# New Trend Mathematics (2nd Edition) Supplementary Exercise

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```Chapter 1
Directed Numbers
Chapter 8
Inequalities
Chapter 2
Basic Algebra
Chapter 9
Algebraic Fractions and Formulae
Chapter 3
Basic Geometry
Chapter 10
Areas and Volumes
Chapter 4
Linear Equations in One Unknown
Chapter 11
Introduction to Trigonometric Ratios
Chapter 5
Percentages
Chapter 12
Polygons
Chapter 6
Statistics in Daily Life
Chapter 13
Measures of Central Tendency
Chapter 7
Algebraic Expressions and Polynomials
Chapter 1
Laws of Integral Indices
Chapter 8
Symmetry and Transformation
Chapter 2
Chapter 9
Introduction to Coordinates
Chapter 3
Study of 3-dimensional Figures
Chapter 10
Statistical Graphs
Chapter 4
Mensuration
Chapter 11
Linear Equations in Two Unknowns
Chapter 5
Theorems Related to Triangles
Chapter 12
Ratio and Rate
Chapter 6
Introduction to Probability
Chapter 13
Angles in Rectilinear Figures
Chapter 1
Estimation and Approximation
Chapter 7
Chapter 2
Measurement and Errors
Chapter 8
Chapter 3
Identities and Factorization
Chapter 9
Coordinate Geometry
Chapter 4
Simultaneous Equations
Chapter 10
Applications of Trigonometry
Chapter 5
Congruent and Similar Triangles
Chapter 11
Use and Misuse of Statistics
Chapter 6
Square Roots and Pythagoras’ Theorem
Chapter 7
1. Without using a calculator, find the square roots of the following numbers.
(a) 289
(b) 900
(c) 0.000 4
(d)
64
529
2. Is each of the following a surd?
Yes
No
□
□
□
□
(a)
32
(b)
484
(c)
3 .6
□
□
□
(d)
18
50
□
3. Without using a calculator, find two consecutive numbers between which each of the following lies.
(a)
30


 30 
 30 
6.1
i.e.
 30 

30 lies between
and
.
180
(b)
 180 


 180 
i.e.
 180 

180 lies between
and
.
4. Find the values of the following numbers. (Give your answers correct to 3 significant figures if
necessary.)
80
(a)


3.62
(c)
4
15
(d)


10 2  32
(e)
1 024
(b)

6.2
152  4 2
(f)

1. Find the value of the unknown in each of the following figures. (Give your answers correct to
3 significant figures if necessary.)
Q
(a)
P
24
18
p
R
p2  (
)2  (
)2
(Pyth. theorem)


 p

A
(b)
15
15
B
a
C
6.3
C
(c)
18
5
c
A
B
P
(d)
17
r
R
32
Q
6.4
2. In the figure, S is the mid-point of QR and PS  QR.
Find the value of p.
P
15
p
Q
S
18
R
3. In the figure, find the area of rectangle ABCD.
A
B
17 cm
15 cm
D
C
6.5
4. In the figure, find the area of pentagon ABCDE. (Give your answer correct to 3 significant figures.)
A
7 cm
E
B
4 cm
C
12 cm
D
6.6
5. The figure shows rhombus ABCD. The lengths of the diagonals AC and BD are 10 cm and 8 cm
respectively. The diagonals are perpendicular to each other and bisect each other at F. Find the
perimeter of rhombus ABCD. (Give your answer correct to 3 significant figures.)
A
B
F
D
C
6.7
6. In the figure, D is a point on AC such that AD  DC .
A
D
6m
19 m
C
B
(a) Find the length of AB. (Give your answer correct to 3 significant figures.)
(b) Find the length of BC. (Give your answer correct to 3 significant figures.)
6.8
1. Determine whether each of the following triangles is a right-angled triangle. If yes, indicate the right
angle.
A
(a)
20
39
B
36
C
AB 2  BC 2 


