for HKCEE in Mathematics [Part 1]

Specimen Copy

for HKCEE in Mathematics [ Part 1]

Name:

Date:

Class:

Class No.:

/ 54

Score:

The following table showing the topic of each question in the test is provided for marking purpose. For convenience, one mark will be given for each correct answer, and no marks will be deducted for wrong answers.

Section A Topic

Question No.

Section B Score

Question No.

Score

Total Score

Numbers, Approximation and Estimation Laws of Indices, Exponential and Logarithmic Functions

30 - 33

¤4

34 - 38

¤5

39 - 43

¤8

Formulas Polynomials, Remainder Theorem and Factor Theorem Identities and Factorization Functions and Graphs Equations Inequalities and Linear Programming Percentages Sequences

1-3

Rates, Ratios and Variations

4-6

¤3

7-9

¤3

Mensuration Basic Trigonometry Applications of Trigonometry Symmetry and Transformation Plane Geometry I ¾ Rectilinear Figures Plane Geometry II ¾ Circles Coordinate Geometry and Locus

10 - 18

44 - 49

¤ 15

Probability

19 - 23

50 - 53

¤9

Statistics

24 - 29

54

¤7

Specimen Copy

FORMULAS FOR REFERENCE Surface area

= 4pr 2

Volume

=

Area of curved surface

= 2prh

Volume

= pr 2 h

Area of curved surface

= prl

Volume

=

PRISM

Volume

= base area ´ height

PYRAMID

Volume

=

SPHERE

CYLINDER

CONE

4 3 pr 3

1 2 pr h 3

1 ´ base area ´ height 3

Specimen Copy

4. If a : b = 3 : 4, which of the following must be true?

The diagrams in this test are not necessarily drawn to scale. Choose the best answer for each question.

SECTION A

B. C. D.

1 4 3 4 1 7 4

II.

4a = 3b

□

I only

B.

II only

C.

I and III only

D.

None of the above

A.

decrease by 44%.

B.

decrease by 20%.

C.

increase by 20%.

D.

increase by 44%.

3rd pattern

A.

3

A.

29

B.

4

B.

31

C.

5

C.

36

D.

7

D.

49

7. In the figure, find the area of the trapezium ABCD correct to 3 significant figures. B 8 cm

21.

B.

25.

C.

28.

D.

36.

□

□

□

3. If the first 4 terms of a sequence are 1, 3, 6, 10, then the 6th term of the sequence is A.

□

6. It is given that y is partly constant and partly varies inversely as x . When x = 4, y = 9. When x = 9, y = 8. Find y when x = 36.

L 2nd pattern

A.

5. It is given that R varies directly as c , where c > 0. If c increases by 44%, then R will

2. In the figure, there are 4 dots in the 1st pattern. For any positive integer n, the (n + 1)th pattern is formed by adding (2n + 3) dots to the n th pattern. Find the number of dots in the 5th pattern.

1st pattern

a = 3 and b = 4

III. a : b = 9 : 16

1. Given that the general term of a sequence is n +1 , find the sum of the first two T(n) = 2n terms of the sequence. A.

I.

65°

A 6 cm

□

D

T10.1

C 2

A.

32.5 cm

B.

55.8 cm 2

C.

61.5 cm 2

D.

68.0 cm 2

□

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II.

8. In the figure, a crane stands vertically on the ground. The length of its arm APB is 50 m. The angles of elevation of A and B from O are 40° and 75° respectively. Find OP correct to 1 decimal place.

B

C

A

50 m

Locus of P

P

A

III.

B

B

A Locus of P

O

A.

10.9 m

A.

I only

B.

16.8 m

B.

II only

C.

30.2 m

C.

I and III only

D.

34.3 m

D.

II and III only

□

9. In the figure, OA = OB. Find the bearing of A from B.

□

11. If the polar coordinates of the points O, A and B are (0, 0°), (24, 240°) and (7, 330°) respectively, then the perimeter of DOAB is

A

N

C

40° O 60°

A.

