MATHEMATICS Compulsory Part PAPER 1 (Sample Paper)
Please stick the barcode label here. HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION Candidate Number MATHEMATICS Compulsory Part PAPER 1 (Sample Paper) Marker’s Use Only Examiner’s Use Only Question-Answer Book Marker No. Examiner No. Marks Marks Time allowed: 2 hours 15 minutes This paper must be answered in English. Question No. 1–2 3–4 INSTRUCTIONS 5–6 1. Write your Candidate Number in the space provided on Page 1. 2. Stick barcode labels in the spaces provided on Pages 1, 3, 5, 7 and 9. 3. This paper consists of THREE sections, A(1), A(2) and B. Each section carries 35 marks. 4. 5. Attempt ALL questions in this paper. Write your answers in the spaces provided in this QuestionAnswer Book. Do not write in the margins. Answers written in the margins will not be marked. 7–8 9 10 11 12 13 Graph paper and supplementary answer sheets will be supplied on request. Write your Candidate Number, mark the question number box and stick a barcode label on each sheet, and fasten them with string INSIDE this book. 14 6. Unless otherwise specified, all working must be clearly shown. 17 7. Unless otherwise specified, numerical answers should be either exact or correct to 3 significant figures. 19 The diagrams in this paper are not necessarily drawn to scale. Total 8. 15 16 18 Checker’s Use Only HKDSE-MATH-CP 1 – 1 (Sample Paper) 47 Checker No. Total ( xy ) 2 and express your answer with positive indices. x−5 y 6 1. Simplify 2. Make b the subject of the formula a (b + 7) = a + b . Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 2 (Sample Paper) 48 (3 marks) (3 marks) Answers written in the margins will not be marked. Answers written in the margins will not be marked. SECTION A(1) (35 marks) Please stick the barcode label here. 3. Factorize (a) 3m 2 − mn − 2n 2 , (b) 3m 2 − mn − 2n 2 − m + n . 4. The marked price of a handbag is $ 560 . It is given that the marked price of the handbag is 40 % higher than the cost. (a) Find the cost of the handbag. (b) If the handbag is sold at $ 460 , find the percentage profit. (4 marks) Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 3 (Sample Paper) 49 Answers written in the margins will not be marked. Answers written in the margins will not be marked. (3 marks) In a football league, each team gains 3 points for a win, 1 point for a draw and 0 point for a loss. The champion of the league plays 36 games and gains a total of 84 points. Given that the champion does not lose any games, find the number of games that the champion wins. (4 marks) 6. Figure 1 shows a solid consisting of a hemisphere of radius r cm joined to the bottom of a right circular cone of height 12 cm and base radius r cm . It is given that the volume of the circular cone is twice the volume of the hemisphere. (a) Find r . (b) Express the volume of the solid in terms of π . (4 marks) Figure 1 Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 4 (Sample Paper) 50 Answers written in the margins will not be marked. Answers written in the margins will not be marked. 5. Please stick the barcode label here. 7. In Figure 2, O is the centre of the semicircle ABCD . If AB // OC and ∠BAD = 38° , find ∠BDC . (4 marks) B C A O D Figure 2 8. In Figure 3, the coordinates of the point A are (−2 , 5) . A is rotated clockwise about the origin O through 90° to A′ . A′′ is the reflection image of A with respect to the y-axis. (a) Write down the coordinates of A′ and A′′ . (b) Is OA′′ perpendicular to A A′ ? Explain your answer. (5 marks) y A(−2, 5) O x Figure 3 Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 5 (Sample Paper) 51 Answers written in the margins will not be marked. Answers written in the margins will not be marked. 