AC 2 

 AB 2  BC 2
□
AC 2
 ABC
(b)
P
30
R
16
34
Q
PR 2  RQ 2 


PQ 2 

 PR 2  RQ 2
□
PQ 2
 PQR
6.9
2. (a) Prove that ABC in the figure is a right-angled
triangle.
B
10.5
A
14
17.5
C
(b) Find the area of ABC.
6.10
3. (a) Find the area of PQR in the figure.
P
Q
24
h
25
7
R
(b) Express the area of PQR in terms of h. Hence find the value of h.
6.11
C
4. In the figure, D is a point on AB.
(a) Prove that ADC  90.
24
A
(b) Find the perimeter of ABC.
6.12
32
D
26
10
B
1. At 12:00 noon, aeroplane A is 800 km due west of the airport, while aeroplane B is 1 000 km due north
of the airport. If aeroplane A and aeroplane B are on the same horizontal line, what is the distance
between these two aeroplanes? (Give your answer correct to the nearest km.)
N
B
1 000 km
A
800 km
Airport
Let x km be the distance between these two aeroplanes.
6.13
2. Given that the perimeter of a square wooden board is
60 cm, find the length of the diagonal of the board. (Give
Diagonal
Perimeter  60 cm
3. Find the perimeter of OAB in the figure.
y
A (0, 20)
B ( 21, 0)
6.14
O
x
4. A car and a bus travel at the speeds of 80 km/h due east and 70 km/h due south respectively from
point A at the same time. What is the distance between the car and the bus after two hours? (Give your
answer correct to the nearest km.)
A
80 km/h
70 km/h
N
6.15
5. A pole of 2.5 m long leans against a vertical wall and its upper end is 1.8 m above the ground. If the
upper end of the pole slides down by 0.2 m, how far will the lower end of the pole move from its
0.2 m
2.5 m
1.8 m
6.16
Simplify the following. (1  4)
1. (a)
(c)
2. (a)
35  32  (
)(
)
 32  (
)(
)
(b)
)(
)
 22  (
)(
)




77  72  (
)
 72  (
)
(d)
58 





44  2(
)
(
 2(
)

)
(b)
135 

63  3(
)
(
 3(
)

)


(c)
26  2 2  (
(d)



252 



6.17
3. (a)
7

169
(
(
)
)
(b)


4. (a)
14

81
(d)
)
)
(
(
)
)
32

225






2  32 
(b)

(
(
)
)




6  12

50

216

12


(c)
(
(


(c)
27

4
(
(
(d)
)
)
42

8  28





6.18

(
(
)
)
Rationalize the denominators of the following. (1  2)
1. (a)
1
1 (


6
6 (
)
)
(b)

(c)
2
2 (


5
5 (
)
)

9

11
(d)

15

10


2. (a)
(c)
5

30
5 (

30 (
)
)
(b)
2 7 2 7 (


21
21 (






81
81

5
5

(
(
(d)
)
)
44

3

)
)

(

(
) (

) (
)
)


6.19
Simplify the following. (3  4)
3. (a) 5 5  2 5
(
14  3 14
(b)

) 5


(c) 16 13  9 13
(d) 2 6 




8  18
4. (a)

54  5 24
(b)

