signature
___________
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)
Teacher’s
Signature
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)
Teacher’s
Signature
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)
Teacher’s
Signature
___________
E-Class Multiple Choice
Self-Test
○
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16
(
Mark:
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)
Book 3B
Lesson Worksheet 10B
(Refer to §10.2)
[In this worksheet, unless otherwise stated, give the answers correct to 3 significant figures if
necessary.]
10.2 Angles of Elevation and Depression
(a) When we look up at an object, the angle
between the line of sight and the horizontal
is called the angle of elevation.
When we look down at an object, the angle
between the line of sight and the horizontal
is called the angle of depression.
e.g. In the figure,
the angle of elevation of P from A = 60°
the angle of depression of B from P = 40°
horizontal
40°
60°
(b) Angle of elevation of B from A
= angle of depression of A from B
Example 1
B
A
P
horizontal
Instant Drill 1
C
D
40 m
6m
E
A
2m
B
F
27 m
horizontal
groun
d
building of height
horizontal ground
In the figure, BC is a lamp post of height 6 m.
A is 2 m away from B. Find the angle of
elevation of C from A.
Sol Let θ be the angle of elevation
C
of C from A.
In △ABC,
6m
6m
tan θ =
θ
2m
A 2m B
θ = 71.6°, cor. to 3 sig. fig.
∴ The required angle of elevation is
71.6°.
In the figure, DE is a
40 m.
F is 27 m away from E. Find the angle of
elevation of D from F.
Sol Let θ be the angle of elevation D
of D from F.
40 m
In △DEF,
θ=
=
17
(
(
)
)
E
θ
27 m
F
1.
2.
R
H
110 m
5.3 m
185 m
A
B
horizontal
groun
d
Q
In the figure, a hot-air balloon H is 110 m
vertically above B. Points A and B are
185 m apart. Find the angle of elevation of
H from A.
S horizontal ground
4.6 m
The figure shows a road sign RS of height
5.3 m. Q is 4.6 m away from S. Find the
angle of depression of Q from R.
Refer to the notations in the figure.
P
In △QRS,
θ=
(
(
)
)
5.3 m
θ
=
Q
∠PRQ =
R
4.6 m
S
alt. ∠s, PR // QS
∴ The required angle of depression is
.
→
○ Ex 10B 1, 2
3.
4.
B
51°
T
600 m
A
sea level
23°
horizontal ground
15 m
S
P
In the figure, a bird B is 600 m from a
buoy A at sea level. The angle of
depression of A from B is 51°. Find the
horizontal distance between A and B.
In the figure, the tree TS casts a shadow PS
of length 15 m on the ground. The angle of
elevation of the sun from P is 23°. Find the
height of the tree.
In △PST,
23° =
(
(
)
)
=
→
○ Ex 10B 3–5
18
Example 2
Instant Drill 2
C
R
P
35°
84 m
9.5 m
A
50 m
64°
1.7 m
S
Q
B
D horizontal ground
In the figure, the height of the flagpole CD is
9.5 m. Ivan stands at point B. His eyes at A are
1.7 m above the ground. The angle of elevation
of C from A is 64°. Find the distance between
Ivan and the flagpole.
Sol With the notations in the figure,
construct AE ⊥ CD.
CE = CD – ED
C
= CD – AB
= (9.5 – 1.7) m
9.5 m
= 7.8 m
A 64° E
In △ACE,
1.7 m
CE
B
D
tan 64° =
AE
CE
AE =
tan 64°
7.8
=
m
tan 64°
= 3.80 m, cor. to 3 sig. fig.
∴ The distance between Ivan and the
flagpole is 3.80 m.
In the figure, PQ and RS are two buildings on
the horizontal ground, and their heights are
50 m and 84 m respectively. The angle of
elevation of R from P is 35°. Find the distance
between the two buildings.
Sol With the notations in the figure,
construct PT ⊥ RS.
RT = (
)–(
)
R
=
P
35°
T 84 m
50 m
In △PRT,
35° =
(
(
)
)
S
Q
=
5. In the figure, the top P of a table is 130 cm above the horizontal
table P
ground. The angle of elevation of P from D is 28°. P and D are 200
cm apart. Find the height of D above the ground.
130 cm
Q
19
200 cm
28°
D
6. In the figure, the heights of a spotlight (X) and the top of a display board (Y)
are 6 m and 3.3 m above the horizontal ground respectively. The horizontal
distance between X and Y is 2.5 m. Find the angle of depression of Y from
X.
X
Y
6m
display
bo
ard
3.3 m
2.5 m
○→ Ex 10B 6–9
 Level Up Question
7. In the figure, A and B are two windows of a building on the horizontal
ground. The angles of elevation of A and B from a point C on the ground
are 42° and 63° respectively. A, B and C lie on the same vertical plane.
Find the distance between A and B.
B
A
63°
C
20
42°
9m
New Century Mathematics (2nd Edition) 3B
10
Applications in Trigonometry
Consolidation Exercise
10B

[In this exercise, give the answers correct to 3 significant figures.]
Level 1
1. In the figure, XY represents a building of height 45 m. K is a point on the horizontal ground and
KY = 80 m. Find the angle of elevation of X from K.
X
45 m
K
Y
80 m
2. In the figure, the bottom B of a balloon is tied to a point C on the horizontal ground by a straight
string of length 6.3 m. If the horizontal distance between B and C is 1.9 m, find the angle of
depression of C from B.
B
6.3 m
1.9 m
C
3. In the figure, ST is a tower of height 90 m. The angle of elevation of T from a point U on the
horizontal ground is 54°. Find the horizontal distance between U and T.
T
90 m
54°
U
S
21
4. In the figure, the top M of a vertical flagpole is tied to a point R on the horizontal ground by a
straight rope of length 8.2 m. If the angle of depression of R from M is 37°, find the height of
the flagpole.
M
37°
8.2 m
R
5. In the figure, a cat stands at P on the horizontal ground and a light bulb is mounted at Q on the
horizontal celling. The horizontal distance between P and Q is 0.9 m. If the angle of depression
of P from Q is 70°, find the distance between P and Q.
Q
P
6. In the figure, Paul’s eyes at P are 1.5 m above the
horizontal ground. He looks at the top Q of a tree. The
angle of elevation of Q from P is 23°. The horizontal
distance between P and Q is 6 m. Find the height of the
tree.
Q
P
23°
1.5 m
6m
7. In the figure, AB and CD represent two buildings on the horizontal ground. The height of
building AB is 160 m and the angle of depression of C from A is 48°.
A
48°
1
160 m
C
B
D
It is given that the distance between A and C is 170 m. Find the height of building CD.
22
8. In the figure, a hawk X and a squirrel Y are 10.8 m and 7.2 m above the horizontal ground
respectively. The distance between X and Y is 9 m. Find the angle of depression of Y from X.
X
9
Y
10.8 m
7.2 m
Level 2
9. In the figure, XY is a lamppost of height 5 m. P and Q are points on the horizontal ground and
PQ = 13 m. PYQ is a straight line. The angle of elevation of X from P is 40°.
X
5m
P
40°
Y
13 m
Q
(a) Find PY.
(b) Find the angle of depression of Q from X.
10. In the figure, SN is a Christmas tree. P and Q are two points on the horizontal ground. The
angles of elevation of S from P and Q are 37° and 52° respectively. If S and P are 26 m apart,
find the distance between N and Q.
S
26 m
P
52°
37°
Q
N
11. In the figure, PQR is a vertical statue. X and R lie
on the horizontal ground. The angles of elevation of
P and Q from X are 29° and 7° respectively. It is
given that the distance between X and Q is 96 m.
(a) Find the distance between X and R.
(b) Find the distance between P and Q.
P
X
23
29°
7°
Q
R
12. In the figure, the height of a vertical cliff BC is 75 m. The angle of elevation of a bird A from C
is 16°. The horizontal distance between A and C is 120 m.
B
A
75 m
16°
C
120 m
horizontal ground
(a) Find the height of the bird above the horizontal ground.
(b) Find the angle of depression of A from B.
13. In the figure, the height of a hill PN is 760 m. X and Y represent the two banks of a river. N, X
and Y lie on the same horizontal line. The angles of depression of X and Y from P are 38° and
29° respectively. Find the width of the river.
P
38°
29°
760 m
N
X
14. The figure shows two buildings AB and CD in a school. C
is tied to A and B with two straight ribbons. The height of
building AB is 25 m and the length of ribbon AC is
30 m. The angle of depression of A from C is 15°.
(a) Find the height of building CD.
(b) Find the angle of elevation of C from B.
Y
C
30 m
15°
A
25 m
B
15. In the figure, XY is the lightning rod of a building YZ and XYZ
is a vertical line. The angles of elevation of X and Y from a
helicopter P are 56° and 23° respectively. The length of the
lightning rod is 47 m.
(a) Find the horizontal distance between the helicopter and
the building.
(b) If XZ = 150 m, find the vertical distance between P and Z.
D
X
56°
47 m
Y
P
23°
Z
24
Answer
Consolidation Exercise 10B
1. 29.4°
2. 72.4°
3. 65.4 m
4. 4.93 m
5. 2.63 m
6. 4.05 m
7. 33.7 m
8. 23.6°
9. (a) 5.96 m
10. 12.2 m
11. (a) 95.3 m
12. (a) 34.4 m
13. 398 m
14. (a) 32.8 m
15. (a) 44.4 m
(b) 35.4°
(b) 41.1 m
(b) 18.7°
(b) 48.5°
(b) 84.1 m
25
F3B: Chapter 10C
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26
Book Example 18
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Consolidation Exercise
Maths Corner Exercise
10C Level 1
Maths Corner Exercise
10C Level 2
Maths Corner Exercise
10C Level 3
Maths Corner Exercise
10C Multiple Choice
E-Class Multiple Choice
Self-Test
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27
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3B
Lesson Worksheet 10C
(Refer to §10.3)
[In this worksheet, unless otherwise stated, give the answers correct to 3 significant figures if
necessary.]
10.3 Bearings
There are 8 major bearings used in daily life:
east (E), south (S), west (W), north (N)
north-east (NE), south-east (SE), south-west (SW), north-west (NW)
10.3A
Compass Bearing
Compass bearing is expressed in the form of Nx°E, Nx°W, Sx°E or Sx°W, where
x° is measured from either the north or the south, and 0 < x < 90.
e.g. The compass bearings of A, B, C and D from O are:
N40°E
N45°W
Example 1
Refer to the figure. Find
the compass bearings of
(a) A from O,
(b) C from O.
N
S50°E
Instant Drill 1
Refer to the figure. Find
the compass bearings of
(a) D from O,
(b) F from O.
A
30°
F N
70°
O
O
55°
Sol (a) The compass bearing C
of A from O is N30°E.
S55°W
65°
N
D
Sol (a) The compass bearing of D from O is
.
A
30°
E
(b) With the notations in
the figure,
θ=
O
(b) With the notations in
N
the figure,
W
55° O
θ = 90° – 55°
θ
= 35°
C
∴ The compass bearing of C from
S O
is S35°W.
F
N
θ
70°
∴ The compass bearing of F fromO O
is
.
It cannot be written
as W55°S.
28
1.
Refer to the figure. Find
the compass bearings of
(a) P from O,
(b) Q from O.
P
2.
N
22°
Refer to the figure. Find
the compass bearings of
(a) R from O,
(b) T from O.
N
T
47°
O
O
81°
15°
R
Q
○→ Ex 10C 1(a)(i), (b)(i), (c)(i), (d)(i)
Example 2
In the figure, the compass
bearing of B from A is N40°E.
(a) Find θ.
(b) Find the compass bearing
N
of A from B.
40°
Sol (a)
(b)
N
B
θ
D
A
θ = 40°
alt. ∠s, CA // DB
The compass bearing of ACfrom B is
Instant Drill 2
In the figure, the compass
bearing of K from H is S68°W.
N
(a) Find α.
(b) Find the compass bearing
α
K
of H from K.
N
H
68°
Sol
S40°W.
3.
In the figure, the compass
bearing of Q from P is
S33°E. Find the compass
bearing of P from Q.
N
Mark ‘
’ at Q first.
4.
N
P
If the compass bearing of F from G is
N76°W, find the compass bearing of G
from F.
Step 1:
33°
F
N
?
G
Q
N
Step 2: Mark ‘
’ at F.
○→ Ex 10C 2, 4
29
10.3BTrue Bearing
True bearing is expressed in the form of y°, where y° is measured from the north in a
clockwise direction and 0 ≤ y < 360. The integral part of y must be written in 3 digits.
e.g. The true bearings of P, Q, R and S from O are:
009°
Example 3
Refer to the figure. Find
the true bearings of
(a) P from O,
(b) Q from O.
097.5°
N
Q
76°
220°
P
25°
300.5°
Instant Drill 3
Refer to the figure. Find
the true bearings of
(a) R from O,
(b) T from O.
N
94°
32°
O
Sol (a) The true bearing of P from O is
025°.
N
(b) With the notations in
the figure,
θ = 360° – 76°
76° θ
Q
= 284°
O
∴ The true bearing of Q from O
is 284°.
R
O
Sol (a) The true bearing of R from O is
T
.
(b) With the notations in
N
the figure,
θ = 180° + (
)
O
=
∴ The true bearing of T 32°
θ
from O is
.
T
5. Refer to the figure. Find the true bearings of
(a) A from O, (b) B from O, (c) C from O,
N
(d) D from O.
O
D
58°
126°
40°
A
C
B
○→ Ex 10C 1(a)(ii), (b)(ii), (c)(ii), (d)(ii)
30
Example 4
In the figure, the true bearing
N
of B from A is 069°.
N
(a) Find x.
x
(b) Find the true bearing of
69°
B
A from B.
Sol (a) x + 69° = 180°
A
N
x = 111°
(b) With the notations in N
the figure,
x
θ = 360° – 111°
69°
B
= 249°
θ
∴ The true bearing ofA A from B
is 249°.
6.
In the figure, the true
Q
bearing of Q from P is 331°.
Find the true bearing of P from
Q.
Instant Drill 4
In the figure, the true bearing
of D from C is 130°.
(a) Find y.
(b) Find the true bearing of
C from D.
Sol (a) y +
=
y=
(b) With the notations in
the figure,
θ=
N
130°
N
C
y
N
D
130°
N
C
y
∴ The true bearing of C from DD
θ
is
.
○→ Ex 10C 3
7.
N
In the figure, the true
bearing of K from H is 245°.
Find the true bearing of H from
K.
N
H
245°
P
K
331°
○→ Ex 10C 5
8.
Refer to the figure. Find
(a) the true bearing of Y
from X,
(b) the compass bearing
of Z from X.
9.
N
200°
X
If the true bearing of B from A is 226°,
(a) find the true bearing of A from B,
(b) find the compass bearing of A from B.
55°
Y
Z
○→ Ex 10C 7
○→ Ex 10C 6
31
10.3C
Practical Problems Involving Bearings
Example 5
In the figure, Sam walks
N
N
70 m
80 m due north from A to B,
C
then walks 70 m due east from B
B to C. Find the true bearing of
80 m
A from C.
A
Instant Drill 5
In the figure, B is 72 m due
N
north of A. B and C are 90 m
apart. Find the compass bearing B
90 m
of C from B.
72 m
Sol From the question, ∠ABC = 90°.
In △ABC,
80 m
tan ∠ACB =
70 m
∠ACB = 48.814°, cor. to 5 sig. fig.
∴ True bearing of A from C
= 270° – 48.814°
= 221°, cor. to 3 sig. fig.
Sol
Example 6
Instant Drill 6
A
N
C
○→ Ex 10C 9, 10
N
N
H
152°
50 m
K
35°
In the figure, point P is 20 km due west of
P
20 km T
town T. If a car moves
from P in the direction
N35°E, find the shortest distance between the
car and town T.
L
d = shortest distance
between T and line L
T
The figure shows a market K which is 50 m
due east of bus stop H. If Jane walks from H in
the direction of 152°, find the shortest distance
between Jane and market K.
Sol With the notations in the figure,
construct KL ⊥ HL.
N 152°
=
β
50 m K
H
β
Sol With the notations in the figure,
construct TM ⊥ PM.
N
θ = 90° – 35° = 55°
M
In △PTM,
35°
TM
θ
sin θ =
P
PT
20 km T
TM = PT sin θ
= 20 sin 55° km
= 16.4 km, cor. to 3 sig. fig.
∴ The shortest distance between the car
and town T is 16.4 km.
L
○→ Ex 10C 11, 12
32
10. Two ships A and B set out from the same pier O at 6:00 a.m. Ship A sails due
south at a speed of 24 km/h while ship B sails due west at a speed of 30 km/h.
Find, at 9:00 a.m. on the same day,
B
(a) the compass bearing of B from A,
(b) the distance between A and B.
N
O
A
○→ Ex 10C 13
 Level Up Questions
11. Refer to the figure.
(a) Find the true bearing of A from B.
(b) Find the compass bearing of B from C.
N
N
B
160°
A
N
C
12. In the figure, ships Q and R are 135 km and 110 km from a pier P
respectively. The bearing of Q from R is N27°W and the bearing of P
from R is S63°W. Find the distance between the two ships.
Q
N
63°
110 km
P
33
27°
135 km
R
New Century Mathematics (2nd Edition) 3B
10
Applications in Trigonometry

