數學新里程(第二版) 附加練習 - 樣章 - ctep.com.hk€¦ · 6.1 2009 中大出版社...

2010

1 2 3 4 5 6

7 8 9 10 11 12 13

1 2 3 4 5 6 7

8 9 10 11 12 13

1 2 3 4 5 6

7 8 9 10 11

6.1

2009

1. Without using a calculator, find the square roots of the following numbers.

(a) 289 (b) 900

(c) 0.000 4 (d) 52964

2. Is each of the following a surd? Yes No

(a) 32

(b) 484

(c) 6.3

(d) 5018

3. Without using a calculator, find two consecutive numbers between which each of the following lies.

(a) 30

30

30

6.2

2009

i.e.

30

30

30 lies between and .

(b) 180

180

180

i.e.

180

180

180 lies between and .

4. () Find the values of the following numbers. (Give your answers correct to 3 significant figures if

necessary.)

(a)

80

(b)

024 1

(c)

62.3

(d)

154

(e)

310 22

(f)

415 22

6.3

2009

1. () Find the value of the unknown in each of the following figures. (Give your answers correct to

3 significant figures if necessary.) (a) Q P

R

p

24

18

) () ( 222

p () (Pyth. theorem)

p

(b) A

15

B Ca

15

6.4

2009

(c)

c

C

5

A

18

B

(d)

r17

32

P

Q

R

6.5

2009

2. S QR PS QR p In the figure, S is the mid-point of QR and PS QR.

Find the value of p.

3. ABCD In the figure, find the area of rectangle ABCD.

Q

P

R

S

p

15

18

D C

A B

17 cm15 cm

6.6

2009

4. ABCDE () In the figure, find the area of pentagon ABCDE. (Give your answer correct to 3 significant figures.)

C D

B

A

12 cm

4 cm

7 cm

E

6.7

2009

5. ABCD AC BD 10 cm 8 cm F ABCD ()

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.)

DB

C

F

A

6.8

2009

6. D AC DCAD In the figure, D is a point on AC such that DCAD .

C

A

B

19 m6 m

D

(a) AB () Find the length of AB. (Give your answer correct to 3 significant figures.)

(b) BC () Find the length of BC. (Give your answer correct to 3 significant figures.)

6.9

2009

1. Determine whether each of the following triangles is a right-angled triangle. If yes, indicate the right

angle. (a)

C

B

A

20

36

39

22

BCAB

2

AC

22 BCAB 2AC ABC

(b)

Q

34

RP 30

16

22

RQPR

2

PQ

22 RQPR 2PQ PQR

6.10

2009

2. (a) ABC Prove that ABC in the figure is a right-angled

triangle.

(b) ABC Find the area of ABC.

C

17.5

BA

14

10.5

6.11

2009

3. (a) PQR Find the area of PQR in the figure.

(b) h PQR h Express the area of PQR in terms of h. Hence find the value of h.

R

25

QP 24

7h

6.12

2009

4. D AB In the figure, D is a point on AB.

(a) 90ADC Prove that 90ADC .

(b) ABC Find the perimeter of ABC.

C

24

D BA 32 10

26

6.13

2009

1. A 800 km B 1 000 km A B ( km)

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.)

1 000 km

800 kmA

B/N

Airport

x km Let x km be the distance between these two aeroplanes.

6.14

2009

2. 60 cm()

Given that the perimeter of a square wooden board is 60 cm, find the length of the diagonal of the board. (Give your answer correct to 3 significant figures.)

3. OAB Find the perimeter of OAB in the figure.

y

x

A (0, 20)

OB (21, 0)

(Perimeter) 60 cm

Diagonal

6.15

2009

4. A 80 km/h 70 km/h ( km)

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.)

80 km/h

70 km/h

A

/ N

6.16

2009

5. 2.5 m 1.8 m 0.2 m()

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 original position? (Give your answer correct to 3 significant figures.)