25.

B.

48.

C.

56.

D.

62.

□

B

A.

N10°W

B.

N20°W

C.

S10°E

D.

S20°W

12. If the length of the line segment joining the points (-1, 2) and (k + 4, k) is 13, then k =

□

10. It is given that the moving point P is equidistant from the straight lines AB and BC. Which of the following may represent a part of the locus of P? I.

A.

5.

B.

7.

C.

-8 or 5.

D.

-10 or 7.

13. Find the slope of the straight line passing through the two points (-6, -5) and (1, 9).

B

A. B.

A

□

C

C.

Locus of P

D.

T10.2

1 2 -2 -

1 2 2

□

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14. In the figure, the straight line L 1 passes through (3, 4). If the slope of the straight 2 line L 2 is - , find c. 9

18. If a > 0, b < 0 and c < 0, which of the following may represent the graph of the straight line ax + by = c ? A.

y

y

c O O

L2

x

L1

A. B. C. D.

10 3 14 3 35 2 22

B.

y

□ y

15. A(-5, 7), B(-k, 8), C(3, 7) and D(6, 2k + 1) are four points. If AB ^ CD, find the slope of CD. -8

B.

-

C.

4

D.

9

D.

y

□ O

16. If (-1, -4) is the mid-point of (a, b) and (b + 1, -1), then a = 7.

B.

4.

C.

2.

D.

-7.

(-3, -4)

B.

(-3, -2)

C.

(3, -8)

D.

(3, 2)

x

□ 19. Which of the following cannot be the probability of an event?

□

17. The straight lines x + 5 y - 2 = 0 and L are perpendicular to each other. If (2, -3) is a point lying on L, which of the following is also a point on L? A.

x

O

1 4

A.

x

O

C.

A.

x

A.

0

B.

1

C. D.

2 2 p

□

□ T10.3

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20. A box contains some cards marked with a three-digit number (from 100 to 499) each without repetition. If a card is drawn randomly from the box, find the probability that two digits of the number on the card drawn are 0. 1 A. 100 1 B. 133 4 C. 399 3 D. 400

23. The figure shows a square dartboard with sides of 50 cm each. A square with sides of 10 cm each is placed at the centre of the dartboard and the remaining region is divided into 4 equal parts. If a dart is thrown randomly at the dartboard, assuming that it will not hit any boundary, find the probability of having the dart hit the shaded region.

□

A.

21. The table below shows the distribution of the monthly salaries of the employees in a company. If an employee is selected randomly, find the probability that the monthly salary of the employee selected is between $8 000 and $11 999 inclusive.

A. B. C. D.

Monthly salary ($)

Number of employees

6 000 - 7 999

5

8 000 - 9 999

8

10 000 - 11 999

12

12 000 - 13 999

10

14 000 - 15 999

7

³ 16 000

3

4 9 5 9 4 15 8 45

B. C. D.

1 2 3 5 12 25 13 25

□

24. Which of the following scatter diagrams may represent a positive relation between x and y? I. y

O

x

II. y

□

O

x

III.

22. The respective probabilities for Laine to have Chinese food, Western food and Japanese food for dinner are 0.7, 0.1 and 0.2, and they cost Laine $50, $60 and $70 respectively. Find the expected value of the expense on dinner.

y

O

x

A.

$35

B.

$55

A.

II only

C.

$60

B.

III only

D.

$70

C.

I and II only

D.

I, II and III

□ T10.4

□

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25. For the five numbers x - 3, x - 3, x, x - 1 and x - 3, which of the following must be true? I.

The mean is x.

II.

The median is x - 3.

28. Which of the following groups of numbers has a different standard deviation as compared with that of {1, 2, 3, 4, 5}?

III. The mode is x - 3. A.

I only

B.

II only

C.

I and III only

D.

II and III only

□

A.

{-1, -2, -3, -4, -5}

B.

{3, 4, 5, 6, 7}

C.