Consolidation Exercise
10C
[ In this exercise, unless otherwise stated, give the answers correct to 3 significant figures if
necessary.]
Level 1
1. Refer to the figure. Find the
(i) compass bearing,
(ii) true bearing
of each of the following points from O.
(a) P
(b) Q
(c) R
(d) S
N
18°
S
P
31°
O
27°
R
43°
Q
2. In the figure, the compass bearing of Q from P is N50°E.
(a) Find θ.
(b) Find the compass bearing of P from Q.
N
N
50°
3. In the figure, the true bearing of H from K is 250°.
(a) Find θ.
(b) Find the true bearing of K from H.
θ
N
N
K
θ
250°
H
4. In the figure, the true bearing of B from A is 115°.
Find the true bearing of A from B.
N
115°
A
B
34
5. Refer to the figure.
(a) Find the compass bearing of P from Y.
(b) Find the true bearing of P from X.
Y
N
56°
144°
P
6. In the figure, A is due west of B, and C is due south of B. It
is given that the compass bearing of C from A is S63°E and
the length of BC is 45 m. Find the length of AC.
N
N
X
B
A
63°
45 m
C
7. In the figure, P, Q and R are the positions of three houses. Q
is 5 km due east of P, and R is 6 km due north of P. Find the
compass bearing of Q from R.
N
R
6 km
N
8. In the figure, A is due north of B, and C is due west of B.
If BC = 180 m and AC = 195 m, find the true bearing of C
from A.
N
N
P
5 km
Q
A
19
C
18
9. In the figure, V is 7.5 km due north of U. A car travels
from U in the direction 320° along a straight road. Find the
shortest distance between the car and V.
B
N
V
7.5 km
320°
35
U
10. In the figure, Ben and Calvin set out from the same point O at
the same time. Ben runs due east at a speed of 7.2 km/h and
Calvin runs due south at a speed of 9.6 km/h. Find the compass
bearing of Calvin from Ben after 4 hours.
N Ben
O
Calvin
Level 2
11. Refer to the figure. Find the compass bearing of Y from Z.
N
Y
N
63°
X
12. Refer to the figure.
(a) Find the compass bearing of P from R.
(b) Find the true bearing of R from Q.
Z
Q
109°
R
N
44°
36°
P
13. In the figure, the true bearing of B from A is 096°,
AC = AB and ∠ABC = 37°. Find the true bearing of
A from C.
N
A
37°
C
36
96°
B
14. In the figure, a race car travels 1.6 km due north from X, then travels 6 km due west. Finally, it
travels 4.8 km due south to Y.
6 km
1.6 km
X
4.8 km
N
Y
(a) Find the compass bearing of Y from X.
(b) Find the distance between X and Y.
15. In the figure, Tom walks 370 m from his home at F in the direction 148° to a cinema at G, then
walks 160 m in the direction 058° to a restaurant at H.
N
148°
F
370 m
H
N
58°
160 m
G
(a) Tom claims that FG ⊥ GH. Do you agree? Explain your answer.
(b) How far does Tom walk along a straight road from the restaurant to his home? In what
direction does he walk?
37
16. In the figure, A and B are 3.9 km apart. O is the centre of a circular lake with diameter 4.5 km. It
is given that OA = OB, and the compass bearings of O from A and B are N40°E and N76°W
respectively.
N
148°
F
370 m
H
N
58°
160 m
G
(a) Find the compass bearing of A from B.
(b) Lily claims that A and B are both located inside the lake. Do you agree? Explain your
answer.
17. In the figure, P, Q and R represent three bus stations. A bus
travels 5 km from P in the direction 072° to Q, and then travels
in the direction 310° to R, which is due north of P. Find the
distance between P and R.
R
N
Q
72°
P
310°
5 km
18. In the figure, a car travels from S to T in the direction N53°W, then travels in the direction
S26°W to U, which is 140 km due west of S. Suppose the speed of the car is constant
throughout the whole journey and the car takes 2 hours to travel from S to T. Find the time taken
for the car to travel from T to U.
N
T
26°
U
140 km
38
N
53°
S
Answer
Consolidation Exercise 10C
1. (a) (i) N31°E
(ii) 031°
(b) (i) S43°E
(ii) 137°
(c) (i) S63°W
(ii) 243°
(d) (i) N72°W
(ii) 288°
2. (a) 50°
(b) S50°W
3. (a) 70°
(b) 070°
4. 295°
5. (a) S56°E
(b) 324°
6. 99.1 m
7. S39.8°E
8. 247°
9. 4.82 km
10. S36.9°W
11. N27°W
12. (a) S80°E
(b) 171°
13. 022°
14. (a) S61.9°W
(b) 6.8 km
15. (a) yes
(b) distance: 403 m,
direction: 305° (or N55.4°W)
16. (a) S72°W
(b) no
17. 5.54 km
18. 1.34 h
39
F3B: Chapter 11A
Date
Task
Lesson Worksheet
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Book Example 3
○
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○
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Problems encountered
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(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
11A Level 1
Maths Corner Exercise
11A Level 2
Maths Corner Exercise
11A Level 3
Maths Corner Exercise
11A Multiple Choice
E-Class Multiple Choice
Self-Test
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Problems encountered
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(Full Solution)
40
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3B
Lesson Worksheet 11A
(Refer to §11.1)
[In this worksheet, unless otherwise stated, leave the radical sign ‘√’ in the answers if necessary.]
11.1 Distance between Two Points
For any two points A(x1 , y1) and B(x2 , y2) in
a rectangular coordinate plane, the distance
between A and B is given by:
y
B ( x 2 , y 2)
AB = ( x2 − x1 )2 + ( y2 − y1 )2
A ( x 1 , y 1)
x
O
Example 1
In each of the following, find the length of the
line segment AB.
y
(a)
(b) A(−3 , 4) y
B(5 , 5)
A(1 , 2)
Instant Drill 1
In each of the following, find the length of the
line segment PQ.
y
y
(a)
(b)
P(2 , 9)
Q(−3 , 5)
x
O
x
O
Q(8 , 1)
O
B(2 , −3)
2
2
Sol (a) AB = (5 − 1) + (5 − 2) units
x
O
x
P(−5 , −2)
2
Sol (a) PQ = [( ) − ( )] + [( ) − ( )]2 units
= 42 + 32 units
= 5 units
= ( ) 2 + ( ) 2 units
=
(b) AB = [2 − (−3)]2 + (−3 − 4) 2 units
(b) PQ = [( ) − ( )]2 + [( ) − ( )]2 units
=
= 52 + (−7) 2 units
= 74 units
1.
Find the length of the line segment AB in
the figure.
2.
Find the length of the line segment CD in
the figure.
y
O
y
A(1 , −1)
C(4 , 3)
x
x
O
B(7 , −6)
D(−3 , −4)
41
In each of the following, find the distance between the two given points. [Nos. 3–4]
3. A(1 , 0), B(9 , 6)
4. C(−6 , 7), D(1 , −2)
Sketch the line segment
AB if you need help.
y
x
O
→
○ Ex 11A 1−8
5.
Find the length of the line segment AB in
the figure.
6.
Find the length of the line segment CD in
the figure.
y
Coordinates
= (−3 , ____)
A
y
−4 −3 −2 −−110
D
10
8
6
4
2
6
5
4
3
2
1
−8 −6 −4 −2 0
−2
−4
C
x
1 2
3
B
x
2 4 6
→
○ Ex 11A 9−12
7. In the figure, A(1 , −2), B(8 , −2) and C(5 , 2) are the
three vertices of △ABC. Find the perimeter of △ABC,
correct to 2 decimal places.
AB = [8 − (
BC =
y
C(5 , 2)
)] units =
x
O
AC =
A(1 , −2)
B(8 , −2)
∴ Perimeter of △ABC
=
→
○ Ex 11A 13−15
42
‘Explain Your Answer’ Question
8. In the figure, A(0 , −2), B(6 , 4) and C(−7 , −3) are the three
vertices of △ABC. Is △ABC an isosceles triangle? Explain
your answer.
y
B(6 , 4)
x
O
A(0 , −2)
C(−7 , −3)
 Level Up Questions
9.
O(0 , 0), P(12 , 0) and Q(7 , 13) are the three vertices of △OPQ. Find the perimeter of
△OPQ, correct to 3 significant figures.
10. Find the area of square ABCD shown in the figure.
y
B(4 , 9)
C
4
A
O
43
D
x
New Century Mathematics (2nd Edition) 3B
11
Coordinate Geometry of Straight Lines