2.5 m

0.2 m

1.8 m

6.17

2009

(1 4)

Simplify the following. (1 4)

1. (a)

) () (3

) () (33

2

25

(b)

) () (2

) () (22

2

26

(c)

) (7

) (77

2

27

(d)

58

2. (a)

2

) (244

) (

) (

(b)

3

) (363

) (

) (

(c)

135

(d)

252

6.18

2009

3. (a)

) () (

1697

(b)

) () (

) () (

427

(c)

8114

(d)

22532

4. (a)

322

(b)

) () (

12216

(c)

) () (

50

126

(d)

) () (

288

42

6.19

2009

(1 2)

Rationalize the denominators of the following. (1 2)

1. (a)

) () (

61

61

(b)

) () (

52

52

(c)

119

(d)

10

15

2. (a)

) () (

305

305

(b)

) () (

2172

2172

(c)

) () (

) () (

) () (

581

581

(d)

3

44

6.20

2009

(3 4)

Simplify the following. (3 4)

3. (a)

5) (5255

(b)

14314

(c)

1391316

(d)

5

6262

4. (a)

188

(b)

5

24554

(c)

12754

(d)

73285

6.21

2009

1.

(a) 24 (b) 36 (c) 110 (d) 450 (e) 784 (f) 1 089

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.

(a) 45

(b) 120

(c) 20098

(d) 22455

(e) 69.1

(f) 8.4

2. Determine whether each of the following is asurd.

(a) 45

(b) 120

(c) 20098

(d) 22455

(e) 69.1

(f) 8.4

3. (a) 81 (b) 196 (c) 484

(d) 144

1

(e) 4

169

(f) 100121


(a) 81 (b) 196 (c) 484

(d) 144

1

(e) 4

169

(f) 100121

6.22

2009

4. ()

(a) 21

(b) 120

(c) 950

(d) 7.0

(e) 6

55

(f) 5

227

4. Find the values of the following numbers. (Give your answers correct to 3 significant figures.)

(a) 21

(b) 120

(c) 950

(d) 7.0

(e) 6

55

(f) 5

227

5.

(a) 22

(b) 35

(c) 60

(d) 140

(e) 380

(f) 480

5. Without using a calculator, find two consecutive numbers between which each of the following lies.

(a) 22

(b) 35

(c) 60

(d) 140

(e) 380

(f) 480

6.

(a) 400361

(b) 729256

(c) 324625

(d) 2.89 (e) 4.84 (f) 0.002 5


(a) 400361

(b) 729256

(c) 324625

(d) 2.89 (e) 4.84 (f) 0.002 5

6.23

2009

7. ()

(a) 22 139

(b) 22 1721

(c) 22 1029

(d) 22 1832

(e) 22 25408

(f) 132435 22

7. Find the values of the following expressions. (Give your answers correct to 3 significant figures.)

(a) 22 139

(b) 22 1721

(c) 22 1029

(d) 22 1832

(e) 22 25408

(f) 132435 22

8. ()

8. Find the values of the unknowns in the following. (Give your answers correct to 3 significant figures.)

(a) B

7

9C A

a

(b)

R

38

Q

P

b

(c)

B

6

c

CA

11

9. ()

9. Find the values of the unknowns in the following. (Give your answers correct to 3 significant figures.)

(a) 10

C

B

A

14

p

(b) Q

511

RP q

(c)

R

1420

PQ r

6.24

2009

10. BDC c ()

10. In the figure, BDC is a straight line. Find the value of c. (Give your answer correct to3 significant figures.)

16

B

c

D C

A

5

11. ABCD ()

11. In the figure, find the length of each side of square ABCD. (Give your answer correct to 3 significant figures.)

DA

CB

10 cm

12. ABC 2cm 364 AB ()

12. In the figure, the area of ABC is 2cm 364 . Find the length of AB. (Give your answer correct to 3 significant figures.)

C

A

B

26 cm

6.25

2009

13. cm 34 AB 15:8: BCAC BC

13. In the figure, cm 34 AB and 15:8: BCAC . Find the length of BC.

C

34 cm

B

A

14. ABCD 14. In the figure, find the perimeter of trapezium ABCD.

C28 cmEB

DA 12 cm

12 cm

15. ABCD E AB 4:1: EBAE cm 20 CD

cm 12 AD ABCD ()

15. The figure shows parallelogram ABCD. E is a point on AB such that 4:1: EBAE . If

cm 20 CD and cm 12 AD , find the area of parallelogram ABCD. (Give your answer correct to 3 significant figures.)