2 6
5






(c) 4 75  12
(d) 5 28  3 7








6.20
5
1. Determine whether each of the following numbers is a perfect square. If yes, find the square roots of
the number.
(a) 24
(b) 36
(c) 110
(d) 450
(e) 784
(f) 1 089
2. Determine whether each of the following is a surd.
(a)
45
(b) 120
(c)
98
200
(d)
55
224
(e)
1.69
(f)
4.8
3. Without using a calculator, find the square roots of the following numbers.
(a) 81
(b) 196
(c) 484
(d)
1
144
169
4
(f)
121
100
(e)
4. Find the values of the following numbers. (Give your answers correct to 3 significant figures.)
(a)
21
(b) 120
(c)
950
(d)
0.7
(e)
55
6
(f)
227
5
6.21
5. Without using a calculator, find two consecutive numbers between which each of the following lies.
35
(a)
22
(b)
(c)
60
(d) 140
(e)
380
(f)
480
6. Without using a calculator, find the square roots of the following numbers.
(a)
361
400
(b)
(c)
625
324
(d) 2.89
(e) 4.84
256
729
(f) 0.002 5
7. Find the values of the following expressions. (Give your answers correct to 3 significant figures.)
(a)
9 2  132
(b)
212  17 2
(c)
29 2  10 2
(d)
32 2  182
(f)
352  24 2  13
(e) 8  40 2  252
8. Find the values of the unknowns in the following. (Give your answers correct to 3 significant figures.)
B
(a)
P
(b)
8
a
9
A
b
R
11
A
B
6.22
C
6
3
7
Q
C
(c)
c
9. Find the values of the unknowns in the following. (Give your answers correct to 3 significant figures.)
A
(a)
(b)
Q
10
5
p
B
(c)
Q
P
r
11
14
20
P
14
q
R
C
R
A
10. In the figure, BDC is a straight line. Find the value of c.
16
c
B
D
C
5
A
11. In the figure, find the length of each side of square ABCD.
D
10 cm
B
12. In the figure, the area of ABC is 364 cm 2. Find the length
C
A
26 cm
B
C
6.23
13. In the figure, AB  34 cm and AC : BC  8 : 15. Find the length of BC.
B
34 cm
C
A
14. In the figure, find the perimeter of trapezium ABCD.
A
12 cm
D
12 cm
B
E
28 cm
C
15. The figure shows parallelogram ABCD. E is a point on AB such that AE : EB  1 : 4 . If CD  20 cm
and AD  12 cm , find the area of parallelogram ABCD. (Give your answer correct to 3 significant
figures.)
20 cm
D
C
12 cm
A
E
B
16. In the figure, PQ  18 cm, QR  12 cm and PS  10 cm.
P
10 cm
S
18 cm
Q
12 cm
R
(a) Find the length of QS.
(b) Find the length of RS.
6.24
17. In the figure, find the perimeter of quadrilateral ABCD. (Give your answer correct to 3 significant figures.)
B
7 cm
A
5 cm
C
13 cm
D
18. In the figure, WX  47, XY  40 and YZ  32.
Z
32
Y
q
p
W
40
47
X
(a) Find the value of p.
(b) Find the value of q.
(c) Find the area of quadrilateral WXYZ.
19. In the figure, find the perimeter of pentagon PQRST. (Give your answer correct to 3 significant
figures.)
T
P
S
25 cm
30 cm
Q
R
7 cm
6.25
20. Determine whether each of the following triangles is a right-angled triangle. If yes, indicate the right
angle.
A
(a)
(b)
A
15
B
7.5
C
40
24
A
(c)
9
18
B
19.5
C
C
32
18
B
21. In the figure, the areas of squares ABCD, DPQR and ARST are 65 cm2 , 90 cm 2 and 155 cm 2
respectively. Determine whether ADR is a right angle.
Q
Area  90 cm
2
R
S
P
Area  155 cm2
D
C
Area  65 cm2
A
T
B
22. (a) Prove that PQR in the figure is a right angle.
P
(b) Find the area of PQR.
34
3
Q
6.26
5
R
23. In the figure, D is a point on BC and BD  DC.
A
(a) Prove that ABC is a right-angled triangle.
39
3 significant figures.)
15
h
B
24. (a) Prove that ABC in the figure is a right-angled
triangle.
A
1.5
2
B
C
2.5
25. In the figure, ADB  DCB.
(b) Find the perimeter of quadrilateral ABCD. (Give
C
P
(b) If 3PQ  4 PR , prove that ABC ~ PQR.
(a) Prove that ABD ~ DBC .
D
36
A
5
Q
R
D
7
74
B
C
26. In the figure, AC and BD intersect at E.
D
(a) Prove that AC  BD.
30
(b) Find the lengths of BC and DC.
(c) Is BCD a right-angled triangle? Explain briefly.
A
9
E
15
16
12
B
C
6.27
27. The lengths of the hour hand and minute hand of a watch are 8 mm and 15 mm respectively. What is