Consolidation Exercise
11A
Level 1
In each of the following, find the distance between the two given points. [Nos. 1−
−6]
(Give the answers correct to 2 decimal places if necessary.)
1. A(−4 , 2), B(0 , 5)
2. C(1 , −1), D(6 , 11)
3. E(−7 , 4), F(2 , −8)
4. P(6 , −5), Q(−9 , 3)
5. R(−10 , 5), S(2 , 3)
6. T(−9 , −17), U(−1 , 0)
In each of the following, find the length of the line segment XY. [Nos. 7−
−9]
(Give the answers correct to 3 significant figures if necessary.)
7.
8.
9.
y
y
6
5
4
3
2
1 X
0
X
Y
x
1 2 3 4 5 6
4
3
2
1
−3 −2 −−110
−2
Y
y
3
2
1
X
x
1 2 3
10. In the figure, A(−3 , 5), B(0 , −2) and C(7 , 3) are the three
vertices of △ABC. Find the perimeter of △ABC, correct to
2 decimal places.
−4 −3 −2 −−110
−2
−3
−4
−5
y
x
1 2 3 4
Y
A(−3 , 5)
C(7 , 3)
x
O
B(0 , −2)
11. In the figure, O(0 , 0), A(4 , −3), B(9 , 1) and C(5 , 7) are the four
vertices of quadrilateral OABC. Find the perimeter of OABC,
correct to 3 significant figures.
y
C(5 , 7)
B(9 , 1)
x
O
A(4 , −3)
44
y
12.In the figure, X(2 , 5), Y(−1 , −2) and Z(5 , −2) are the three vertices
of △XYZ. Is △XYZ an isosceles triangle? Explain your answer.
X(2 , 5)
O
x
Y(−1 , −2)
Z(5 , −2)
13.In the figure, A(−4 , 0), B(−2 , 4), C(5 , 9) and D(0 , 2) are the four
vertices of quadrilateral ABCD. Is ABCD a kite? Explain your
answer.
y
C(5 , 9)
B(−2 , 4)
D(0 , 2)
14. In the figure, the coordinates of P and R are (−7 , 2) and (5 , −3)
respectively. If PQ is parallel to the x-axis and PQ = PR, find the
coordinates of Q.
x
A(−4 , 0) O
y
P(−7 , 2)
Q
x
O
R(5 , −3)
Level 2
15. A(5 , 3), B(−1 , 2) and C(2 , −3) are the three vertices of △ABC. Arrange the lengths of the
three sides of △ABC in ascending order.
16.P(4 , 8), Q(−7 , 5) and R(9 , −1) are the three vertices of △PQR. Which vertex, P, Q or R, is
closest to the origin O? Explain your answer.
17.A(−6 , 5), B(4 , 4), C(5 , −3) and D(−8 , −2) are the four vertices of quadrilateral ABCD. Which
diagonal, AC or BD, is longer? Explain your answer.
18. P(a , 0), Q(2 , 4) and O(0 , 0) are the three vertices of △OPQ. If PQ = PO, find the value of a.
19.In the figure, A(1 , 1), B(10 , 7), C(6 , 0) and D(3 , −2) are the four vertices of a trapezium, where
AB // DC and AD ⊥ AB. Find the area of trapezium ABCD.
y
B(10 , 7)
O
A(1 , 1)
C(6 , 0)
D(3 , −2)
45
x
20.In the figure, A(−3 , −2), B(1 , 2) and C(4 , −1) are the three
vertices of △ABC.
(a) Prove that △ABC is a right-angled triangle.
(b) Find the area of △ABC.
y
B(1 , 2)
x
O
C(4 , −1)
A(−3 , −2)
y
21.In the figure, the coordinates of P and R are (−1 , 0) and (8 , 6)
respectively. Q is a point on the x-axis and PQ = QR.
(a) Find the coordinates of Q.
(b) Find the area of △PQR.
R(8 , 6)
P(−1 , 0)
O
22.In the figure, the coordinates of Q and R are (8 , 9) and (5 , −4)
respectively. P is a point on the y-axis and PQ = PR.
(a) Find the coordinates of P.
(b) Prove that △PQR is a right-angled triangle.
x
Q
y
Q(8 , 9)
P
x
O
23. In the figure, ABCD is a square, where A is on the y-axis. The
coordinates of B and D are (7 , 11) and (−24 , −6) respectively.
(a) Find the coordinates of A.
(b) Find the perimeter and area of ABCD.
R(5 , −4)
y
C
B(7 , 11)
x
O
D(−24 , −6)
46
A
Answer
Consolidation Exercise 11A
1. 5 units
2. 13 units
3. 15 units
4. 17 units
5. 12.17 units
6. 18.79 units
7. 5 units
8. 5.39 units
9. 8.60 units
10. 26.42 units
11. 27.2 units
12. yes
13. yes
14. (6 , 2)
15. BC < AB < CA
16. Q
17. AC
18. 5
19. 26 sq. units
20. (b) 12 sq. units
 11 
, 0
2

21. (a) 
(b)
39
sq. units
2
22. (a) (0 , 4)
23. (a) (0 , −13)
(b) perimeter = 100 units,
area = 625 sq. units
47
F3B: Chapter 11B
Date
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Consolidation Exercise
Maths Corner Exercise
11B Level 1
Maths Corner Exercise
11B Level 2
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48
Teacher’s
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Maths Corner Exercise
11B Level 3
Maths Corner Exercise
11B Multiple Choice
E-Class Multiple Choice
Self-Test
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49
(
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Mark:
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Book 3B
Lesson Worksheet 11B
(Refer to §11.2)
11.2ASlope Formula
For any two points A(x1 , y1) and B(x2 , y2) (where x1 ≠ x2)
in a rectangular coordinate plane, the slope m of the line
AB is given by:
y − y1
m= 2
x2 − x1
Note: The slope of a horizontal line is 0, while the
slope of a vertical line is undefined.
y
B ( x 2 , y 2)
A ( x 1 , y 1)
x
O
y
Slope is
undefi
ned.
Slope = 0
x
O
Example 1
Find the slope of the line AB in the figure.
A(−4 , 4)
Instant Drill 1
Find the slope of the
line PQ in the figure.
y
y
P(6 , 4)
x
B(2 , 1)
Q(−2 , −3)
x
O
O
1− 4
2 − (−4)
−3
=
6
1
=−
2
Sol Slope of AB =
1.
Sol Slope of PQ =
(
(
)−(
)−(
)
)
=
y
Find the slope of the
line RS in the figure.
2.
R(1 , 7)
S(6 , 2)
Find the slope of the
line CD in the figure.
y
D(−1 , 7)
C(−5 , 1)
x
If a line slopes O
upwards from left
to right, slope of
the line > 0.
x
O
50
3.
Find the slope of the
line PQ in the figure.
4.
y
P(6 , 5)
Find the slope of the
line MN in the figure.
y
O
x
M(2 , −1)
x
O
If a lineNslopes
(5 , −8)
downwards from left
to right, slope of the
line < 0.
Q(1 , −3)
→
○ Ex 11B 2, 3
5.
Given that the slope of the line passing
through P(−8 , h) and Q(h − 8 , 6) is 5,
find the value of h.
6.
Given that the slope of the line passing
through A(1 , k) and B(3k + 1 , −2) is −
2
,
3
find the value of k.
Slope of PQ = (
(
(
)−(
)−(
)
)
=(
)
)
=
→
○ Ex 11B 9, 10
Example 2
Prove that the three points A(3 , 1), B(5 , 3) and
C(9 , 7) lie on the same straight line.
Sol Slope of AB
Sketch:
3 −1
y
=
same slope?
5−3
C(9 , 7)
2
=
B(5 , 3)
2
A(3 , 1)
=1
x
O
Slope of BC
7−3
=
9−5
4
=
We can also
4
say that
=1
A, B and
∵ Slope of AB = slope of BC
C are
∴ The three points A, B and C lie on the
same straight line.
Instant Drill 2
Prove that the three points P(0 , 2), Q(2 , −2)
and R(5 , −8) lie on the same straight line.
Sol Slope of PQ
Sketch:
(
)−(
)
y
=
(
)−(
)
P(0 , 2)
=
Slope of QR
(
)−(
=
(
)−(
=
51
O
)
)
x
Q(2 , −2)
same
slo
pe?
R(5 , −8)
7.
Prove that the three points A(−8 , −4),
B(0 , 0) and C(2 , 1) lie on the same
straight line.
Prove that the three points R(2 , −1),
S(−6 , 1) and T(6 , −2) are collinear.
8.
→
○ Ex 11B 13, 14
11.2B Inclination
For a straight line ℓ with inclination θ :
slope of ℓ = tan θ
y
ℓ
θ (inclination)
Note: If θ = 0°, the line is a horizontal line.
x
O
θ is the angle measured
anticlockwise from the
positive x-axis to ℓ .
9. Complete the following tables.
(Give the answers correct to 3 significant figures.)
Slope of ℓ = tan
(a)
(i)
(ii)
(iii)
Inclination θ
30°
40°
75°
Slope θof ℓ
(b)
(i)
(ii)
(iii)
For 0° < θ < 90°,
the slope of ℓ (increases /
decreases) with θ.
Slope of ℓ
1.5
8
15.5
Inclination θ
tan θ
= 1.5
θ
→
○ Ex 11B 4, 5
52
Example 3
L is a straight line passing through two points
A(−4 , −3) and B(1 , 7). Find
(a) the slope of L,
(b) the inclination of L, correct to the nearest
degree.
7 − (−3)
Sol (a) Slope of L =
1 − (−4)
10
=
5
=2
(b) Let θ be the inclination of L.
Slope of L = tan θ
2 = tan θ
θ = 63°, cor. to the nearest
degree
∴ The inclination of L is 63°.
Instant Drill 3
L is a straight line passing through two points
P(1 , 2) and Q(6 , 4). Find
(a) the slope of L,
(b) the inclination of L, correct to the nearest
degree.
(
)−(
)
Sol (a) Slope of L =
(
)−(
)
10. L is a straight line passing through two
points R(2 , −4) and S(12 , 1). Find
(a) the slope of L,
(b) the inclination of L, correct to the
nearest 0.1°.
11. C(−8 , −10) and D(−4 , −3) lie on a
straight line L. Find
(a) the slope of L,
(b) the inclination of L, correct to the
nearest 0.1°.
=
(b) Let θ be the inclination of L.
→
○ Ex 11B 6−8
53
‘Explain Your Answer’ Question
12. In each of the following, which line has a greater slope? Explain your answer.
(a) line ℓ1 : passing through two points (2 , −7) and (7 , −1)
line ℓ 2 : passing through two points (4 , 2) and (8 , 6)
(b) line ℓ 3 : passing through two points (2 , −2) and (10 , −3)
line ℓ 4 : passing through two points (−3 , −5) and (7 , −10)
We can compare the steepness of
lines by considering their
slopes:
(i) For lines with positive slopes,
slope , steepness 
(ii) For lines with negative slopes,
slope , steepness 
 Level Up Question
13. In the figure, A is a point on the y-axis. A straight line L
passes through A and B(−7 , −3).
(a) Find the inclination of L.
(b) Find the slope of L.
(c) Find the coordinates of A.
y
L
A
135°
x
O
B(−7 , −3)
54
New Century Mathematics (2nd Edition) 3B
11
Coordinate Geometry of Straight Lines