38° 9. In Figure 4, the pie chart shows the distribution of the numbers of traffic accidents occurred in a city in a year. In that year, the number of traffic accidents occurred in District A is 20% greater than that in District B . District B District A x° 72° 120° District C 30° District D The distribution of the numbers of traffic accidents occurred in the city Figure 4 (a) Find x . (b) Is the number of traffic accidents occurred in District A greater than that in District C ? Explain your answer. (5 marks) Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 6 (Sample Paper) 52 Answers written in the margins will not be marked. Answers written in the margins will not be marked. District E Please stick the barcode label here. Section A(2) (33 marks) 10. (a) Find the quotient when 5 x 3 + 12 x 2 − 9 x − 7 is divided by x 2 + 2 x − 3 . (b) Let g ( x ) = (5 x 3 + 12 x 2 − 9 x − 7) − (ax + b) , where a and b are constants. It is given that (2 marks) g ( x) is divisible by x 2 + 2 x − 3 . (i) Write down the values of a and b . (ii) Solve the equation g ( x) = 0 . Answers written in the margins will not be marked. Answers written in the margins will not be marked. (4 marks) Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 7 (Sample Paper) 53 In a factory, the production cost of a carpet of perimeter s metres is $ C . It is given that C is a sum of two parts, one part varies as s and the other part varies as the square of s . When s = 2 , C = 356 ; when s = 5 , C = 1 250 . Find the production cost of a carpet of perimeter 6 metres. (4 marks) (b) If the production cost of a carpet is $ 539 , find the perimeter of the carpet. (2 marks) Answers written in the margins will not be marked. (a) Answers written in the margins will not be marked. 11. Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 8 (Sample Paper) 54 Please stick the barcode label here. 12. Figure 5 shows the graph for John driving from town A to town D ( via town B and town C ) in a morning. The journey is divided into three parts: Part I (from A to B ), Part II (from B to C ) and Part III (from C to D ). C 18 B 4 A 0 8:00 8:11 8:30 Time Figure 5 (a) For which part of the journey is the average speed the lowest? Explain your answer. (2 marks) (b) If the average speed for Part II of the journey is 56 km / h , when is John at C ? (2 marks) (c) Find the average speed for John driving from A to D in m / s . (3 marks) Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 9 (Sample Paper) 55 Answers written in the margins will not be marked. Distance travelled (km) Answers written in the margins will not be marked. D 27 13. In Figure 6, the straight line L1 : 4 x − 3 y + 12 = 0 and the straight line L 2 are perpendicular to each other and intersect at A . It is given that L1 cuts the y-axis at B and L 2 passes through the point (4 , 9) . y L2 L1 A B x Figure 6 (a) Find the equation of L 2 . (b) Q is a moving point in the coordinate plane such that AQ = BQ . Denote the locus of Q by Γ . (3 marks) (i) Describe the geometric relationship between Γ and L 2 . Explain your answer. (ii) Find the equation of Γ . (6 marks) Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 10 (Sample Paper) 56 Answers written in the margins will not be marked. Answers written in the margins will not be marked. O Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 11 (Sample Paper) 57 Answers written in the margins will not be marked. Answers written in the margins will not be marked. SECTION B (35 marks) 15. The seats in a theatre are numbered in numerical order from the first row to the last row, and from left to right, as shown in Figure 7. The first row has 12 seats. Each succeeding row has 3 more seats than the previous one. If the theatre cannot accommodate more than 930 seats, what is the greatest number of rows of seats in the theatre? M K K 29 28 3rd row 2nd row 44 K 14 13 1st row 26 27 45 11 12 2 1 Figure 7 Answers written in the margins will not be marked. Answers written in the margins will not be marked. (4 marks) Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 13 (Sample Paper) 59 17. A researcher defined Scale A and Scale B to represent the magnitude of an explosion as shown in the following table: Scale A Formula M = log 4 E B N = log8 E Answers written in the margins