{5, 10, 15, 20, 25}

D.

{1, 1, 2, 2, 3, 3, 4, 4, 5, 5}

□

29. The box-and-whisker diagram below shows the distribution of the time taken by 40 students to finish a 100 m race.

26. The stem-and-leaf diagram below shows the amount of fund raised by a group of students for a charitable organization in a lottery tickets selling event. If the range of the amount of fund raised is $450, find the mode.

11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 Time (second)

Leaves ($10)

Which of the following must be true?

0

56789

I.

1

023456789

All students finished the race in not more than 15.5 seconds.

2 3 4

23356788 0147 268

II.

5

ab c

The number of students who finished the race between 12.5 seconds and 13.5 seconds is more than the number of students who finished the race between 13.5 seconds and 14 seconds.

Stem ($100)

A.

$230

B.

$280

III. The mean of the time taken to finish the race is 13.5 seconds.

C.

$500

A.

I only

D.

$230 and $280

B.

I and II only

C.

II and III only

D.

I, II and III

□

Cumulative frequency

27. The cumulative frequency curve below shows the distribution of the ages of 40 students. Find the inter-quartile range of the ages of these 40 students.

SECTION B 1

40

30.

20

1

A.

10

B.

A.

1.3

B.

1.5

C.

2.2

D.

3.7

1

m4 ×m2

=

(25m 2 ) 2

30

0

□

15

16

17 Age

18

19

C.

1 1 5m 4

1 7 5m 8

. .

1 1 25m 4

.

7

D.

□ T10.5

m8 . 5

□ Go on to the next page

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31. If a > 0, which of the following must be true? 1 1 I. a+ = a (1 + ) a a 1 1 1 II. a= (a + )(a - ) a a a 1 1 III. a + = (a + 1) a a A.

I only

B.

I and III only

C.

II and III only

D.

I, II and III

D.

1 O

log ab .

B.

(log a)(log b) .

C.

log(a + b) .

D.

log a + log b .

□

□

A.

2

B.

4

C.

5

D.

8

□

A.

y

y x

O

B. O

B.

y

x

1

y

O

x

O x

1

C.

C.

□

35. Which of the following shaded regions may represent the solutions of 2 x - y - 1 ³ 0 ?

33. Which of the following may represent the graph of y = log 3 x ? A.

x

34. If x is a positive integer, how many solutions ì4 x + 5 £ 21 does í have? î6 x + 1 ³ 5 x - 2

32. If 10c = a + b , then c = A.

y

y

y

x

O -1 O

x

D.

y

O

x

□ T10.6

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36. Which region in the following figure may ìx - 4 y £ 4 represent the solutions of í ? î x ³ -6 y

I

A.

4.

B.

9.

C.

17.

D.

22.

□

IV O -1

-6

39. If the 3rd term and the 4th term of an arithmetic sequence are 13 and 18 respectively, then the 7th term of the sequence is

x

4 III

II

A.

Region I

B.

Region II

C.

Region III

D.

Region IV

B.

28.

C.

31.

D.

33.

□

40. The n th term of the arithmetic sequence 17, 14, 11, L is 2x - y = 0

x

O

2x + y = 0

A.

ì2 x - y > 0 í2 x + y > 0 î

B.

ì2 x - y < 0 í2 x + y > 0 î

C.

ì2 x - y > 0 í2 x + y < 0 î

D.

ì2 x - y < 0 í2 x + y < 0 î

A.

14 + 3n.

B.

17 + 3n.

C.

17 - 3n.

D.

20 - 3n.

□

41. If the n th term of a geometric sequence is 2 - n , find the first term and the common 3 ratio.

Which of the following systems of inequalities has its solutions represented by the shaded region in the figure?

First term

□

A.

-2

B.

-2

C.

-

D.

Common ratio

2 3 2 3

1 3 1 3 1 3 1 3 -

□

42. Which of the following could be a geometric sequence / geometric sequences?

38. If (x, y) is a point lying in the shaded region (including the boundary), then the greatest value of 2 x + 3 y + 5 is y (-4, 3)

23.