Consolidation Exercise
11B
Level 1
1. Name all the line segment(s) in the figure satisfying each of the
following conditions.
(a) The slope is positive.
(b) The slope is negative.
(c) The slope is 0.
(d) The slope is undefined.
y
B
C
O
E
D
2. Find the slope of the line AB in each of the following figures.
y
(a)
(b)
A
x
y
A(−6 , 4)
B(2 , 7)
B(4 , 2)
A(−4 , −3)
x
O
x
O
3. In each of the following, find the slope of the straight line passing through the two given points.
(a) A(0 , 6), B(2 , 0)
(b) P(5 , −2), Q(9 , −1)
(c) X(−3 , 4), Y(1 , −4)
4. In each of the following, find the inclination of the line with the given slope, correct to the
nearest 0.1°.
9
(a) Slope = 5
(b) Slope = 0.5
(c) Slope =
4
5. In each of the following, find the slope of the line L, correct to 2 decimal places.
y
(a)
(b)
y
L
L
65°
O
138°
x
O
55
x
6. L is a straight line passing through the points P(1 , −8) and Q(4 , 3).
(a) Find the slope of L.
(b) Find the inclination of L, correct to the nearest 0.1°.
7. Find the inclination of the line XY in each of the following figures.
(Give the answers correct to the nearest 0.1°.)
(a)
(b)
y
y
Y(4 , 10)
Y(5 , 9)
O
x
X(−4 , 1)
X(−3 , −5)
x
O
8. Given that the slope of the line passing through
and D(n + 3 , −5) is −2, find the value of n.
C(−4 , 3 − 4n)
9. If P(−k , −5) and Q(3k , 3) lie on a straight line with inclination 45°, find the value of k.
In each of the following, determine whether the three given points lie on the same straight line.
[Nos. 10−11]
10. A(−6 , −5), B(0 , −1), C(3 , 1)
11. P(2 , 11), Q(4 , 7), R(10 , −3)
12.Refer to the figure.
(a) Find the inclination of L2.
(b) Find the slope of L2, correct to 2 decimal places.
y
140°
115°
L2
13.Consider three points A(6 , −1), B(8 , 3) and C(−5 , 2). Among AB,
BC and CA, which one has the greatest slope? Explain your
answer.
O
x
L1
Level 2
14. In each of the following, find the slope of the straight line passing through the two given points.
(a) A(−1.3 , 2.8), B(0.75 , 6.9)
 11 1 
4 5
(b) C  , −  , D  , 
2
7
7 2
(c) E(4a , 3a), F(0 , −a), where a ≠ 0
15. In each of the following, find the inclination of the straight line passing through the two given
points.
(Give the answers correct to the nearest 0.1° if necessary.)
(a) P(−2.6 , −1.3), Q(4.4 , 3.7)
(b) R(−5 , 3 ), S(−4 , 2 3 )
(c) T(−c , −c), U(0 , c), where c ≠ 0
56
16. In the figure, A is a point on the y-axis. If the slope of the straight line
2
passing through A and B(6 , −6) is − , find the coordinates of A.
3
y
O
x
A
B(6 , −6)
17. The slope of a straight line L passing through (2 , 9) is 3. If L cuts the x-axis at A and cuts the
y-axis at B, find the coordinates of A and B.
y
18.In the figure, P(1 , −3), Q and R(9 , 1) are three points lying on the
same straight line.
(a) If Q lies on the x-axis, find the coordinates of Q.
(b) Does the straight line pass through (6 , −1)? Explain your answer.
R(9 , 1)
x
Q
O
P(1 , −3)
19. The inclination of a straight line L passing through (4 , 3 ) is 30°.
(a) Find the coordinates of the point where L cuts the x-axis.
(b) If P(a , 3 3 ) is a point lying on L, find the value of a.
20. Consider three points A(1 , k), B(5 , 7) and C(−2 , −5). It is given that the slope of AB is
3
.
2
(a) Find the value of k.
(b) Which line segment, AB, AC or BC, is the steepest? Explain your answer.
21.In the figure, L1 is a straight line passing through the points
A(−2 , −5) and B(9 , 6). The angle between the lines L1 and L2
is 20°.
(a) Find the inclination of L1.
(b) Find the slope of L2, correct to 2 decimal places.
y
L2 L
1
B(9 , 6)
20°
x
O
A(−2 , −5)
22.In the figure, L1 and L2 are straight lines pass through A(4 , 1)
and B(−2 , 0) respectively. L1 and L2 intersect at C(8 , p). It is
given that the slope of L1 is twice the slope of L2.
(a) Find the value of p.
(b) Is the inclination of L1 twice that of L2? Explain your answer.
y
C(8 , p)
B(−2 , 0)
O
57
L1
L2
A(4 , 1)
x
Answer
Consolidation Exercise 11B
1. (a) CB, DA, EA (b) BA, CA
(c) DE
(d) CD
5
1
2. (a)
(b) −
3
5
1
3. (a) −3
(b)
(c) −2
4
4. (a) 78.7° (b) 26.6° (c) 66.0°
5. (a) 2.14
(b) 0.90
11
6. (a)
(b) 74.7°
3
7. (a) 65.0°
(b) 41.6°
8. −1
9. 2
10. yes
11. no
12. (a) 25°
(b) 0.47
13. AB
14. (a) 2
(b) −3
(c) 1
15. (a) 35.5° (b) 60°
(c) 63.4°
16. (0 , −2)
17. A(−1 , 0), B(0 , 3)
18. (a) (7 , 0)
(b) no
19. (a) (1 , 0)
(b) 10
20. (a) 1
(b) AC
21. (a) 45°
(b) 2.14
22. (a) 5
(b) no
58
F3B: Chapter 11C
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Consolidation Exercise
Maths Corner Exercise
11C Level 1
Maths Corner Exercise
11C Level 2
Maths Corner Exercise
11C Level 3
Maths Corner Exercise
11C Multiple Choice
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59
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
E-Class Multiple Choice
Self-Test
○
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60
(
Mark:
_________
)
Book 3B
Lesson Worksheet 11C
(Refer to §11.3)
11.3AParallel Lines
In the figure, if m1 = m2,
then L1 // L2.
1. In each of the following, determine whether the straight lines L1 and L2 are parallel.
Slope of L1
Slope of L2
(a)
3
3
(yes / no)
(b)
2
−2
(yes / no)
1
(yes / no)
(c)
5
5
(d)
−4
−4
(yes / no)
2.
Refer to the figure.
y
(1 , 3)
(−3 , −1)
3.
Refer to the figure.
y
L1
L2: slope = 1
L1
x
(−4 , 3)
O
O
(a) Find the slope of L1.
(b) Determine whether L1 and L2 are
parallel.
L2: slope = −2
x
(2 , −1)
(a) Find the slope of L1.
(b) Determine whether L1 and L2 are
parallel.
(a) Slope of L1 =
(b) ∵ Slope of L1 (= / ≠) slope of L2
∴ L1 and L2 (are / are not) parallel.
→
○ Ex 11C 1, 3
61
In the figure, if L1 // L2,
then m1 = m2.
Example 1
In the figure, L1 is parallel to L2.
y
L1
Instant Drill 1
In the figure, L1 is parallel to L2.
L2
L2
y
L1
(−5 , 5)
(4 , 6)
x
O
x
O
(1 , −4)
(−1 , −4)
(a) Find the slope of L2.
(b) Find the slope of L1.
6 − (−4)
Sol (a) Slope of L2 =
4 − (−1)
10
=
5
=2
(b) ∵ L1 // L2
∴ Slope of L1 = slope of L2
=2
4. In the figure, L1 is parallel to L2.
(a) Find the slope of L2.
(b) Find the slope of L1.
Sol
(a) Slope of L2 =
(b) ∵ L1 // L2
∴ Slope of L1 =
5. In the figure, L1 is parallel to L2.
y
y
L1
(−2 , 7)
L2
(6 , 9)
O
x
L2
(5 , −2)
L1
O
(a) Find the slope of L1.
(b) Find the slope of L2.
Find the slope of L1.
x
Coordinates of
the origin O
= (___ , ___)
→
○ Ex 11C 5, 7
62
6. In the figure, L1 is parallel to L2.
(a) Find the slope of L2.
(b) Find the value of a.
y
C(−2 , 5)
B(5 , 6)
x
O
D(−7 , −5)
L2
A(0 , a)
L1
→
○ Ex 11C 9, 12
11.3B Perpendicular Lines
In the figure, if m1 × m2 = −1,
then L1 ⊥ L2.
7. In each of the following, determine whether the straight lines L1 and L2 are perpendicular.
Slope of L1
Slope of L2
(a)
4
−4
(yes / no)
(b)
1
−1
(yes / no)
1
(c)
2
(yes / no)
2
1
(d)
3
(yes / no)
−
3
8. Refer to the figure.
(a) Find the slope of L1.
(b) Determine whether L1 and L2 are perpendicular.
y
L2: slope = 2
L1
(−2 , 2)
x
O
(4 , −1)
63
In the figure, if L1 ⊥ L2,
then m1 × m2 = −1.
Example 2
In the figure, L1 is perpendicular to L2.
y
Instant Drill 2
In the figure, L1 is perpendicular to L2.
L1
(3 , 9)
y
L2
x
O
L1
(4 , 7)
O
(2 , −1)
x
L2
(a) Find the slope of L1.
(b) Find the slope of L2.
9−0
Sol (a) Slope of L1 =
3−0
9
=
3
=3
(b) ∵ L1 ⊥ L2
∴ Slope of L1 × slope of L2 = −1
3 × slope of L2 = −1
1
Slope of L2 = −
3
(a) Find the slope of L1.
(b) Find the slope of L2.
9. In the figure, L1 is perpendicular to L2.
10. In the figure, L1 is perpendicular to L2.
Find the slope of L2.
y
L2
L1
Sol
(a) Slope of L1 =
(b) ∵ L1 ⊥ L2
∴ Slope of L1 × slope of L2 = (
=
(4 , 7)
L2
y
L1
O
(−6 , −3)
)
x
(2 , 1)
x
O
(−8 , −5)
(a) Find the slope of L1.
(b) Find the slope of L2.
→
○ Ex 11C 6, 8
64
11. In the figure, L1 passes through C(1 , −3) and D(6 , 7).
L2 cuts the y-axis at A and passes through B(6 , 2). It is
given that L1 and L2 are perpendicular.
(a) Find the slope of L1.
(b) Find the coordinates of A.
y
L1
D(6 , 7)
L2
A
B(6 , 2)
x
O
C(1 , −3)
→
○ Ex 11C 10, 13
‘Explain Your Answer’ Question
12. L1 is a straight line passing through A(2 , −3) while L2 is another straight line with slope
L1 // L2, does the point B(−2 , 3) lie on L1? Explain your answer.
∵ L1 // L2
∴ Slope of L1 =
Slope of AB =
∵ Slope of AB (= / ≠) slope of L1
∴ The point B (lies / does not lie) on L1.
65
9
. If
4
Step 1: Find the slope of
L1.
Step 2: Check if slope of
AB = slope of L .
 Level Up Questions
13. O(0 , 0), B(2 , 3), C(6 , 1) and D(4 , −2) are the vertices of a quadrilateral. It is given that
BC // OD. Determine whether OBCD is a parallelogram.
Sketch OBCD
first.
14. X(−9 , −2), Y(0 , −8) and Z(8 , 4) are the vertices of a triangle.
(a) Find the slopes of the three sides of △XYZ.
(b) Hence, prove that △XYZ is a right-angled triangle and state which angle is a right angle.
66
New Century Mathematics (2nd Edition) 3B
11
Coordinate Geometry of Straight Lines

Consolidation Exercise
11C
Level 1
1. In each of the following, determine whether L1 and L2 are parallel.
y
(a)
(b)
y
L2 : slope = 1
L1
(2 , 1)
O
L1
(−5 , 4)
(7 , 1)
x
L2
x
O
(3 , −5)
(−6 , −2)
(−3 , −4)
2. In each of the following, determine whether L1 and L2 are perpendicular.
(a)
(b)
y
y
L2 : Slope =
7
L2
3
2
x
0
8
(−3 , −2)
(5 , −6)
x
0
5
L1
L1
3. Consider four points A(−4 , 3), B(7 , 6), C(−6 , −9) and D(5 , −6).
(a) Find the slopes of AB and CD.
(b) What is the relationship between AB and CD?
4. Consider four points P(−4 , 0), Q(1 , 6), R(−8 , 7) and S(4 , −3).
(a) Find the slopes of PQ and RS.
(b) What is the relationship between PQ and RS?
In each of the following, find the slope of the line L. [Nos. 5−6]
y
5. (a)
(b)
y
L
L
(2 , 9)
(7 , 2)
(−2 , −4)
O
x
O
67
x
(5 , −3)
6. (a)
(b)
y
y
L
x
0
−7
L
(7 , 4)
(1 , 1)
O
x
(1 , −10)
Find the unknown in each of the following figures. [Nos. 7−8]
7. (a)
(b)
1 y
L2 : slope = −
3
(4 , 1)
L1
O
L1
x
x
O
(a , −3)
8. (a)
y
(−6 , b)
(b)
y
2
5
L2 : slope =
y
L1
L2 : slope =
(−3 , d)
L1
0
−
4
(c , −6)
x
3
4
x
O
(6 , −7)
L2 : slope = −2
9. Consider four points P(−3 , −2), Q(n , 1), R(3 , 5) and S(1 , −1). Find the value of n in each of
the following cases.
(a) PQ // RS
(b) PQ ⊥ RS
y
10.In the figure, L1 passes through A(−5 , −7) and B(3 , −4) while L2
passes through C(−2 , −3). It is given that L1 // L2.
(a) Find the slope of L1.
(b) Suppose L2 cuts the x-axis at P. Find the coordinates of P.
L2
L1 A(−5 , −7)
68
x
C(−2 , −3) O
B(3 , −4)
11.In the figure, L is perpendicular to the line passing through P(−1 , 6)
and Q(−4 , −1). R(−7 , −2) is a point lying on L.
(a) Find the slope of PQ.
(b) Find the coordinates of the point where L cuts the y-axis.
y
P(−1 , 6)
Q(−4 , −1)
x
O
R(−7 , −2)
L
12. A(−5 , 2), B(7 , 6), C(8 , 4) and D(2 , 2) are the vertices of a
quadrilateral.
(a) Find the slopes of AB, BC, CD and AD.
(b) Name all the parallel sides of quadrilateral ABCD.
13. X(2 , 11), Y(5 , −1) and Z(9 , 0) are the vertices of a triangle.
(a) Find the slopes of the three sides of △XYZ.
(b) Hence, prove that △XYZ is a right-angled triangle and state which angle is a right angle.
Level 2
14.In the figure, two perpendicular lines L1 and L2 intersect at
A(−3 , −6). It is given that L1 cuts the y-axis at B(0 , −7), while
L2 cuts the y-axis at C.
(a) Find the coordinates of C.
(b) Find the area of △ABC.
y
C
O
L1
A(−3 , −6)
x
B(0 , −7)
L2
15.The figure shows two parallel lines L1 and L2. L1 passes through
P(−1 , 7) and Q(1 , 2) while L2 passes through R(6 , 1). Does L2
passes through the point (4 , 5)? Explain your answer.
y
L1
L2
P(−1 , 7)
Q(1 , 2)
O
69
R(6 , 1)
x
16.In the figure, L1 passes through C(−1 , 8) and D while L2 passes
through A(−7 , 6) and B(1 , 2). D lies on the x-axis and L1 ⊥ L2.
(a) Find the coordinates of D.
(b) If K(−2 , a) is a point on L1, find the value of a.
y
C(−1 , 8)
L2
A(−7 , 6)
B(1 , 2)
D
x
O
L1
3 
17.In the figure, P(3 , −2), Q(13 , 6) and R  , 7  are the three
2 
vertices of △PQR. PQ cuts the x-axis at T.
(a) Find the coordinates of T.
(b) Is RT the corresponding altitude of △PQR if PQ is taken as its
base? Explain your answer.
18.The figure shows three points A(4 , −4), B(10 , 4) and C(8 , −7).
Suppose D is a point on the y-axis such that AD ⊥ AB.
(a) Find the coordinates of D.
(b) Prove that the three points D, A and C lie on the same straight
line.
y
3 
R  , 7
2 
Q(13 , 6)
T
O
x
P(3 , −2)
y
B(10 , 4)
O
x
A(4 , −4)
C(8 , −7)
19. A(−5 , k), B(k , 1), C(6 , 5) and D(−3 , 8) are the vertices of a quadrilateral. It is given that
AB // DC.
(a) Find the value of k.
(b) Is ABCD a parallelogram? Explain your answer.
y
20.In the figure, P(−3 , −1), Q(a , b), R(8 , 6) and S(3 , 9) are the four
vertices of parallelogram PQRS.
(a) Find the values of a and b.
(b) Is PQRS a rectangle? Explain your answer.
S(3 , 9)
R(8 , 6)
P(−3 , −1)
O
Q( a , b )
70
x
21.In the figure, ABCD is a right-angled trapezium. The coordinates of
 9 
A, C and D are (2 , 0), (3 , 5) and  − , 0  respectively. It is given
 2 
that DC // AB and ∠DCB = 90°. Find the coordinates of B.
y
C(3 , 5)
B
 9  O
D − , 0
 2 
22. In the figure, OAB is a triangle. The coordinates of B are (−1 , 3 )
and the inclination of OA is 30°.
(a) Prove that △OAB is a right-angled triangle and state which
angle is a right angle.
(b) If AB is parallel to the x-axis, find the coordinates of A.
(Leave the radical sign ‘√’ in the answer.)
71
x
A(2 , 0)
y
B( − 1 , 3 )
A
30°
O
x
Answer
Consolidation Exercise 11C
1. (a) yes
2. (a) no
(b) no
(b) yes
3
3
3. (a) AB:
, CD:
(b) AB // CD
11
11
6
5
4. (a) PQ: , RS: −
(b) PQ ⊥ RS
5
6
2
5. (a)
(b) −4
3
4
6. (a) −2
(b)
5
7. (a) 9
(b) −3
8. (a) −4
(b) 5
9. (a) −2
(b) −12
3
(b) (6 , 0)
10. (a)
8
7
11. (a)
(b) (0 , −5)
3
1
1
12. (a) AB: , BC: −2, CD: , AD: 0
3
3
(b) AB // CD
1
11
13. (a) XY: −4, YZ: , XZ: −
4
7
(b) ∠Y
14. (a) (0 , 3)
(b) 15 sq. units
15. no
16. (a) (−5 , 0)
(b) 6
11