D C

A E B

12 cm

20 cm

6.26

2009

16. cm 18 PQ cm 12 QR cm 10 PS

16. In the figure, cm 18 PQ , cm 12 QR and cm 10 PS .

P

18 cm

S

Q R

10 cm

12 cm

(a) QS

(b) RS

()

(a) Find the length of QS. (b) Find the length of RS. (Give your answers correct to 3 significant

figures.)

17. ABCD ()

17. In the figure, find the perimeter of quadrilateral ABCD. (Give your answer correctto 3 significant figures.)

C

D

BA

5 cm

7 cm

13 cm

18. 47WX 40XY 32YZ 18. In the figure, 47WX , 40XY and 32YZ .

X

YZ

W

q

p

47

32

40

6.27

2009

(a) p

(b) q

(c) WXYZ

()

(a) Find the value of p. (b) Find the value of q. (c) Find the area of quadrilateral WXYZ. (Give your answers correct to 3 significant

figures.)

19. PQRST ()

19. In the figure, find the perimeter of pentagon PQRST. (Give your answer correct to 3 significant figures.)

P S

RQ

25 cm

7 cm

30 cm

T

20.

20. Determine whether each of the following triangles is a right-angled triangle. If yes, indicate the right angle.

(a)

B

24

32

40

C

A (b) 15

189

B

C

A (c) 7.5

18

19.5

A

C

B

6.28

2009

21. ABCDDPQR ARST 2cm 65 2cm 90 2cm 155 ADR

21. In the figure, the areas of squares ABCD, DPQR and ARST are 2cm 65 , 2cm 90 and

2cm 155 respectively. Determine whether ADRis a right angle.

(Area) 90 cm2

SR

TA

B

C

P

D

Q

(Area) 65 cm2

(Area) 155 cm2

22. (a) PQR

(b) PQR

22. (a) Prove that PQR in the figure is a right angle.

(b) Find the area of PQR.

3

Q R

P

5

34

23. D BC DCBD 23. In the figure, D is a point on BC and DCBD .

B36D C

A

15 h39

(a) ABC

(b) h ()

(a) Prove that ABC is a right-angled triangle. (b) Find the value of h. (Give your answer

correct to 3 significant figures.)

6.29

2009

24. (a) ABC (b) PRPQ 43 PQRABC ~

24. (a) Prove that ABC in the figure is a right-angled triangle.

(b) If PRPQ 43 , prove that PQRABC ~ .

C

2

B Q R

P

A

2.5

1.5

25. DCBADB 25. In the figure, DCBADB .

D

5

A

B

C

7

74

(a) DBCABD ~

(b) ABCD ()

(a) Prove that DBCABD ~ . (b) Find the perimeter of quadrilateral ABCD.

(Give your answer correct to 3 significant figures.)

26. AC BD E 26. In the figure, AC and BD intersect at E.

B C

A

E

D

15

30

16

9

12

6.30

2009

(a) BDAC

(b) BC DC

(c) BCD

(a) Prove that BDAC . (b) Find the lengths of BC and DC. (c) Is BCD a right-angled triangle? Explain

briefly.

27. 8 mm 15 mm

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 oclock?

12

6

8 mm

15 mm9 3

28. 5 m 3.5 m ()

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.)

5 m3.5 m

6.31

2009

29. B OA 29. In each of the following figures, find the coordinates of B and the length of OA.

(a)

x

y

A (12, 16)

O B

(b)

x

y

A (12, 9)

O

B

30. 6.4 m D B A C B

m 6.3 DC ()

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 m 6.3 DC , find the height of the lamp post. (Give your answer correct to 1 decimal place.)

3.6 m

6.4 m

D C

B

A

31. 5 m 3 m 7 m ()

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.)

7 m

3 m

5 m

6.32

2009

32. ABC 32. In the figure, find the perimeter of ABC.

x

y

A (9, 6)

O

B (3, 6)

C (3, 3)

33. 20 m A C D CBD m 30 ADAC C D ()

33. In the figure, the lighthouse is 20 m high. Thetop A of the lighthouse is connected to points Cand D on the ground by two wires, where CBDis a straight line. If AC AD 30 m, find the distance between C and D. (Give your answer correct to 3 significant figures.)