will not be marked. Answers written in the margins will not be marked. It is given that M and N are the magnitudes of an explosion on Scale A and Scale B respectively while E is the relative energy released by the explosion. If the magnitude of an explosion is 6.4 on Scale B , find the magnitude of the explosion on Scale A . (5 marks) Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 15 (Sample Paper) 61 18. In Figure 8(a), ABC is a triangular paper card. D is a point lying on AB such that CD is perpendicular to AB . It is given that AC = 20 cm , ∠CAD = 45° and ∠CBD = 30° . C 20 cm 45° 30° A B D Figure 8(a) (a) Find, in surd form, BC and BD . (3 marks) (b) The triangular paper card in Figure 8(a) is folded along CD such that ∆ ACD lies on the horizontal plane as shown in Figure 8(b). Answers written in the margins will not be marked. Answers written in the margins will not be marked. B C A D Figure 8(b) (i) If the distance between A and B is 18 cm , find the angle between the plane BCD and the horizontal plane. (ii) Describe how the volume of the tetrahedron from 40° to 140° . Explain your answer. A BCD varies when ∠ ADB increases (5 marks) Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 16 (Sample Paper) 62 Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 17 (Sample Paper) 63 Answers written in the margins will not be marked. Answers written in the margins will not be marked. 19. In Figure 9, the circle passes through four points A , B , C and D . PQ is the tangent to the circle at C and is parallel to BD . AC and BD intersect at E . It is given that AB = AD . P B C E Q A D Answers written in the margins will not be marked. (a) (b) (i) Prove that ∆ A BE ≅ ∆ ADE . (ii) Are the in-centre, the orthocentre, the centroid and the circumcentre of ∆ A BD collinear? Explain your answer. (6 marks) A rectangular coordinate system is introduced in Figure 9 so that the coordinates of A , B and D are (14 , 4) , (8 , 12) and (4 , 4) respectively. Find the equation of the tangent PQ . (7 marks) Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 18 (Sample Paper) 64 Answers written in the margins will not be marked. Figure 9 END OF PAPER Answers written in the margins will not be marked. HKDSE-MATH-CP 1 – 19 (Sample Paper) 65 Answers written in the margins will not be marked. Answers written in the margins will not be marked. HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION MATHEMATICS Compulsory Part PAPER 2 (Sample Paper) Time allowed: 1 hour 15 minutes 1. Read carefully the instructions on the Answer Sheet. Stick a barcode label and insert the information required in the spaces provided. 2. When told to open this book, you should check that all the questions are there. Look for the words ‘END OF PAPER’ after the last question. 3. All questions carry equal marks. 4. ANSWER ALL QUESTIONS. You are advised to use an HB pencil to mark all the answers on the Answer Sheet, so that wrong marks can be completely erased with a clean rubber. 5. You should mark only ONE answer for each question. If you mark more than one answer, you will receive NO MARKS for that question. 6. No marks will be deducted for wrong answers. Not to be taken away before the end of the examination session HKDSE-MATH-CP 2 – 1 (Sample Paper) 66 There are 30 questions in Section A and 15 questions in Section B. The diagrams in this paper are not necessarily drawn to scale. Choose the best answer for each question. Section A 1. 2. 3. (3a) 2 ⋅ a 3 = A. 3a 5 . B. 6a 6 . C. 9a 5 . D. 9a 6 . If 5 − 3m = 2n , then m = A. n . B. 2n − 5 . 3 C. −2n + 5 . 3 D. −2n + 15 . 3 a 2 − b 2 + 2b − 1 = A. (a − b − 1)(a + b − 1) . B. (a − b − 1)(a + b + 1) . C. (a − b + 1)(a + b − 1) . D. (a − b + 1)(a − b − 1) . HKDSE-MATH-CP 2 – 2 (Sample Paper) 67 4. 5. 6. 