(b) no
17. (a)  , 0 
2


18. (a) (0 , −1)
19. (a) 4
(b) yes
20. (a) a = 2, b = −4 (b) yes
21. (a) (5 , 2)
22. (a) ∠AOB
(b) (3 , 3 )
72
F3B: Chapter 11D
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 15
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 16
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 17
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 18
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
11D Level 1
Maths Corner Exercise
11D Level 2
Maths Corner Exercise
11D Level 3
Maths Corner Exercise
11D Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
Complete and Checked
Problems encountered
Skipped
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
(Full Solution)
73
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3B
Lesson Worksheet 11D
(Refer to §11.4)
11.4AMid-point Formula
If M(x , y) is the mid-point of the line segment
joining the points A(x1 , y1) and B(x2 , y2), then
x +x
y + y2
x = 1 2 and y = 1
.
2
2
Example 1
In the figure, M is the mid-point of the line
segment AB. Find the coordinates of M.
Instant Drill 1
In the figure, M is the mid-point of the line
segment CD. Find the coordinates of M.
y
y
A(0 , 4)
M
D(0 , 8)
M
x
O
Sol
1.
x
C(−10 , 0)
B(5 , 0)
Let (x , y) be the coordinates of M.
0+5
x=
= 2.5
2
4+0
y=
=2
2
∴ The coordinates of M are (2.5 , 2).
In the figure, M is the mid-point of the line
segment EF. Find the coordinates of M.
Sol
2.
y
F(4 , 3)
O
Let (x , y) be the coordinates of M.
(
)+(
)
x=
=
(
)
(
)+(
)
y=
=
(
)
∴ The coordinates of M are (
,
).
In each of the following, find the
coordinates of the mid-point of the line
segment AB.
(a) A(3 , 8), B(−3 , 6)
(b) A(−7 , 1), B(−5 , −3)
x
O
M
E(−2 , −5)
→
○ Ex 11D 1−5
74
3.
In the figure, P is the mid-point of AB.
Find the values of a and b.
y
4.
In the figure, P is the mid-point of AB.
Find the values of a and b.
B( − 6 , b ) y
B(7 , b)
P(4 , 5)
P(2 , 1)
x
O
A(a , 1)
x
O
A(a , −5)
Consider the x-coordinate of P.
(
)+(
)
4=
2
=
Consider the y-coordinate of P.
(
)+(
)
(
)=
2
=
→
○ Ex 11D 6, 7
 11.4B Section Formula
If P(x , y) is a point on the line segment
joining the points A(x1 , y1) and B(x2 , y2)
such that AP : PB = r : s, then
sx + rx2
sy + ry2
x= 1
and y = 1
.
r+s
r+s
Example 2
In the figure, P is a point
on AB. Find the coordinates
of P.
y
We say that P
divides AB
internally in the
ratio r : s.
Instant Drill 2
In the figure, P is a point on AB. Find the
coordinates of P.
A(1 , 9)
1 :2
P
3
y
:
1
P
A(−6 , 1)
x
O
B(4 , 0)
Sol Let (x , y) be the coordinates of P.
2(1) + 1( 4) 2 + 4
x=
=
=2
1+ 2
3
2(9) + 1(0) 18 + 0
y=
=
=6
1+ 2
3
∴ The coordinates of P are (2 , 6).
B(6 , 5)
Sol
75
x
O
Let (x , y) be the coordinates of P.
( )( ) + ( )( )
x=
=
( )+( )
( )( ) + ( )( )
y=
=
( )+( )
∴ The coordinates of P are (
,
).
5.
In the figure, P is a point on the line
6.
segment AB such that AP : PB = 1 : 4. Find
the coordinates of P.
y
A(−8 , 0)
P
If a point P divides the line segment
joining A(9 , 8) and B(−6 , −2) internally in
the ratio 2 : 3, find the coordinates of P.
Based on the given
information, we can
sketch this:
y
x
O
B(2 , −5)
O
→
○ Ex 11D 8−11
7.
In the figure, P is a point on the line
8.
segment AB such that AP : PB = 1 : 2. Find
the value of b.
In the figure, P is a point on the line
segment AB such that AP : PB = 3 : 1. Find
the coordinates of B.
y
y
A(6 , 5)
A(1 , 6)
P(4 , 4)
x
O
O
x
B(b , 0)
P(3 , −1)
B
Consider the x-coordinate of P.
2( ) + 1( )
4=
( )+( )
=
We may do the checking by substituting
the answer into the formula:
2( ) + 1( )
x-coordinate of P =
( )+( )
=
→
○ Ex 11D 12, 13
76
x
9. In the figure, P and Q are the points on AB such that they divide
AB into three equal parts.
A(−3 , −4)
B(3 , 5)
Q
P
(a) Find AP : PB.
(b) Find the coordinates of P.
1
:1
:
1
:1 + (
y
B(3 , 5)
Q
1
O
P
)
x
A(−3 , −4)
(c) Find the coordinates of Q.
→
○ Ex 11D 14, 15
‘Explain Your Answer’ Question
10. The figure shows two line segments APB and PQ. P is
the mid-point of AB.
(a) Find the coordinates of P.
(b) Is PQ perpendicular to AB? Explain your answer.
y
Q(6 , 6)
A(−3 , 5)
(a) x-coordinate of P =
P
x
O
B(7 , −3)
y-coordinate of P =
(b) Slope of AB =
Slope of PQ =
Recall:
If AB ⊥ PQ, then
slope of AB × slope of PQ =
(
).
∴ PQ (is / is not) perpendicular to AB.
77
 Level Up Questions
11. In the figure, a line segment runs from B(8 , 9)
to cut the y-axis at M, and to cut the x-axis at A.
If M is the mid-point of AB, find the coordinates
of A and M.
y
B(8 , 9)
M
A
O
x
12. P(3 , a) is a point on the line segment joining A(−1 , a + 1) and B(b , 2a). If P divides AB in the
ratio 1 : 3, find the values of a and b.
78
New Century Mathematics (2nd Edition) 3B
11
Coordinate Geometry of Straight Lines

Consolidation Exercise
11D
Level 1
1. In each of the following figures, M is the mid-point of the line segment AB. Find the coordinates
of M.
(a)
(b)
y
y
B(−3 , 6)
A(1 , 3)
M
B(5 , 1)
M O
x
O
x
A(−7 , −8)
In each of the following, find the coordinates of the mid-point of the line segment XY. [Nos. 2−4]
2. X(3 , −6), Y(11 , 0)
3. X(−7 , 2), Y(4 , −10)
4. X(−1.5 , 1.1), Y(−7.5 , 4.7)
5. In each of the following figures, P is the mid-point of the line segment AB. Find the values of
r and s.
(a)
(b)
y
y
O
A(−11 , r)
x
B(r , s)
x
O
P(−2 , −3)
A(−2 , −7)
B(s , −10)
P(3 , −5)
6. In each of the following, M is the mid-point of the line segment joining P and Q. Find the
coordinates of Q.
(a) P(−8 , −3), M(−7 , 4)
5
2


(b) P(−2 , 9), M  , 6 
79

7. In each of the following figures, P is a point on AB. Find the coordinates of P.
y
(a)
(b)
y
A(−5 , 4)
B(10 , 7)
P
A(−2 , −1)
O
x
O
x
P
B(1 , −5)
AP : PB = 1 : 3
AP : PB = 2 : 1

In each of the following, find the coordinates of a point P which divides the line segment XY
internally in the given ratio. [Nos. 8−9]
8. X(−4 , 1), Y(8 , −5), XP : PY = 1 : 2

9. X(2 , −1), Y(9 , 6), XP : PY = 4 : 3

10. In each of the following figures, P is a point on AB. Find the values of r and s.
(a)
(b)
y
y
B(7 , 5)
A(−4 , r)
P(2 , 3)
A(r , s)
O
P(−1 , −3)
x
x
O
B(s , −7)
AP : PB = 1 : 4
AP : PB = 2 : 1

11. In each of the following, P is a point which divides the line segment CD internally in the given
ratio. Find the coordinates of D.
(a) C(1 , 12), P(5 , 4), CP : PD = 4 : 5
(b) C(−3 , −5), P(9 , 1), CP : PD = 6 : 1

12.In the figure, X and Y are points on the line segment joining
A(−4 , −7) and B(14 , 5). If AX : XY : YB = 1 : 1 : 1, find the
coordinates of X and Y.
y
B(14 , 5)
Y
x
O
X
A(−4 , −7)
80
Level 2
13. The line segment joining A(−6 , k) and B(k + 3 , −5) cuts the x-axis at M. If M is the mid-point
of AB, find
(a) the value of k,
(b) the coordinates of M.
14.In the figure, Q is a point on the y-axis. The line segment joining
P(−16 , −5) and Q cuts the x-axis at M. If M is the mid-point of
PQ, find the coordinates of Q and M.
y
Q
M
x
O
P(−16 , −5)

15. P(2c − 1 , 2) is a point on the line segment joining A(9 , c + 4)
and B(3d , 1 − 3c), and P divides AB internally in the ratio 1 : 2. Find the values of c and d.

16.In the figure, B is a point on the x-axis. The line segment joining
A(−9 , 5) and B cuts the y-axis at P. If AP : PB = 3 : 2, find the
coordinates of P and B.
y
A(−9 , 5)
P
B
x
O

17.In the figure, X and Y are points on the line segment joining
A(−5 , −15) and B(9 , 6). If AX : XY : YB = 2 : 1 : 4, find the
coordinates of X and Y.
y
B(9 , 6)
x
O
Y
X
A(−5 , −15)

18.In the figure, the line segment joining A(3 , −10) and B(11 , 6)
cuts the x-axis at P.
(a) Find AP : PB.
(b) Using the result of (a), find the coordinates of P.
19.In the figure, A(6 , 7) is a vertex of △ABC. M(1 , 4) and N(5 , 1)
are the mid-points of AB and AC respectively.
(a) Find the coordinates of B and C.
(b) Suppose P is the mid-point of BC. Does P lie on the y-axis?
Explain your answer.
y
B(11 , 6)
O
x
P
A(3 , −10)
y
A(6 , 7)
M(1 , 4)
N(5 , 1)
x
B
O
C
81
Answer
Consolidation Exercise 11D
1. (a) (3 , 2)
2. (7 , −3)
 3

3.  − , − 4 
 2

(b) (−5 , −1)
4. (−4.5 , 2.9)
5. (a) r = 8, s = −3 (b) r = 4, s = 7
6. (a) (−6 , 11)
(b) (7 , 3)
7. (a) (1 , 1)
(b) (−1 , −2)
8. (0 , −1)
9. (6 , 3)
10. (a) r = −8, s = −1 (b) r = −2, s = 11
11. (a) (10 , −6)
(b) (11 , 2)
12. X(2 , −3), Y(8 , 1)
13. (a) 5
(b) (1 , 0)
14. Q(0 , 5), M(−8 , 0)
15. c = 3, d = −1
16. P(0 , 2), B(6 , 0)
17. X(−1 , −9), Y(1 , −6)
18. 5 : 3
(b) (8 , 0)
19. (a) B(−4 , 1), C(4 , −5)
(b) yes
82
F3B: Chapter 11E
Date
Task
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Book Example 20
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Book Example 21
○
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(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
11E Level 1
Maths Corner Exercise
11E Level 2
Maths Corner Exercise
11E Level 3
Maths Corner Exercise
11E Multiple Choice
E-Class Multiple Choice
Self-Test
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(Full Solution)
83
Teacher’s
Signature
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)
Teacher’s
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)
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)
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)
Mark:
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 Book 3B
Lesson Worksheet 11E
(Refer to §11.5B)
[In this worksheet, use analytic approach to complete the proofs.]
11.5B Prove Geometric Properties by Analytic Approach
Analytic approach in geometry:
prove geometric properties by introducing a rectangular coordinate system
Example 1
In the figure, OABC is a
square. Let (0 , a) be the
coordinates of A.
A(0 , a)
B
O
C
(a) Express the coordinates of B and C in
terms of a.
Check if m1 × m2 = −1.
(b) Prove that the two diagonals OB and AC
are perpendicular to each other.
Sol (a) The coordinates of B are (a , a).
The coordinates of C are (a , 0).
a−0
(b) Slope of OB =
=1
a−0
0−a
Slope of AC =
= −1
a−0
∵ Slope of OB × slope of AC
= 1 × (−1)
= −1
∴ OB ⊥ AC
1.
Instant Drill 1
In the figure, OABC is a
rectangle. Let (p , q) be
the coordinates of B.
y
Refer to Example 1. Prove that
∠BOC = 45°.
Recall:
x
y
B(p , q)
A
O
C
x
(a) Express the coordinates of A and C in
terms of p and q.
(b) Prove that the two diagonals OB and AC
bisect each other.
Sol (a) The coordinates of A are
(
,
).
The coordinates of C are (
,
).
(b) Coordinates of the mid-point of OB
=
Coordinates of the mid-point of AC
=
∵ The two diagonals OB and AC
(have / do not have) the same
mid-point.
∴
2.
For a straight line with
inclination θ, slope =
tan θ.
Refer to Instant Drill 1. Prove that
the two diagonals are equal in length.
Distance =
( x2 − x1 ) 2 + ( y2 − y1 ) 2
→
○ Ex 11E 1−3
84
3. In the figure, O(0 , 0), A(h , k) and B are the vertices of a
triangle. M is the mid-point of OB and AM is a vertical line.
Prove that △OAB is an isosceles triangle.
y
A ( h , k)
x
M
O : Express
Step 1
the B
coordinates of B in
terms of h.
Step 2: Check if it has two
equal sides by using
→
○ Ex 11E 6
 Level Up Question
4.
In the figure, B is a point on AO such that AB : BO = 1 : 2.
C is a point on AD such that BC is a horizontal line. Prove
that AC : CD = 1 : 2.
y
A(a , p)
B
O
85
C
D(a , 0)
x
New Century Mathematics (2nd Edition) 3B
11
Coordinate Geometry of Straight Lines