30 m30 m 20 m

A

C B D

34. ABCD ABCD 2m 98.0 BCAB 2 AC ()

34. The figure shows a rectangular top ABCD of a desk. If the area of ABCD is 2m 98.0 and

BCAB 2 , find the length of diagonal AC. (Give your answer correct to 3 significant figures.)

CD

BA

6.33

2009

35. 130 m 74 m

35. The length and width of a field are 130 m and 74 m respectively.

74 m

130 m

A D

CB

Field

A C ( A B C) ( A C) (

)

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. 2cm 125 ()

36. Given that the area of a square wooden board is 2cm 125 , find the length of the diagonal of

the board. (Give your answer correct to3 significant figures.)

(Area) 125 cm2

Diagonal

37. A 60 m / 3 B A B 425 m 385 m

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.

/ N

425 m385 m

B A

6.34

2009

(a) B

(b) B 5 C C ()

(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. PR ()

38. In the figure, find the length of PR. (Give your answer correct to 3 significant figures.)

P (2, 4) Q (4, 4)

S (4, 2) R (2, 2)

Ox

y

39. ()

39. Find the area of the polygon in the figure. (Give your answer correct to 3 significant figures.)

1 cm2 cm

3 cm

4 cm

40.

(a) 72

(b) 45

(c) 317

(d) 510

(e) 411

(f) 96

40. Simplify the following.

(a) 72

(b) 45

(c) 317

(d) 510

(e) 411

(f) 96

6.35

2009

41.

(a) 28

(b) 99

(c) 104

(d) 140

(e) 264

(f) 756


(a) 28

(b) 99

(c) 104

(d) 140

(e) 264

(f) 756

42.

(a) 643

(b) 10027

(c) 4980

(d) 14432

(e) 50216

(f) 784135


(a) 643

(b) 10027

(c) 4980

(d) 14432

(e) 50216

(f) 784135

43.

(a) 546

(b) 4510

(c) 4260

(d) 5

125

(e) 15

180

(f) 33

243


(a) 546

(b) 4510

(c) 4260

(d) 5

125

(e) 15

180

(f) 33

243

6.36

2009

44.

(a) 752015

(b) 18

12540

(c) 54

7021

(d) 10512

315


(a) 752015

(b) 18

12540

(c) 54

7021

(d) 10512

315

45.

(a) 103

(b) 136

(c) 6

27

(d) 21

70

(e) 42

18

(f) 8

36

45. Rationalize the denominators of the following.

(a) 103

(b) 136

(c) 6

27

(d) 21

70

(e) 42

18

(f) 8

36

6.37

2009

46.

(a) 2

15

(b) 120

6

(c) 45

147

(d) 5

1082

(e) 5433

(f) 18154


(a) 2

15

(b) 120

6

(c) 45

147

(d) 5

1082

(e) 5433

(f) 18154

47.

(a) 764

(b) 1532

(c) 4811

(d) 12825

(e) 72

135

(f) 30

168


(a) 764

(b) 1532

(c) 4811

(d) 12825

(e) 72

135

(f) 30

168

6.38

2009

48.

(a) 31334

(b) 7278

(c) 210310

(d) 5512

(e) 116119

(f) 3152

31514


(a) 31334

(b) 7278

(c) 210310

(d) 5512

(e) 116119

(f) 3152

31514

49.

(a) 272123

(b) 60154

(c) 639285

(d) 50128

(e) 203458

(f) 5431502


(a) 272123

(b) 60154

(c) 639285

(d) 50128

(e) 203458

(f) 5431502

6.39

2009

39

6A (6.1) 1. (a) 17 1717 or 17 (b) 30 3030 or 30 (c) 0.02 0.020.02 or 0.02

(d) 238

238

238

or 238

2. (a) Yes (b) No (c) Yes (d) No

3. (a) 5 6 Between 5 and 6 (b) 13 14 Between 13 and 14

4. (a) 8.94 (b) 32 (c) 1.90 (d) 0.516 (e) 10.4 (f) 14.5

6B (6.3) 1. (a) 30 (b) 21.2 (c) 17.3 (d) 27.1

2. 12

3. 120 cm2

4. 82.1 cm2

5. 25.6 cm

6. (a) 18.0 m (b) 21.7 m

6C (6.9) 1. (a) No (b) PRQYes, PRQ

2. (b) 73.5

3. (a) 84 (b) Area 72.6 ;