7. Let p and q be constants. If x 2 + p ( x + 5) + q ≡ ( x − 2)( x + 5) , then q = A. −25 . B. −10 . C. 3. D. 5. Let f ( x) = x 3 + 2 x 2 − 7 x + 3 . When f ( x) is divided by x + 2 , the remainder is A. 3. B. 5. C. 17 . D. 33 . Let a be a constant. Solve the equation ( x − a)( x − a − 1) = ( x − a) . A. x = a +1 B. x=a+2 C. x = a or x = a + 1 D. x = a or x = a + 2 Find the range of values of k such that the quadratic equation x 2 − 6 x = 2 − k has no real roots. A. k < −7 B. k > −7 C. k < 11 D. k > 11 HKDSE-MATH-CP 2 – 3 (Sample Paper) 68 8. In the figure, the quadratic graph of y = f ( x) intersects the straight line L at A(1, k ) and B(7 , k ) . Which of the following are true? I. The solution of the inequality f ( x) > k is x < 1 or x > 7 . II. The roots of the equation f ( x) = k are 1 and 7 . III. The equation of the axis of symmetry of the quadratic graph of y = f ( x) is x = 3 . A. I and II only B. I and III only C. II and III only D. I , II and III y y = f ( x) A O 9. 10. L B x The solution of 5 − 2 x < 3 and 4 x + 8 > 0 is A. x > −2 . B. x > −1 . C. x >1 . D. −2 < x < 1 . Mary sold two bags for $ 240 each. She gained 20% on one and lost 20% on the other. After the two transactions, Mary A. lost $ 20 . B. gained $ 10 . C. gained $ 60 . D. had no gain and no loss. HKDSE-MATH-CP 2 – 4 (Sample Paper) 69 11. Let an be the nth term of a sequence. If a1 = 4 , a 2 = 5 and a n + 2 = an + an +1 for any positive integer n , then a10 = 12. 13. 14. A. 13 . B. 157 . C. 254 . D. 411 . If the length and the width of a rectangle are increased by 20% and x % respectively so that its area is increased by 50% , then x = A. 20 . B. 25 . C. 30 . D. 35 . If x , y and z are non-zero numbers such that 2 x = 3 y and x = 2 z , then ( x + z ) : ( x + y ) = A. 3:5 . B. 6:7 . C. 9:7 . D. 9 : 10 . It is given that z varies directly as x and inversely as y . When x = 3 and y = 4 , z = 18 . When x = 2 and z = 8 , y = A. 1. B. 3. C. 6. D. 9. HKDSE-MATH-CP 2 – 5 (Sample Paper) 70 15. 16. The lengths of the three sides of a triangle are measured as 15 cm , 24 cm and 25 cm respectively. If the three measurements are correct to the nearest cm , find the percentage error in calculating the perimeter of the triangle correct to the nearest 0.1% . A. 0.8% B. 2.3% C. 4.7% D. 6.3% In the figure, O is the centre of the circle. C and D are points lying on the circle. OBC and BAD are straight lines. If OC = 20 cm and OA = AB = 10 cm , find the area of the shaded region BCD correct to the nearest cm 2 . D 2 A. 214 cm B. 230 cm 2 O C. 246 cm 2 D. 270 cm 2 A B 17. C The figure shows a right circular cylinder, a hemisphere and a right circular cone with equal base radii. Their curved surface areas are a cm2 , b cm2 and c cm2 respectively. r 2 r r Which of the following is true? A. a<b<c B. a<c<b C. c<a<b D. c<b<a HKDSE-MATH-CP 2 – 6 (Sample Paper) 2r r 71 r 18. In the figure, ABCD is a parallelogram. T is a point lying on AB such that DT is perpendicular to AB . It is given that CD = 9 cm and AT : TB = 1 : 2 . If the area of the parallelogram ABCD is 36 cm 2 , then the perimeter of the parallelogram ABCD is A. 26 cm . B. 28 cm . C. 30 cm . D. 32 cm . D A 19. 20. C B T sin θ cos(270° − θ ) + = cos 60° tan 45° A. sin θ . B. 3 sin θ . C. 2 sin θ − cos θ . D. 2 sin θ + cos θ . In the figure, AB = 1 cm , BC = CD = DE = 2 cm and EF = 3 cm . Find the distance between A and F correct to the nearest 0.1 cm . A. 7.2 cm B. 7.4 cm C. 8.0 cm D. 8.1 cm A B D C E 21. F In the figure, ABCD is a semi-circle. If BC = CD , then ∠ ADC = A. 118° . B. 121° . C. 124° . D. 126° . C D A HKDSE-MATH-CP 2 – 7 (Sample Paper) 72 28° B 22. In the figure, O is the centre of the circle ABCDE . If ∠ ABE = 30° and ∠CDE = 105° , then ∠ AOC = A. 120° . B. 135° . D E 105° C. 150° . D. 165° . O A C 30° B 23. In the figure, ABCD is a parallelogram. F is a point lying on AD . BF produced and CD produced meet at E . If CD : DE = 2 : 1 , then AF : BC = E A. 1: 2 . B. 2:3 . C. 3: 4 . D. 8:9 . F A D C B 24. In the figure, ABC is a straight line. If BD = CD and AB = 10 cm , find BC correct to the nearest cm . A. 8 cm B. 13 cm C. 14 cm D. 15 cm D 40° C HKDSE-MATH-CP 2 – 8 (Sample Paper) 20° B 73 A 25. 