Consolidation Exercise
11E
Level 1
1. In the figure, ABC is a triangle. O is the mid-point of AC. Let
(a , 0) and (0 , b) be the coordinates of A and B respectively.
(a) Express the coordinates of C in terms of a.
(b) Hence, prove that △ABC is an isosceles triangle.
2. In the figure, O(0 , 0), A(0 , 2a), B(2a , 2a) and C(2a , 0) are
the vertices of a square. D and E are the mid-points of OA and
AB respectively.
(a) Express the coordinates of D and E in terms of a.
(b) Hence, prove that OE ⊥ DC.
3. In the figure, O(0 , 0), P(a , 0), Q(a , b) and R(0 , b) are the
vertices of a rectangle. X and Y are the points on OP and RQ
respectively such that OX = YQ. Let (c , 0) be the coordinates
of X.
(a) Express the coordinates of Y in terms of a, b and c.
(b) Hence, prove that RX // YP.
y
B(0 , b)
y
A(0 , 2a)
86
x
O
A(a , 0)
E
B(2a , 2a)
D
C(2a , 0)
O
y
Y
R(0 , b)
X(c , 0)
O
4. In the figure, O(0 , 0), D(0 , b), E(a , b) and F(a , 0) are the
vertices of a rectangle. M and N are the points on DO and EF
respectively such that EN = NF and DE // MN // OF.
(a) Express the coordinates of N in terms of a and b.
(b) Hence, prove that M is the mid-point of DO.
C
x
Q( a , b )
P(a , 0)
x
y
D(0 , b)
E(a , b)
M
N
O
F(a , 0)
x
5. In the figure, O(0 , 0), P(b , c), Q(a + b , c) and R(a , 0) are the
vertices of a parallelogram. X and Y are the mid-points of PO and
QR respectively. Prove that PQ // XY by analytic approach.
y
Q( a + b , c )
P ( b , c)
X
Level 2
6. In the figure, O(0 , 0), P(b , c), Q(a + b , c) and R(a , 0) are the
vertices of a quadrilateral, where OP = PQ. Prove that OR = RQ by
analytic approach.
Y
y
P ( b , c)
7. In the figure, O(0 , 0), A(6a , 0) and B(6b , 6c) are the vertices of
△OAB. P is the mid-point of AB. Y is a point on OP such that
OY : YP = 2 : 1.
(a) Express the coordinates of Y in terms of a, b and c.
(b) If Q is the mid-point of OA, prove by analytic approach that
(i) BY : YQ = 2 : 1,
(ii) B, Y and Q lie on the same straight line.
x
y
B(6b , 6c)
Y
O
P
x
A(6a , 0)
Q
A
X
W
B
D
Z
Y
C
87
Q( a + b , c )
R(a , 0)
O
8. In the figure, ABCD is a kite, where AB = AD and CB = CD. W, X, Y
and Z are the mid-points of AB, AD, BC and DC respectively. Prove
that WX // BD // YZ and BD = WX + YZ by analytic approach.
x
R(a , 0)
O
Answer
Consolidation Exercise 11E
1. (a) (−a , 0)
2. (a) D(0 , a), E(a , 2a)
3. (a) (a − c , b)


b
2
4. (a)  a , 
7. (a) (2a + 2b , 2c)
88
F3B: Chapter 12A
Date
Task
Lesson Worksheet
Progress
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Book Example 1
○
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○
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Book Example 2
○
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Complete
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Book Example 3
○
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Complete
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Book Example 4
○
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Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
12A Level 1
Maths Corner Exercise
12A Level 2
Maths Corner Exercise
12A Level 3
Maths Corner Exercise
12A Multiple Choice
E-Class Multiple Choice
Self-Test
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(Full Solution)
89
Teacher’s
Signature
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)
Teacher’s
Signature
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(
)
Teacher’s
Signature
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)
Teacher’s
Signature
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)
Mark:
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Book 3B
Lesson Worksheet 12A
(Refer to §12.1B–C)
12.1BPossible Outcomes of an Activity and Outcomes Favourable to an Event
1. List ALL the possible outcomes of each of the following activities.
(The first two have been done for you as examples.)
Activity
All possible outcomes
(a)
Toss a coin.
head, tail
(b)
Choose a letter from the word ‘MATHS’.
‘M’, ‘A’, ‘T’, ‘H’, ‘S’
(c)
Record the gender of a person.
(d)
Choose a digit from the number ‘630’.
2. Complete the table below.
(The first one has been done for you as an example.)
Activity
Event
(a)
Play the game
‘rock–paper–scissors’.
Paper is thrown.
(b)
Choose a letter from
the word ‘SINE’.
‘I’ is chosen.
(c)
Throw a dice to obtain
a number.
An odd number
is obtained.
12.1C
All possible outcomes
rock, paper, scissors
Outcome(s)
favourable
to the event
paper
Definition of Probability
Suppose all the possible outcomes in an activity are equally likely to occur (i.e.
they are equally likely outcomes). Then the probability of an event E, denoted by
P(E), is defined as:
number of outcomes favourable to E
P(E) =
total number of possible outcomes
Note: (a) For any event E, 0 ≤ P(E) ≤ 1.
(b) If E is an impossible event, then P(E) = 0.
(c) If E is a certain event, then P(E) = 1.
90
Example 1
‘Chosen randomly’ means that all the
possible outcomes are equally likely to
chosen.randomly from the word
A letter isbechosen
‘ADD’. Find the probability of choosing a
letter ‘D’.
Sol
Instant Drill 1
A letter is chosen randomly from the word
‘TERRY’. Find the probability of choosing
each of the following letters.
(a) ‘E’
(b) ‘R’
Sol (a)
Total number of possible outcomes =
Total number of possible outcomes = 3
T
A
D
D
E
R
R
Y
Circle the favourable outcome(s), i.e. ‘E’.
Number of favourable outcomes =
Number of favourable outcomes = 2
There are 3 letters in the word ‘ADD’.
There are 2 ‘D’s in the word.
2
∴ P(‘D’ is chosen) =
3
There are
‘TERRY’.
There is/are
letters in the word
‘E’(s) in the word.
(
)
∴ P(‘E’ is chosen) =
(
)
(b)
T
E
R
R
Y
Circle the favourable outcome(s), i.e. ‘R’.
Number of favourable outcomes =
There is/are
∴ P(‘
3.
A digit is chosen randomly from the date
‘01-10-2019’. What is the probability that
it is
(a) a ‘0’?
(b) an odd number?
4.
‘R’(s) in the word.
(
)
’ is chosen) =
(
)
A fair dice is thrown. Find the probability
of getting
(a) a ‘5’,
(b) an even number.
A fair dice means that it has a uniform
weight.
All the possible outcomes are equally
likely outcomes.
∴
○→ Ex 12A 1–4
91
Example 2
4# is a 2-digit number, where # is an integer
from 0 to 9 inclusive. Find the probability that
the 2-digit number is a multiple of 5.
Sol
4# can be:
40
41
42
43
44
45
46
47
Multiples of 5
48
49
Instant Drill 2
3 is a 2-digit number, where is an integer
from 1 to 9 inclusive. Find the probability that
the 2-digit number is a multiple of 3.
Sol
3 can be:
31
Total number of possible outcomes =
Only
are multiples of 3.
∴ Number of favourable outcomes =
Total number of possible outcomes = 10
Only 40 and 45 are multiples of 5.
∴ Number of favourable outcomes = 2
2
P(a multiple of 5) =
10
1
=
5
5.
♦6 is a 2-digit number, where ♦ is an
integer from 2 to 7 inclusive. Find the
probability that the 2-digit number is
greater than 39.
32
P(
6.
)=
In a school, the ages of 8 teachers are as
follows:
24, 27, 29, 29, 33, 37, 38, 43
If a teacher is chosen at random from
them, find the probability that the age of
the teacher is an odd number.
○→ Ex 12A 5–10
7. In a group of 20 students, 5 of them are boys. If a student is randomly selected from the
group, what is the probability that the student selected is not a boy?
○→ Ex 12A 11
92
8. A box contains 6 apples, 4 oranges and 5 mangoes. If a piece of fruit is drawn at random from
the box, find the probability that the fruit drawn is
(a) an apple,
(b) not a mango,
(c) not a kiwi fruit,
(d) a peach.
○→ Ex 12A 12–14
 Level Up Question
9. In the figure, there are only white balls and black balls in a bag. A ball
is randomly drawn from the bag. Find the probability that the ball
drawn is
(a) a black ball or a white ball,
(b) a white ball,
(c) a green ball,
(d) a white ball with a number.
93
1
3
2
New Century Mathematics (2nd Edition) 3B
12
Introduction to Probability
Consolidation Exercise
12A