225

hh

4. (b) 108

6D (6.13) 1. 1 281 km

2. 21.2 cm 3. 70 70 units

4. 213 km

5. 0.186 m

6E (6.17) 1. (a) 39 (b) 8 (c) 7343 (d) 625

2. (a) 112 (b) 73 (c) 153 (d) 76

3. (a) 13

7 (b) 233 (c)

914 (d)

1524

4. (a) 8 (b) 23 (c) 56 (d)

43

6F (6.19)

1. (a) 66 (b)

552 (c)

11119 (d)

2103

2. (a) 66 (b)

332 (c)

559 (d)

3332

3. (a) 57 (b) 144 (c) 137 (d) 5

68

4. (a) 25 (b) 613 (c) 318 (d) 77

6A (6.21) 1. (a) No (b) 6 6Yes, 6 or 6 (c) No (d) No (e) 28 28Yes, 28 or 28 (f) 33 33Yes, 33 or 33

2. (a) Yes (b) Yes (c) No (d) Yes (e) No (f) Yes

3. (a) 9 99 or 9 (b) 14 1414 or 14 (c) 22 2222 or 22

(d) 121

121

121

or 121

(e) 2

13

213

2

13 or

213

(f) 1011

1011

1011

or 1011

4. (a) 4.58 (b) 11.0 (c) 30.8 (d) 0.837 (e) 3.03 (f) 6.74

5. (a) 4 5 Between 4 and 5 (b) 5 6 Between 5 and 6 (c) 7 8 Between 7 and 8 (d) 11 12 Between 11 and 12 (e) 19 20 Between 19 and 20 (f) 21 22 Between 21 and 22

6. (a) 2019

2019

2019

or 2019

(b) 2716

2716

2716

or 2716

6.40

2009

(c) 1825

1825

1825

or 1825

(d) 1.7 1.71.7 or 1.7 (e) 2.2 2.22.2 or 2.2 (f) 0.05 0.050.05 or 0.05

7. (a) 15.8 (b) 27.0 (c) 27.2 (d) 26.5 (e) 39.2 (f) 12.5

6B (6.23) 8. (a) 11.4 (b) 8.54 (c) 12.5

9. (a) 9.80 (b) 9.80 (c) 14.3

10. 12.5

11. 7.07 cm

12. 38.2 cm

13. 30 cm

14. 72 cm

15. 226 cm2

16. (a) 15.0 cm (b) 19.2 cm

17. 35.0 cm

18. (a) 51.2 (b) 20.4 (c) 1 120

19. 87.5 cm

6C (6.27) 20. (a) ABCYes, ABC (b) No (c) BACYes, BAC

21. Yes

22. (b) 7.5

23. (b) 23.4

25. (b) 38.8 26. (b) 34 ,20 DCBC (c) No

6D (6.30) 27. 17 mm

28. 3.57 m

29. (a) B(12, 0) 20OA B(12, 0) 20OA units (b) B(0, 9) 15OA B(0, 9) 15OA units

30. 10.6 m

31. 7.90 m 32. 36 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 7.21 units

39. 6.41 cm2

6E (6.34) 40. (a) 28 (b) 25 (c) 1717

(d) 10100 (e) 121 (f) 6296 1

41. (a) 72 (b) 113 (c) 262

(d) 352 (e) 662 (f) 216

42. (a) 83 (b)

1033 (c)

754

(d) 32 (e)

536 (f)

28153

43. (a) 18 (b) 215 (c) 706

(d) 5 (e) 32 (f) 3

44. (a) 150 (b) 3

50 (c) 357

(d) 21

6F (6.36)

45. (a) 10103 (b)

13136 (c)

269

(d) 3

2110 (e) 7423 (f) 29

46. (a) 230 (b)

105 (c)

15157

(d) 51512 (e)

22 (f)

3302

47. (a) 778 (b)

15304 (c)

1233

(d) 16

25 (e) 430 (f)

5352

48. (a) 317 (b) 710 (c) 2105

(d) 511 (e) 113 (f) 154

49. (a) 312 (b) 156 (c) 737

(d) 23 (e) 518 (f) 6

數學新里程(第二版) 附加練習 - 樣章 - ctep.com.hk€¦ · 6.1 2009 中大出版社...

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