26. In the figure, the two 6-sided polygons show A. a rotation transformation. B. a reflection transformation. C. a translation transformation. D. a dilation transformation. If the point (− 4 , 3) is rotated anti-clockwise about the origin through 180° , then the coordinates of its image are A. ( −3 , − 4) . B. (3 , 4 ) . C. (− 4 , − 3) . D. (4 , − 3) . 27. The box-and-whisker diagram below shows the distribution of the scores (in marks) of the students of a class -------------------------------------------------------------------------------------------------------in a test. -------- Score (marks) 30 40 50 60 70 80 90 If the passing score of the test is 50 marks, then the passing percentage of the class is ------------------------------------------------------------------------------ -----------A. 25% . B. 50% . ------------ -----------C. 70% . -----------D. 75% . HKDSE-MATH-CP 2 – 9 (Sample Paper) 74 Section B 31. 32. 1 x −1 + = 2 − x ( x − 2) 2 A. −3 . (2 − x) 2 B. 1 . (2 − x) 2 C. −2 x + 3 . (2 − x) 2 D. 2x − 3 . (2 − x) 2 The graph in the figure shows the linear relation between x and log 5 y . If y = ab x , then a = A. 1. B. 2. C. 5. D. 25 . log 5 y 2 O 33. 34. 10100100010012 = A. 212 + 210 + 137 . B. 212 + 210 + 273 . C. 213 + 211 + 137 . D. 213 + 211 + 273 . If k is a real number, then 4k − A. 3k + 6i . B. 3k − 6i . C. 5k + 6i . D. 5k − 6i . HKDSE-MATH-CP 2 – 11 (Sample Paper) 6 + ki = i 76 x 35. 0 ≤ x ≤ 6 Which of the triangular regions in the figure may represent the solution of 0 ≤ y ≤ 3 ? x ≤ 2 y A. ∆OAC B. ∆OBD C. ∆OCE D. ∆ODF y 3 F D 2 E 1 C A O 36. 37. B x 1 3 2 5 4 6 If the 3rd term and the 6th term of an arithmetic sequence are 18 and − 6 respectively, then the 2nd term of the sequence is A. −8 . B. 10 . C. 26 . D. 34 . If the figure shows the graph of y = f ( x) and the graph of y = g( x) on the same rectangular coordinate system, then y A. g ( x ) = f ( x − 2) − 3 . y = f ( x) 38. B. g ( x ) = f ( x − 2) + 3 . C. g ( x ) = f ( x + 2) − 3 . D. g ( x ) = f ( x + 2) + 3 . y = g( x) x O (−2 , − 3) In the figure, y = A. x sin 77° . sin 56° B. x sin 47° . sin 56° C. x sin 56° . sin 77° D. x sin 77° . sin 47° 47° y 56° HKDSE-MATH-CP 2 – 12 (Sample Paper) 77 x 39. 40. Peter invests $P at the beginning of each month in a year at an interest rate of 6% per annum, compounded monthly. If he gets $10 000 at the end of the year, find P correct to 2 decimal places. A. 806.63 B. 829.19 C. 833.33 D. 882.18 The figure shows a cuboid ABCDEFGH . If the angle between the triangle ACE and the plane ABCD is θ , then tan θ = E H A. 2. B. 3 . 2 C. 5 . 2 D. 41. 42. F G 6 cm D C 3 cm 12 . 5 A B 4 cm In the figure, A , B and C are points lying on the circle. TA is the tangent to the circle at A . The straight line CBT is perpendicular to TA . If BC = 6 cm , find the radius of the circle correct to the nearest 0.1 cm . A. 3.2 cm B. 3.9 cm C. 4.2 cm D. 4.7 cm T B 20° A C Let a be a constant and −90° < b < 90° . If the figure shows the graph of y = a cos( x° + b) , then A. a = −3 and b = −40° . B. a = −3 and b = 40° . C. a = 3 and b = −40° . D. a = 3 and b = 40° . HKDSE-MATH-CP 2 – 13 (Sample Paper) y 3 O 78 140 x Please stick the barcode label here. PP-DSE MATH CP PAPER 1 HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION Candidate Number PRACTICE PAPER MATHEMATICS Compulsory Part PAPER 1 Question-Answer Book (2¼ hours) This paper must be answered in English INSTRUCTIONS 1. After the announcement of the start of the examination, you should first write your Candidate Number in the space provided on Page 1 and stick barcode labels in the spaces provided on Pages 1, 3, 5, 7, 9 and 11. 2. This paper consists of THREE sections, A(1), A(2) and B. 3. Attempt ALL questions in this paper. Write your answers in the spaces provided in this QuestionAnswer Book. Do not write in the margins. Answers written in the margins will not be marked. 4. Graph paper and supplementary answer sheets will be supplied on request. Write your Candidate Number, mark the question number box and stick a barcode label on each sheet, and fasten them with string INSIDE this book. 5. Unless otherwise specified, all working must be clearly shown. 