Level 1
1. A letter is chosen randomly from the word ‘EXPERIENCE’. Find the probability of choosing
each of the following letters.
(a) ‘I’
(b) ‘E’
2. Benny selects a digit from his staff number ‘39769’ at random. Find the probability that the
digit selected is an odd number.
3. A fair dice is thrown. Find the probability of getting
(a) a ‘4’,
(b) a number less than 4.
4. ☆9 is a 2-digit number, where ☆ is an integer from 1 to 9 inclusive. Find the probability that
the 2-digit number is
(a) greater than 70,
(b) a prime number,
(c) an even number.
5. The scores of eight students in a test are 49, 25, 74, 36, 58, 43, 65 and 85 respectively. If a
student is chosen at random, find the probability that the score of the student chosen
(a) is a square number,
(b) has the tens digit 3 greater than the units digit.
6. A card is selected at random from the playing cards shown in the figure.
What is the probability of selecting each of the following cards?
(a) a ‘5’
(b) a face card
(Note: A face card is a ‘J’, a ‘Q’ or a ‘K’.)
(c) a diamond
7. There are 10 boys in a group of 16 students. If a student is chosen at random from the group,
find the probability that the student chosen is a girl.
8. An inspector visits a restaurant on a day at random in April. Suppose there are 9 public holidays
in April. Find the probability that the visit is not on a public holiday.
9. An integer is randomly selected from the 90 integers 1 to 90. What is the probability that the
integer selected is
(a) a multiple of 9?
(b) not a multiple of 9?
94
10. A box contains 15 white chocolates and 25 dark chocolates. If a chocolate is chosen at random
from the box, find the probability of each of the following events.
(a) A dark chocolate is chosen.
(b) A white chocolate is chosen.
11. There are 5 oranges, 3 apples and 4 pears in a refrigerator. Kenneth takes out a piece of fruit at
random. Find the probability that the fruit taken out is
(a) an apple,
(b) not a pear,
(c) a lemon.
12. Sam has n books, and 24 of them are storybooks. If he selects a book at random, the probability
of selecting a storybook is 0.3. Find the value of n.
Level 2
13. A card is drawn randomly from a pack of 52 playing cards. What is the probability of drawing
(a) a ‘10’,
(b) a black ‘J’,
(c) a diamond or a club,
(d) any card with number from ‘3’ to ‘7’ inclusive.
14. There are 5 green stone marbles, 4 blue stone marbles, 2 blue glass marbles and 4 yellow glass
marbles in a bag. If a marble is drawn at random from the bag, find the probability of drawing
(a) a green stone marble,
(b) a blue marble,
(c) a yellow marble or a stone marble,
(d) a stone marble or a glass marble.
15. A number is randomly selected from the 40 integers 1 to 40. What is the probability that the
number selected is
(a) an even square number?
(b) a multiple of 7 or a multiple of 8?
Number of students
16. The figure shows the distribution of the favourite fruit of a group of students.
Favourite fruit of a group of students
15
10
5
0
Apple Orange Mango Peach
Fruit
If a student is selected at random from the group, what is the probability that the favourite fruit
of the student is
(a) orange?
(b) apple or peach?
(c) not apple?
95
17. The stem-and-leaf diagram below shows the ages of the employees in a company.
Ages of the employees in a company
Stem(10) Leaf (1)
2 3 4 5 6 7 9
3 1 2 3 3 4 7 8
4 1 5 8 9 9
5 6 8
If an employee is randomly selected from the company, find the probability that the employee
(a) is younger than 28,
(b) is older than 43,
(c) is aged between 35 and 57.
18. Janet has 120 books, in which 30 are cookery books, 36 are comic books, 42 are textbooks and
the rest are travel books. She takes a book at random. What is the probability that the book
taken is
(a) a comic book?
(b) a travel book?
(c) neither a comic book nor a textbook?
19. In a group of people, the numbers of males and females who are smokers or non-smokers are
shown in the table below.
Smokers
Non-smokers
18
72
Number of males
12
108
Number of females
(a) If a person in the group is chosen randomly, what is the probability that the person is
(i) a male non-smoker?
(ii) a smoker?
(b) Henry claims that if a person is chosen randomly from the group, the probability of
choosing a female is more than that of choosing a male. Do you agree? Explain your
answer.
96
Answer
Consolidation Exercise 12A
1. (a)
2.
1
10
(b)
2
5
(b)
1
2
4
5
1
6
1
4. (a)
3
3
5. (a)
8
1
6. (a)
8
3
7.
8
7
8.
10
1
9. (a)
9
5
10. (a)
8
1
11. (a)
4
12. 80
1
13. (a)
13
1
14. (a)
3
3
15. (a)
40
13
16. (a)
30
1
17. (a)
4
3
18. (a)
10
3. (a)
19. (a) (i)
(b)
5
9
(c) 0
(b)
(b)
1
4
1
4
(c)
3
8
8
9
3
(b)
8
(b)
(b)
2
3
1
26
2
(b)
5
(b)
3
10
3
(b)
10
1
(b)
10
(b)
12
35
(c) 0
1
5
(d)
2
13
13
(c)
(d) 1
15
1
(b)
4
4
(c)
5
2
(c)
5
7
(c)
20
1
(ii)
7
(c)
(b) yes
97
F3B: Chapter 12B
Date
Task
Lesson Worksheet
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Book Example 6
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Book Example 7
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Book Example 8
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Book Example 9
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Book Example 10
○
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○
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(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
12B Level 1
Maths Corner Exercise
12B Level 2
○
○
○
Complete and Checked
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Skipped
○
○
○
○
○
Complete and Checked
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Complete and Checked
Problems encountered
(Full Solution)
98
Teacher’s
Signature
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(
)
Teacher’s
Signature
___________
Maths Corner Exercise
12B Level 3
Maths Corner Exercise
12B Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
○
○
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○
Skipped
Complete and Checked
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Complete and Checked
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Problems encountered
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99
(
)
Teacher’s
Signature
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(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3B
12.2A
I.
Lesson Worksheet 12B
(Refer to §12.2)
Methods for Listing Possible Outcomes
Tree Diagram
Example 1
A letter is randomly chosen from each of the
two words ‘UP’ and ‘POP’. Find the
probability of each of the following events.
(a) Two ‘P’s are obtained.
(b) Only one ‘P’ is obtained.
Sol The tree diagram below shows all the
possible outcomes.
1st letter 2nd letter Outcome
P . . . . . . UP
U
O . . . . . . UO
P . . . . . . UP
Instant Drill 1
Two fair coins are tossed. Find the probability
of each of the following events.
(a) Two heads are obtained.
(b) Only one head is obtained.
Sol Let H stand for a head and T for a tail.
The tree diagram below shows all the
possible outcomes.
1st coin
2nd coin
Outcome
H ......
H
......
......
P . . . . . . PP
O . . . . . . PO
P . . . . . . PP
Total number of possible outcomes = 6
(a)
P
T
......
Total number of possible outcomes =
(a)
The favourable outcome is
Which of them are favourable
outcomes?
UP
UO UP
PP
HH.
PO
PP
Number of favourable outcomes =
∴ P(two heads)
Number of favourable outcomes = 2
∴ P(two ‘P’s are obtained)
2
=
6
1
=
3
(b)
=
(b)
The favourable outcomes
are
.
Number of favourable outcomes =
∴ P(only one head is obtained)
Which of them are favourable
outcomes?
=
UP of
UO
UP PPoutcomes
PO PP
Number
favourable
=3
∴ P(only one ‘P’ is obtained)
3
=
6
1
=
2
100
1.
Box X contains a red pen and a blue pen.
Box Y contains a red pen, a green pen and
a blue pen. Roy randomly chooses a pen
from each box. Find the probability of
each of the following events.
(a) Two blue pens are chosen.
(b) The pens chosen are of the same
colour.
2.
There are three true or false questions in a
quiz. If Donna answers each question by
choosing ‘true’ or ‘false’ at random, find
the probability that ‘true’ is chosen
(a) three times,
(b) only once,
(c) at least two times.
Let
stand for a red pen,
stand
for a blue pen and
stand for a green
pen.
The tree diagram below shows all the
possible outcomes.
Box X
Box Y
Outcome
Total number of possible outcomes =
(a) Number of favourable outcomes =
∴ P(two blue pens)
=
(b)
The favourable outcomes
are
.
○→ Ex 12B 1–5
101
II. Tabulation
Instant Drill 2
A bag contains two red blocks and two green
blocks. Two blocks are randomly drawn from
the bag at the same time. Find the probability
of each of the following events.
(a) Two red blocks are drawn.
(b) The two blocks drawn are of different
colours.
Sol The table below lists all the possible
outcomes.
Number on the 2nd ball
Sol Let R1, R2 stand for the two red blocks,
and G1, G2 stand for the two green blocks.
The table below lists all the possible
outcomes.
2nd block
1
2
3
1

(1 , 2)
(1 , 3)
2
(2 , 1)

(2 , 3)
3
(3 , 1)
(3 , 2)

R1
1st block
Number on
the 1st ball
Example 2
A box contains 3 balls marked with 1, 2 and 3
respectively. Two balls are randomly drawn
from the box at the same time. Find the
probability of each of the following events.
(a) The numbers on the two balls are both odd
numbers.
(b) One of the balls drawn is marked with 2.
‘’ denotes that the
outcome
is impossible.
G1

R1R2
R1G1
G2

R2

G1

Total number of possible outcomes =
(a) The favourable outcomes
The favourable outcomes are:
(3 , 1)
are
.
Number of favourable outcomes =
∴ P(two red blocks)
Number of favourable outcomes = 2
∴ P(both are odd numbers)
2
=
6
1
=
3
(b)
R2
G2
Total number of possible outcomes = 6
(a)
(1 , 3)
R1
=
(b)
Circle the favourable
outcomes in the table.
The favourable outcomes are:
(1 , 2) (2 , 1) (2 , 3) (3 ,
2)
Number of favourable outcomes = 4
∴ P(one of the numbers is 2)
4
=
6
2
=
3
Number of favourable outcomes =
∴ P(two blocks of different colours)
=
102
There are two boys and two girls in a
room. Two of them are chosen at random.
(a) List all the possible outcomes in a
table.
(b) Hence, find the probability of each of
the following events.
(i) The two children chosen are of
different genders.
(ii) No boys are chosen.
4.
A letter is randomly chosen from each of
the two words ‘SUN’ and ‘MOON’.
(a) List all the possible outcomes in a
table.
(b) Hence, find the probabilities of the
following events.
(i) Two ‘N’s are chosen.
(ii) At least one ‘N’ is chosen.
(iii) Two ‘O’s are chosen.
(a) Let
stand for the two
boys, and
stand for
the two girls.
The table below lists all the possible
outcomes.
2nd child
1st child
3.
(b)
○→ Ex 12B 6–10
103
12.2BGeometric Probability
(a) The probability obtained by considering measures of geometric figures,
such as lengths, areas or volumes, is called a geometric probability.
(b) If an event E happens in a certain region of a geometric figure, then
measure of the region in which E happens
P(E) =
same measure of the whole figure
5. Refer to the line segment ABC on the right. If a point on AC
is selected at random, what is the probability that the point
lies on BC?
P(the point lies on BC) =
2 cm
A
5 cm
C
B
the region in which the event
length of (happens )
length of (
=
)
the whole
figure
○→ Ex 12B 11
Example 3
Instant Drill 3
8 cm
4 cm
6 cm
15 cm
The figure shows a target formed by two
squares. Ada throws a dart at random and it hits The figure shows a target formed by two
the target. Find the probability that the dart hits circles. Carol shoots an arrow at random and it
the smaller square.
hits the target. Find the probability that the
arrow hits the smaller circle.
Sol
Region in which the
event happens
Whole figure
the smaller square
Sol P(
the larger square
=
P(hitting the smaller square)
area of the smaller square
=
area of the larger square
4 2 cm 2
Do not write:
= 2
2
P(hitting the smaller
8 cm
square)
1
=
4 cm
4
=

8 cm
104
)
7.
6.
The figure shows an equilateral triangular
target, in which all the small triangles are
identical. Billy throws a dart randomly and
it hits the target. Find the probability that
the dart hits the shaded region.
The figure shows a circular
lucky wheel. A player turns
the wheel once at random.
Find the probability that the
pointer falls in the shaded
sector.
80°
Consider the
arc
length
of the
○→ Ex 12B 12–14
‘Explain Your Answer’ Question
8. Box X contains 2 gold coins and 1 silver coin. Box Y contains 1 gold coin and 2 silver coins.
Teresa draws one coin from each box at random. She claims that the probability of drawing
1
one gold coin and one silver coin is greater than . Do you agree? Explain your answer.
2
105
 Level Up Questions
9. Two fair dice are thrown. Find the probability that the sum of the two numbers is 7.
10. The figure shows a target formed by three concentric circles.
Kevin throws a dart randomly and it hits the target. Find the
probability of each of the following events.
(a) The dart hits the grey bullseye.
(b) The dart hits the dotted region.
106
2 cm
10 cm
4 cm
New Century Mathematics (2nd Edition) 3B
12
Introduction to Probability
Consolidation Exercise
12B

Level 1
Use tree diagrams to solve the following problems. [Nos. 1–4]
1. James tosses a fair coin two times. Find the probability that he gets a head and then a tail.
2. David makes two basketball shots. Assume that the probabilities of the shots being successful
and unsuccessful are equal. Find the probability that he makes one unsuccessful shot only.
3. Patrick has a red tie, a blue tie and a green tie. He wears one of the three ties at random every
day. Find the probability that he wears the same tie in two successive days.
4. In a shop, there are apple juice, orange juice and pineapple juice only. Karen and Carman each
buy one kind of juice at random from the shop. What is the probability that
(a) both of them buy pineapple juice?
(b) only one of them buys orange juice?
Use the method of tabulation to solve the following problems. [Nos. 5–8]
5. A letter is randomly selected from each of the two words ‘SUM’ and ‘US’. Find the probability
that the two letters selected are different.
6. There are three candidates P, Q and R in an election. Calvin and Ben each choose one candidate
at random. What is the probability that
(a) they both choose candidate Q?
(b) they choose different candidates?
7. There are two boxes X and Y. Each box contains one apple and two lemons. If a piece of fruit is
randomly taken out from each box at the same time, find the probability that an apple and a
lemon are taken out.
8. There are three boys and four girls in a class. The ages of the boys are 3, 5 and 6 respectively,
and the ages of the girls are 3, 4, 5 and 7 respectively. If one boy and one girl are selected at
random to answer a question, find the probability that they
(a) have the same age,
(b) have an age difference of 2.
107
9. The figure shows a line segment XY of length 12 cm.
N is a point on XY such that NY = 8 cm. If a point K is
selected randomly from XY, find the probability that K
lies on NY.
X
N
8 cm
12
Y
10. The figure shows a square target of side 1 m. A circular region of diameter 60 cm
is in the
1m
middle of the target. Jacky shoots an arrow
that
at random and it hits the target. Find the probability, in terms of π,
the arrow hits the circular region.
60 cm
11. The figure shows the circular wheel in a game. Fanny turns
the wheel once at random and wins the prize indicated by the
pointer. Find the probability that she wins
(a) prize C,
(b) prize D.
Prize D
Prize A
129°
132°
Prize B
Prize C
Level 2
12. George has a brown dog, a grey dog and two white dogs. He chooses two dogs at random. Find
the probability that
(a) both dogs chosen are white,
(b) one of the dogs chosen is brown.
13. A drawer contains two red socks and two green socks. Ken takes two socks randomly from the
drawer at the same time. Find the probability that
(a) the colours of the two socks taken are different,
(b) at most one green sock is taken.
14. A fair coin is tossed three times. Find the probability of getting
(a) three heads,
(b) at most one tail.
15. Two fair dice are thrown. Find the probability of each of the following events.
(a) The sum of the two numbers obtained is less than 5.
(b) The difference of the two numbers obtained is a multiple of 2.
108
16. Amy has four cards and Ben has three cards as shown below. Each of them draws one of his/her
own cards at random.
Amy’s cards
Ben’s cards
Find the probability of each of the following events.
(a) The cards drawn are in the same suit.
(b) The sum of the numbers on the cards drawn is at least 10.
17.There are 5 pens in a case including 2 different red pens, 1 blue pen and 2 different green pens.
A pen is drawn at random and put back into the case. Then, a pen is drawn at random from the
case again. Find the probability that
(a) a red pen and a blue pen are drawn,
(b) only one of the pens drawn is green,
(c) the pens drawn are different.
18. The figure shows a target formed by three concentric circles.
The diameter of smallest circle is 80 cm, and the widths of the
rings of regions B and C are 30 cm and 20 cm respectively.
Paul throws a dart at random and it hits the target. He can win a
prize according to the region where the dart hits.
Region A
Region B
Region C
A coupon of
A roll of
A pen
Prize
$50
toilet paper
(a) Find the probability that Paul wins a coupon.
(b) Susan claims that the probability that Paul wins a roll of toilet
paper is lower than that of a pen. Do you agree? Explain your
answer.
109
20 cm
30 cm
Region A
Region B
Region C
Answer
Consolidation Exercise 12B
1
4
1
2.
2
1
3.
3
1.
4. (a)
5.
(b)
4
9
1
9
(b)
2
3
1
6
(b)
1
3
2
3
6. (a)
7.
1
9
4
9
8. (a)
2
3
9π
10.
100
11
11. (a)
30
1
12. (a)
6
2
13. (a)
3
1
14. (a)
8
1
15. (a)
6
1
16. (a)
4
4
17. (a)
25
16
18. (a)
81
9.
(b)
(b)
(b)
(b)
(b)
(b)
(b)
12
25
1
40
1
2
5
6
1
2
1
3
2
3
(c)
4
5
(b) yes
110
F3B: Chapter 12C
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 11
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 12
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
12C Level 1
Maths Corner Exercise
12C Level 2
Maths Corner Exercise
12C Level 3
Maths Corner Exercise
12C Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
Complete and Checked
Problems encountered
Skipped
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
(Full Solution)
111
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3B
Lesson Worksheet 12C
(Refer to §12.3)
12.3 Experimental Probability
(a) A probability found by deductive reasoning is called
a theoretical probability. A probability based on
relative frequencies found from experiments is called
an experimental probability.
(b) Experimental probability of an event E
number of times that event E happens
=
number of trials
In Lesson Worksheets
12A and 12B, all
probabilities are
theoretical probabilities.
(c) When the number of trials is very large,
experimental probability ≈ theoretical probability
Example 1
A coin is tossed many times and the results are
as follows:
Head
Tail
Outcome
15
35
Frequency
Find the experimental probability of getting
(a) a head,
(b) a tail.
Sol Total frequency = 15 + 35 = 50
(a) Experimental probability of getting a
head
15
=
50
3
=
10
(b) Experimental probability of getting a
tail
35
=
50
7
=
10
Instant Drill 1
A drawing pin is thrown many times and the
results are as follows:
Outcome
The tip
points up
The tip lands
on the ground
52
28
Frequency
Find the experimental probability that the tip
(a) points up,
(b) lands on the ground.
Sol Total frequency = (
)+(
)=
(a) Experimental probability that the tip
points up
=
(b) Experimental probability that the tip
lands on the ground
=
112
1.
A group of students is randomly chosen
from a school. The numbers of hats they
have are recorded as follows:
1
2
3
Number of hats
2.
27 13
5
Number of students
If a student is randomly chosen from the
school, find the experimental probability
that the student has
(a) 3 hats,
(b) at least 2 hats.
A group of customers is randomly chosen
from a restaurant. The set meals they
ordered are recorded as follows:
A
B
C
D
Set meal
38
21
29
32
Frequency
If a customer is randomly chosen from the
restaurant, find the experimental probability
that the customer ordered
(a) set meal B,
(b) set meal A or D.
(b) Frequency of ‘A’ =
Frequency of ‘D’ =
Frequency of ‘A or D’ = (
(
)
=
)+
○→ Ex 12C 1–6
 Level Up Question
3. 300 books are selected at random from a library. Their languages are recorded as follows:
Chinese
English Japanese
Others
Language
126
x
55
14
Frequency
(a) Find x.
(b) If a book is selected at random from the library, find the experimental probability that it is
(i) an English book,
(ii) not a Chinese book.
113
New Century Mathematics (2nd Edition) 3B
12
Introduction to Probability