the distance between the tips of these two hands when it is 9 o’clock?
12
15 mm
9
8 mm
3
6
28. A ladder of 5 m long leans against a vertical wall. If the upper end of the ladder is 3.5 m above the
ground, what is the distance between the lower end of the ladder and the wall? (Give your answer
correct to 3 significant figures.)
5m
3.5 m
29. In each of the following figures, find the coordinates of B and the length of OA.
y
(a)
y
(b)
A (12, 16)
x
O
B
O
B
x
6.28
A (12,  9)
30. One end of a rope of 6.4 m long is fixed at point D on the ground, while the other end is at point B on
a lamp post. The distances from point B to top A and to bottom C of the lamp post are the same.
If DC  3.6 m , find the height of the lamp post. (Give your answer correct to 1 decimal place.)
A
B
6.4 m
D 3.6 m C
31. In the figure, two vertical poles are 5 m apart and the height of the shorter pole is 3 m. If the distance
between the tops of the poles is 7 m, what is the height of the longer pole? (Give your answer correct
to 3 significant figures.)
7m
3m
5m
32. In the figure, find the perimeter of ABC.
y
A (9, 6)
B (3, 6)
x
O
C (3, 3)
6.29
33. In the figure, the lighthouse is 20 m high. The top A of the lighthouse is connected to points C and D
on the ground by two wires, where CBD is a straight line. If AC  AD  30 m, find the distance
between C and D. (Give your answer correct to 3 significant figures.)
A
30 m
C
30 m
20 m
D
B
34. The figure shows a rectangular top ABCD of a desk. If the area of ABCD is 0.98 m 2 and AB  2 BC,
find the length of diagonal AC. (Give your answer correct to 3 significant figures.)
A
B
D
C
35. The length and width of a field are 130 m and 74 m respectively.
A
74 m
D
Field
B
C
130 m
The following are two routes to walk from point A to point C.
Route 1: Walking along the sides of the field (i.e. A  B  C).
Route 2: Walking diagonally across the field (i.e. A  C).
What is the difference in length between the above two routes? (Give your answer correct to
3 significant figures.)
36. Given that the area of a square wooden board is 125 cm 2, find the length of the diagonal of the board.
Diagonal
Area  125 cm2
6.30
37. Oscar walked due west from point A at a speed of 60 m / min. After 3 minutes, he reached point B. A tree
is 425 m and 385 m away from points A and B respectively.
B
A
N
425 m
385 m
(a) Prove that point B is due north of the tree.
(b) If Oscar continued to walk due west from point B at the same speed for 5 minutes, he reached
point C finally. Find the distance between point C and the tree. (Give your answer correct to
3 significant figures.)
38. In the figure, find the length of PR. (Give your answer correct to 3 significant figures.)
y
Q (4, 4)
P (2, 4)
x
O
S (4, 2)
R (2, 2)
39. Find the area of the polygon in the figure. (Give your answer correct to 3 significant figures.)
4 cm
3 cm
1 cm
2 cm
6.31
40. Simplify the following.
(a)
27
(b)
54
(c)
173
(d)
105
(e)
114
(f)
69
41. Simplify the following.
(a)
28
(b)
99
(c)
104
(d)
140
(e)
264
(f)
756
42. Simplify the following.
(a)
3
64
(b)
27
100
(c)
80
49
(d)
32
144
(e)
216
50
(f)
135
784
43. Simplify the following.
(a)
6  54
(b)
10  45
(c)
60  42
(d)
125
5
(e)
180
15
(f)
243
3 3
6.32
44. Simplify the following.
(a)
15  20  75
(b)
40  125
18
(c)
21  70
54
(d)
315
12  105
(b)
6
13
45. Rationalize the denominators of the following.
(a)
3
10
(c)
27
6
(d)
70
21
(e)
18
42
(f)
36
8
46. Rationalize the denominators of the following.
(a)
15
2
(b)
6
120
(c)
147
45
(d)
2 108
5
(f)
4 15
18
(e)
3 3
54
6.33
47. Rationalize the denominators of the following.
(a)
64
7
(b)
32
15
(c)
11
48
(d)
25
128
(e)
135
72
(f)
168
30
48. Simplify the following.
(a) 4 3  13 3
(b) 8 7  2 7
3 10
2
(d) 12 5  5
(c)
10 
(e) 9 11  6 11
(f)
14 15 2 15