6. Unless otherwise specified, numerical answers should be either exact or correct to 3 significant figures. 7. The diagrams in this paper are not necessarily drawn to scale. 8. No extra time will be given to candidates for sticking on the barcode labels or filling in the question number boxes after the ‘Time is up’ announcement. 香港考試及評核局 保留版權 Hong Kong Examinations and Assessment Authority All Rights Reserved 2012 PP-DSE-MATH-CP 1−1 1 *A030e001* (m 5 n −2 ) 6 1. Simplify 2. Make a the subject of the formula m 4 n −3 and express your answer with positive indices. 5+b = 3b . 1− a (3 marks) (3 marks) Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−2 2 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. SECTION A(1) (35 marks) Please stick the barcode label here. 3. Factorize (a) 9 x 2 − 42 xy + 49 y 2 , (b) 9 x 2 − 42 xy + 49 y 2 − 6 x + 14 y . 4. The cost of a chair is $ 360 . If the chair is sold at a discount of 20 % on its marked price, then the percentage profit is 30 % . Find the marked price of the chair. (4 marks) Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−3 3 Go on to the next page Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. (3 marks) The ratio of the capacity of a bottle to that of a cup is 4 : 3 . The total capacity of 7 bottles and 9 cups is 11 litres. Find the capacity of a bottle. (4 marks) 6. In a polar coordinate system, the polar coordinates of the points A , B and C are (13 , 157°) , (14 , 247°) and (15 , 337°) respectively. (a) Let O be the pole. Are A , O and C collinear? Explain your answer. (b) Find the area of ∆ ABC . (4 marks) Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−4 4 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. 5. Please stick the barcode label here. 7. In Figure 1, BD is a diameter of the circle ABCD . If AB = AC and ∠BDC = 36° , find ∠ABD . (4 marks) A D 36° B Answers written in the margins will not be marked. Answers written in the margins will not be marked. C Figure 1 Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−5 5 Go on to the next page Page total The coordinates of the points A and B are (−3 , 4) and (−2 , − 5) respectively. A′ is the reflection image of A with respect to the y-axis. B is rotated anticlockwise about the origin O through 90° to B ′ . Write down the coordinates of A′ and B ′ . (b) Let P be a moving point in the rectangular coordinate plane such that P is equidistant from A′ and B′ . Find the equation of the locus of P . (5 marks) Answers written in the margins will not be marked. (a) Answers written in the margins will not be marked. 8. Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−6 6 Page total SECTION A(2) (35 marks) 10. Let f ( x) be a polynomial. When f ( x) is divided by x − 1 , the quotient is 6 x 2 + 17 x − 2 . It is given that f (1) = 4 . (3 marks) (b) Factorize f ( x) . (3 marks) Answers written in the margins will not be marked. Find f (−3) . Answers written in the margins will not be marked. (a) Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−8 8 Page total Please stick the barcode label here. Let $ C be the cost of manufacturing a cubical carton of side x cm . It is given that C is partly constant and partly varies as the square of x . When x = 20 , C = 42 ; when x = 120 , C = 112 . (a) Find the cost of manufacturing a cubical carton of side 50 cm . (4 marks) (b) If the cost of manufacturing a cubical carton is $ 58 , find the length of a side of the carton. (2 marks) Answers written in the margins will not be marked. Answers written in the margins will not be marked. 11. Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−9 9 Go on to the next page Page total 12. Figure 2 shows the graphs for Ada and Billy running on the same straight road between town P and town Q during the period 1:00 to 3:00 in an afternoon. Ada runs at a constant speed. It is given that town P and town Q are 16 km apart. Distance from town P (km) Q 16 12 Ada Billy 2 1:00 1:32 2:03 2:18 3:00 Time Figure 2 (a) How long does Billy rest during the period? (2 marks) (b) How far from town P do Ada and Billy meet during the period? (3 marks) (c) Use average speed during the period to determine who runs faster. Explain your answer. (2 marks) Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−10 10 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. P 0 Answers written in the margins will not be marked. Answers written in the margins will not be marked. Please stick the barcode label here. Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−11 11 Go on to the next page Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−13 13 Go on to the next page Page total 14. In Figure 3, OABC is a circle. It is given that AB produced and OC produced meet at D . D B C Answers written in the margins will not be marked. Figure 3 (a) Write down a pair of similar triangles in Figure 3. (b) Suppose that ∠ AOD = 90° . A rectangular coordinate system, with (2 marks) O as the origin, is introduced in Figure 3 so that the coordinates of A and D are (6 , 0) and (0 , 12) respectively. If the ratio of the area of ∆ BCD to the area of ∆OAD is 16 : 45 , find (i) the coordinates of C , (ii) the equation of the circle OABC . (7 marks) Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−14 14 Page total Answers written in the margins will not be marked. A O Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−15 15 Go on to the next page Page total 1 in the form of a + bi , where a and b are real numbers. 1 + 2i (a) Express (b) The roots of the quadratic equation x 2 + px + q = 0 are (2 marks) 10 10 and . Find 1 + 2i 1 − 2i p and q , (ii) the range of values of r such that the quadratic equation x 2 + px + q = r has real roots. (5 marks) Answers written in the margins will not be marked. (i) Answers written in the margins will not be marked. 17. Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−18 18 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−19 19 Go on to the next page Page total 18. Figure 4 shows a geometric model ABCD in the form of tetrahedron. It is found that ∠ACB = 60° , AC = AD = 20 cm , BC = BD = 12 cm and CD = 14 cm . B C A D (a) Find the length of AB . (2 marks) (b) Find the angle between the plane ABC and the plane ABD . (4 marks) (c) Let P be a movable point on the slant edge AB . Describe how ∠CPD varies as P moves from A to B . Explain your answer. (2 marks) Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−20 20 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. Figure 4 Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−21 21 Go on to the next page Page total 19. The amount of investment of a commercial firm in the 1st year is $ 4 000 000 . The amount of investment in each successive year is r % less than the previous year. The amount of investment in the 4th year is $ 1 048 576 . (a) Find r . (2 marks) (b) The revenue made by the firm in the 1st year is $ 2 000 000 . The revenue made in each successive year is 20 % less than the previous year. (ii) Will the total revenue made by the firm exceed $ 10 000 000 ? Explain your answer. (iii) The manager of the firm claims that the total revenue made by the firm will exceed the total amount of investment. Do you agree? Explain your answer. (10 marks) Answers written in the margins will not be marked. Find the least number of years needed for the total revenue made by the firm to exceed $ 9 000 000 . Answers written in the margins will not be marked. (i) Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−22 22 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−23 23 Go on to the next page Page total END OF PAPER Answers written in the margins will not be marked. PP-DSE-MATH-CP 1−24 24 Page total Answers written in the margins will not be marked. Answers written in the margins will not be marked. PP-DSE MATH CP HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY PAPER 2 HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION PRACTICE PAPER MATHEMATICS Compulsory Part PAPER 2 (1¼ hours) INSTRUCTIONS 1. Read carefully the instructions on the Answer Sheet. After the announcement of the start of the examination, you should first stick a barcode label and insert the information required in the spaces provided. No extra time will be given for sticking on the barcode label after the ‘Time is up’ announcement. 