□

y

37.

A.

I.

30, 32, 34, 36, L

II.

101, 1 001, 10 001, 100 001, L

III. -1, -5, 25, 125, L (-1, 2)

O

(2, -1)

x

(3, -3)

T10.7

A.

I only

B.

II only

C.

I and III only

D.

II and III only

□ Go on to the next page

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43. The sum of the first 10 terms of the geometric sequence 10 368, 5 184, 2 592, L is A.

20 695.5.

B.

20 715.75.

C.

20 725.875.

D.

20 756.25.

47. Which of the following may represent the graph of the circle x 2 + y 2 + 8 x - 10 y + 5 = 0 ? A.

y

□

O

B.

x

y

44. P(-6, 5) and Q(3, -4) are two points. If R is a point lying on PQ such that 4PR = 5RQ, then the coordinates of R are A.

(-1, 0).

B.

(-2, 1). 13 40 (- , ). 3 9 14 41 (- , ) . 3 9

C. D.

x

O

C.

y

□ x

O

45. If the coordinates of the centre of a circle are (-1, 6) and its diameter is 4, then the equation of the circle is A.

x 2 + y 2 - 2 x + 12 y + 21 = 0.

B.

x 2 + y 2 - 2 x + 12 y + 33 = 0 .

C.

x 2 + y 2 + 2 x - 12 y + 21 = 0. 2

2

x + y + 2 x - 12 y + 33 = 0.

D.

D.

O

□

y

2 O

x

A.

x 2 + y 2 - 10 = 0.

B.

x 2 + y 2 - 6 x + 2 y = 0.

C.

x 2 + y 2 - 2 x + 6 y = 0.

D.

x 2 + y 2 + 6 x - 2 y = 0.

x

□

48. Consider the circle x 2 + y 2 - 2 x + 4 y - 32 = 0 . Find the equation of the chord whose mid-point is (-1, 3).

46. In the figure, the equation of the circle is

-6

y

A.

2 x - 5 y + 17 = 0

B.

3x - 7 y + 24 = 0

C.

5x + 2 y - 1 = 0

D.

7x + 3y - 2 = 0

□

49. Find the area of the region bounded by the circle x 2 + y 2 - 4 x + 6 y - 2 = 0 .

□

A.

T10.8

15

B.

15p

C.

2 15p

D.

15p

□

Specimen Copy

50. In a shooting game, the probability for Jim 1 to hit the target is whilst the probability 3 1 for Gene to hit the target is . If each of 4 them shoots once, find the probability that only one of them will hit the target. 1 A. 2 1 B. 12 5 C. 12 7 D. 12

53. Ivy and Marie each throws a dart. The probability for Ivy to hit the target is 0.3 and the probability for Marie to hit the target is 0.4. Find the probability that at most one dart hits the target.

□

0.88

B.

0.58

C.

0.46

D.

0.3

□

54. The mean, median and mode of n numbers are a, b and c respectively. If 4 is deducted from each of the numbers, what will be their new mean, median and mode?

51. The figure shows that Macy has 3 ways to leave town A and Charles has 3 ways to leave town B. Macy and Charles leave town A and town B respectively at the same time. If they select their ways randomly, find the probability that they will meet on their ways. Town D

(Macy) Town A

A.

A.

Mean a

Median b

Mode c

B.

a

b-4

c-4

C.

a-4

b

c

D.

a-4

b-4

c-4

□

- END OF TEST -

(Charles) Town B

Town C

A. B. C. D.

1 2 1 9 2 9 4 9

□

52. A graduation examination consists of three sections. A student can get the graduation certificate by passing at least two sections. If the probabilities for Helen to pass the three sections are 0.5, 0.6 and 0.7 respectively, find the probability for her to get the graduation certificate. A.

0.21

B.

0.44

C.

0.65

D.

0.94

□ T10.9

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