Consolidation Exercise
12C
Level 1
1. The test results of 24 students chosen randomly from S3 students in a school are shown below.
Fail
Pass
Result
15
9
Frequency
If a student is chosen randomly from S3 students in the school, what is the experimental
probability that the student
(a) fails the test?
(b) passes the test?
2. The genders of a group of people chosen randomly from a city are shown below.
Male
Female
Gender
84
66
Frequency
If a person is selected randomly from the city, what is the experimental probability that the
person is a female?
3. Five brands of rice cookers are sold in a shop. The following table shows the brands of rice
cookers bought by 240 customers chosen randomly from the shop.
A
B
C
D
E
Brand
33
63
27
72
45
Frequency
If a customer buying a rice cooker in the shop is chosen randomly, find the experimental
probability that the rice cooker bought by the customer
(a) is brand C,
(b) is not brand E.
4. 500 adults are chosen randomly from a city. The numbers of credit cards owned by them are as
follows.
0
1
2
3
4
5
Number of credit cards
72
145
100
83
64
36
Frequency
If an adult is chosen randomly from the city, find the experimental probability that the adult
(a) does not have any credit card,
(b) has at most 3 credit cards.
5. A dice is thrown many times and the results are as follows.
1
2
3
4
5
Number obtained
19
58
37
21
44
Frequency
Find the experimental probability of getting
(a) an odd number,
(b) a number less than 5.
114
6
71
6. 50 students are chosen at random from a school. Their favourite sports are recorded as follows.
Football
Basketball
Swimming
Others
Favourite sport
12
13
n
18
Frequency
(a) Find the value of n.
(b) If a student is chosen at random from the school, find the experimental probability that the
student’s favourite sport is basketball or swimming.
Level 2
7. Three coins are tossed together 400 times. In 6% of the times, no heads are obtained, and in
7
25
of the times, only one head is obtained.
(a) Find the number of times that
(i) no heads are obtained,
(ii) only one head is obtained.
(b) If the three coins are tossed together once more, find the experimental probability that at
least two heads are obtained.
8. A sample of students from a school is chosen randomly and asked about their favourite
countries. The results are shown in the following bar chart.
Number of students
Favourite countries of a sample of students
30
20
10
0
China
Japan USA
Countries
Others
If a student is chosen at random from the school, find the experimental probability that the
student’s favourite country is
(a) Japan,
(b) USA or others.
9. Yesterday, a scientist caught 76 tortoises from a lake. After making a mark on the shell of each
tortoise, they were put back into the lake. Today, he catches 285 tortoises from the lake and
finds that 31 of them have marks on their shells. Estimate the number of tortoises in the lake,
correct to the nearest integer.
10. There are 483 gold coins, silver coins and copper coins altogether in a bag. Daniel draws a coin
from the bag at random, records the result and then puts it back into the bag. He repeats the
process 150 times and the results are shown in the table below.
Gold coin
Silver coin Copper coin
Result
60
72
18
Frequency
(a) When a coin is drawn from the bag at random, find the experimental probability that the
coin drawn is not a copper coin.
(b) Daniel guesses that there are about 190 gold coins in the bag. Is his guess reasonable?
Explain your answer.
115
11. Jessie throws a dice 800 times and the results are as follows.
1
2
3
4
5
6
Number obtained
185
88
172
94
129
132
Frequency
(a) Jessie claims that the dice is fair. Is her claim reasonable? Explain your answer.
(b) Suppose Nick throws the dice 96 more times. Estimate the number of times that he obtains
an even number, correct to the nearest integer.
116
Answer
Consolidation Exercise 12C
1. (a)
2.
5
8
(b)
3
8
(b)
13
16
11
25
9
80
18
4. (a)
125
2
5. (a)
5
3. (a)
6. (a) 7
7. (a) (i) 24
33
(b)
50
2
8. (a)
5
9. 699
22
10. (a)
25
11. (a) no
(b)
4
5
27
50
2
(b)
5
(ii) 112
(b)
(b)
1
3
(b) yes
(b) 38
117
F3B: Chapter 12D
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 13
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 14
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 15
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
12D Level 1
Maths Corner Exercise
12D Level 2
Maths Corner Exercise
12D Level 3
Maths Corner Exercise
12D Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
Complete and Checked
Problems encountered
Skipped
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
(Full Solution)
118
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3B
12.4A
Lesson Worksheet 12D
(Refer to §12.4)
Expected Number of Occurrences
If the probability of an event is p, then we expect after n trials, this event will occur
np times.
Example 1
A fair coin is tossed once.
(a) Find the probability that a head is obtained.
(b) If the fair coin is tossed 500 times,
estimate the number of times of getting a
head.
1
Sol (a) P(a head) =
2
(b) Estimated number of times
1
= 500 ×
2
= 250
Instant Drill 1
A fair dice is thrown once.
(a) Find the probability of getting a ‘2’.
(b) If the fair dice is thrown 60 times, estimate
the number of times of getting a ‘2’.
Sol (a) P(getting a ‘2’) =
(b) Estimated number of times
=(
)
=
2.
1.
)×(
The probability that a candidate passes a
public examination is 0.71. This year,
30 000 candidates sit in the examination.
Estimate the number of candidates who
(a) pass the examination,
(b) fail the examination.
The figure shows a circular wheel which is
divided into 5 equal sectors. If it is turned
200 times, find the expected number of
times that the pointer stops at the shaded
sectors.
○→ Ex 12D 4–6
○→ Ex 12D 1–3
119
12.4BConcept of Expected Values
Consider an activity with n possible outcomes, and the values obtained from the
possible outcomes are x1, x2, …, xn respectively. If the probabilities of the
occurrences of these possible outcomes are p1, p2, …, pn respectively, then
expected value for the activity = x1p1 + x2p2 + … + xnpn
Example 2
A purse contains ten $1 coins, six $2 coins and
four $5 coins. A coin is randomly drawn from
the purse.
(a) Complete the following table.
Coin
Probability
Instant Drill 2
3 cards are marked with ‘2’, 5 cards are
marked with ‘3’ and 7 cards are marked with
‘6’. A card is drawn from them randomly.
(a) Complete the following table.
Number on the card Probability
$1
2
$2
3
$5
6
(b) Find the expected value of the coin drawn.
Sol (a) Total number of coins in the purse
= 10 + 6 + 4
= 20
All the possible outcomes and the
corresponding probabilities are as
follows:
Coin
Probability
$1
$2
(b) Find the expected value of the number on
the card drawn.
Sol (a) Total number of cards
=
All the possible outcomes and the
corresponding probabilities are as
follows:
Number on the card Probability
10 1
=
20 2
2
6
3
=
20 10
3
4 1
=
20 5
(b) Expected value of the coin drawn
3
1
 1
= $1 × + 2 × + 5 × 
10
5
 2
= $2.1
$5
6
(b) Expected value of the number on the
card drawn
=
120
3.
There are 4 pencils A, B, C and D in a
case. Their prices are $10, $5, $3 and $1
respectively. Tommy chooses a pencil
randomly from the case. Find the expected
price of the pencil chosen.
4.
List all the possible outcomes and find their
corresponding theoretical probabilities
first.
Box M contains 2 black balls and 1 white
ball. Box N contains 2 white balls. Wilson
randomly draws a ball from each box. If
the two balls drawn are of the same colour,
he will get $20; otherwise he will lose $5.
Find the expected amount that Wilson can
You may use a tree diagram or a
obtain.
table to list all the possible
outcomes.
○→ Ex 12D 7–9
 Level Up Question
5. In the lucky draw of a bookshop, the probabilities of winning a $500 coupon, a $100 coupon
and a $10 coupon are 0.04, 0.1 and 0.5 respectively. Anna takes part in the lucky draw
7 times. Find the expected total value of coupons won by Anna.
121
New Century Mathematics (2nd Edition) 3B
12
Introduction to Probability
Consolidation Exercise
12D

Level 1
1. A fair dice is thrown 120 times. Estimate the number of occurrences of each of the following
events.
(a) The number 5 is obtained.
(b) A number less than 4 is obtained.
2. In a school, students are assigned randomly to the Red, Yellow, Green, Blue and Purple
Houses. If there are 780 students in the school, estimate the number of students assigned to Blue
House or Purple House.
3. On a farm, the probability that an egg contains double yolks is 0.002 5. If the farm produces
1 600 eggs today, estimate the number of double-yolk eggs produced.
4. In a city, the probability that a newborn baby being a boy is 0.48. If there are 75 000 newborn
babies this year, estimate the number of newborn baby girls.
5. Suppose the probability that a kettle produced by a factory being defective is 0.045. If it is
expected that 144 kettles produced today are defective, estimate the total number of kettles
produced today.
6. In a lucky draw, the probabilities that Alex wins a coupon of value $1 000, $200 and $40 are
0.03, 0.22 and 0.75 respectively. Find the expected value of the coupon that he wins.
7. Peter draws a card at random from a pack of 52 playing cards. If he can get $4, $2, $5 and $1
for a club, a diamond, a heart and a spade drawn respectively, find the expected amount he can
get.
Level 2
8. A bag contains one red ball, three green balls and six blue balls. Hilary draws a ball at random
from the bag. If she can get $4, $1 and nothing for a red ball, a green ball and a blue ball drawn
respectively, find the expected amount she can get.
9. In a game, Zoe throws a fair dice once. Four points will be awarded if a factor of 6 is obtained,
while five points will be deducted for other results. Find the expected value of the points
obtained by throwing the dice once.
10. A bag contains a $1 coin, a $5 coin and a $10 coin. Two coins are drawn randomly from the bag
at the same time.
(a) Use a tree diagram or the method of tabulation to list all the possible outcomes.
(b) Hence, find the expected total value of the coins drawn.
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11. In a game, a participant draws a card at random from a pack of 52 playing cards. A prize will be
awarded according to the following table.
a face card
an ‘A’
any other card
Card drawn
$7.8
$32.5
$1.3
Prize
If Tom plays the game 15 times, find the expected value of the total amount he will get.
(Note: A face card is a ‘J’, a ‘Q’ or a ‘K’.)
12. The target in the figure is formed by three concentric circles. The radius of the smallest circle is
10 cm, and the widths of the two rings B and C are 10 cm and 20 cm respectively.
C
B
A
John pays $16 for throwing a dart once. A prize is awarded according to the following table.
A
B
C
Region
$100
$20
$4
Prize
If John throws a dart randomly and it hits the target, is the game favourable to John? Explain
your answer.
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Answer
Consolidation Exercise 12D
1. (a) 20
2. 312
3. 4
4. 39 000
5. 3 200
6. $104
7. $3
8. $0.7
9. 1
32
10. (b) $
3
11. $78
12. no
(b) 60
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