3
3
49. Simplify the following.
(a) 3 12  2 27
(b) 4 15  60
(c) 5 28  9 63
(d)
(e) 8 45  3 20
(f) 2 150  3 54
6.34
128  50
Worksheet 6A (page 6.1)
1. (a) 17 or 17
(c) 0.02 or 0.02
Worksheet 6F
(b) 30 or 30
8
8
(d)
or 
23
23
(page 6.19)
1. (a)
6
6
(b)
2 5
5
(c)
9 11
11
(d)
3 10
2
6
6
(b)
2 3
3
(c)
9 5
5
(d)
2 33
3
8 6
5
2. (a) Yes
(c) Yes
(b) No
(d) No
2. (a)
3. (a) Between 5 and 6
(b) Between 13 and 14
3. (a) 7 5
(b) 4 14
(c) 7 13
(d)
4. (a) 8.94
(c) 1.90
(e) 10.4
(b) 32
(d) 0.516
(f) 14.5
4. (a) 5 2
(b) 13 6
(c) 18 3
(d) 7 7
Worksheet 6B
Build-up Exercise 6A (page 6.21)
(page 6.3)
1. (a) 30
(c) 17.3
(b) 21.2
(d) 27.1
2. 12
3. 120 cm 2
4. 82.1 cm 2
5. 25.6 cm
6. (a) 18.0 m
1. (a) No
(c) No
(e) Yes, 28 or  28
(b) Yes, 6 or  6
(d) No
(f) Yes, 33 or  33
2. (a) Yes
(c) No
(e) No
(b) Yes
(d) Yes
(f) Yes
3. (a) 9 or  9
(b) 14 or  14
1
1
(d)
or 
12
12
11
11
(f)
or 
10
10
(c) 22 or  22
(b) 21.7 m
(e)
Worksheet 6C
(page 6.9)
(b) Yes,  PRQ
1. (a) No
2. (b) 73.5
(b) Area 
3. (a) 84
25h
; h  6.72
2
4. (b) 108
13
13
or 
2
2
4. (a) 4.58
(c) 30.8
(e) 3.03
(b) 11.0
(d) 0.837
(f) 6.74
5. (a) Between 4 and 5
(c) Between 7 and 8
(e) Between 19 and 20
(b) Between 5 and 6
(d) Between 11 and 12
(f) Between 21 and 22
19
19
or 
20
20
25
25
(c)
or 
18
18
(e) 2.2 or 2.2
6. (a)
Worksheet 6D
(page 6.13)
1. 1 281 km
2. 21.2 cm
7. (a) 15.8
(c) 27.2
(e) 39.2
3. 70 units
4. 213 km
(b)
16
16
or 
27
27
(d) 1.7 or 1.7
(f) 0.05 or 0.05
(b) 27.0
(d) 26.5
(f) 12.5
5. 0.186 m
Build-up Exercise 6B
Worksheet 6E
(page 6.17)
1. (a) 9 3
(b) 8
(c) 343 7
(d) 625
2. (a) 2 11
(b) 3 7
(c) 3 15
(d) 6 7
7
3. (a)
13
3 3
(b)
2
(c)
4. (a) 8
(b) 3 2
(c)
(page 6.22)
8. (a) 11.4
(c) 12.5
(b) 8.54
9. (a) 9.80
(c) 14.3
(b) 9.80
10. 12.5
14
9
6
5
4 2
(d)
15
(d)
3
4
11. 7.07 cm
12. 38.2 cm
13. 30 cm
6.35
35
14. 72 cm
44. (a) 150
15. 226 cm 2
16. (a) 15.0 cm
(b) 19.2 cm
(d)
(b)
50
3
(c)
7 5
3
1
2
17. 35.0 cm
18. (a) 51.2
(c) 1 120
(b) 20.4
Build-up Exercise 6F
19. 87.5 cm
3 10
10
(b)
6 13
13
(c)
10 21
3
(e)
3 42
7
(f) 9 2
30
2
(b)
5
10
(c)
7 15
15
12 15
5
(e)
2
2
(f)
2 30
3
47. (a)
8 7
7
(b)
4 30
15
(c)
33
12
(d)
5 2
16
(e)
30
4
(f)
2 35
5
5 10
2
45. (a)
Build-up Exercise 6C
(page 6.26)
20. (a) Yes,  ABC
(c) Yes,  BAC
(d)
(b) No
46. (a)
21. Yes
(d)
22. (b) 7.5
(page 6.33)
9 6
2
23. (b) 23.4
25. (b) 38.8
26. (b) BC  20, DC  34
(c) No
48. (a) 17 3
(b) 10 7
(c)
27. 17 mm
(d) 11 5
(e) 3 11
(f) 4 15
28. 3.57 m
49. (a) 12 3
(b) 6 15
(c) 37 7
(e) 18 5
(f)
Build-up Exercise 6D
(page 6.28)
29. (a) B (12, 0), OA  20 units
(b) B (0,  9), OA  15 units
(d) 3 2
30. 10.6 m
31. 7.90 m
32. 36 units
33. 44.7 m
34. 1.57 m
35. 54.4 m
36. 15.8 cm
37. (b) 488 m
38. 7.21 units
39. 6.41 cm 2
Build-up Exercise 6E
40. (a) 8 2
(d) 100 10
41. (a) 2 7
(d) 2 35
(page 6.32)
(b) 25
(c) 17 17
(e) 121
(f) 1 296 6
(b) 3 11
(c) 2 26
(e) 2 66
(f) 6 21
42. (a)
3
8
(b)
3 3
10
(c)
4 5
7
(d)
2
3
(e)
6 3
5
(f)
3 15
28
43. (a) 18
(d) 5
(b) 15 2
(c) 6 70
(e) 2 3
(f) 3
6.36