2. When told to open this book, you should check that all the questions are there. Look for the words ‘END OF PAPER’ after the last question. 3. All questions carry equal marks. 4. ANSWER ALL QUESTIONS. You are advised to use an HB pencil to mark all the answers on the Answer Sheet, so that wrong marks can be completely erased with a clean rubber. You must mark the answers clearly; otherwise you will lose marks if the answers cannot be captured. 5. You should mark only ONE answer for each question. If you mark more than one answer, you will receive NO MARKS for that question. 6. No marks will be deducted for wrong answers. 香港考試及評核局 保留版權 Not to be taken away before the end of the examination session Hong Kong Examinations and Assessment Authority All Rights Reserved 2012 PP-DSE-MATH-CP 2−1 1 There are 30 questions in Section A and 15 questions in Section B. The diagrams in this paper are not necessarily drawn to scale. Choose the best answer for each question. Section A 1. 2. 3. x 3 (2 x + x) = A. 3x 4 . B. 2x 5 . C. 3x 5 . D. 2x 6 . If 3a + 1 = 3(b − 2) , then b = A. a +1 . B. a+3 . C. a+ 7 . 3 D. a− 5 . 3 p2 − q2 − p − q = A. ( p + q)( p − q − 1) . B. ( p + q )( p + q − 1) . C. ( p − q)( p − q + 1) . D. ( p − q)( p + q − 1) . PP-DSE-MATH-CP 2−2 2 4. 5. 6. 7. Let m and n be constants. If m( x − 3) 2 + n( x + 1) 2 ≡ x 2 − 38 x + 41 , then m = A. −4 . B. −1 . C. 3. D. 5. Let f ( x) = x 4 − x 3 + x 2 − x + 1 . When f ( x) is divided by x + 2 , the remainder is A. −2 . B. 0. C. 11 . D. 31 . Let k be a constant. If the quadratic equation 3x 2 + 2k x − k = 0 has equal roots, then k = A. −3 . B. 3. C. −3 or 0 . D. 0 or 3 . In the figure, the x-intercepts of the straight lines L1 and L 2 are 5 while the y-intercepts of the straight lines L2 and L3 are 3 . Which of the following are true? I. The equation of L1 is x = 5 . y 3 . 5 III. The point (2 , 3) lies on L3 . II. The slope of L2 is A. I and II only B. I and III only C. II and III only D. I , II and III PP-DSE-MATH-CP 2−3 L3 3 O 5 x L1 3 L2 Go on to the next page 8. 9. 10. 11. The figure shows the graph of y = a x 2 − 2 x + b , where a and b are constants. Which of the following is/are true? y I. a>0 II. b < 0 III. ab < 1 O x 2 y = a x − 2x + b A. I only B. II only C. I and III only D. II and III only The solution of 4 x > x − 3 or 3 − x < x + 7 is A. x > −2 . B. x < −2 . C. x > −1 . D. x < −2 or x > −1 . John buys a vase for $ 1 600 . He then sells the vase to Susan at a profit of 20% . At what price should Susan sell the vase in order to have a profit of 20% ? A. $ 2 240 B. $ 2 304 C. $ 2 400 D. $ 2 500 If the circumference of a circle is increased by 40% , then the area of the circle is increased by A. 18% . B. 20% . C. 40% . D. 96% . PP-DSE-MATH-CP 2−4 4 12. 13. Let α and β be non-zero constants. If (α + β ) : (3α − β ) = 7 : 3 , then α : β = A. 5:9 . B. 9:5 . C. 19 : 29 . D. 29 : 19 . If z varies directly as x and inversely as y 2 , which of the following must be constant? A. B. C. D. 14. 15. x y2z z xy 2 yz x2 xz y2 0.009049999 = A. 0.00905 (correct to 3 decimal places). B. 0.00905 (correct to 3 significant figures). C. 0.00905 (correct to 6 decimal places). D. 0.00905 (correct to 6 significant figures). In the figure, O is the centre of the sector OABC . If the area of ∆OAC is 12 cm 2 , find the area of the segment ABC . A. 3(π − 2) cm 2 B. 3(π − 1) cm 2 C. 6(π − 2) cm 2 D. 6(π − 1) cm 2 C B O PP-DSE-MATH-CP 2−5 5 A Go on to the next page 17. The figure shows a right circular cone of height 8 cm and slant height 17 cm . Find the volume of the circular cone. A. 255π cm3 B. 345π cm 3 C. 480π cm 3 D. 600π cm3 17 cm 8 cm 16. In the figure, ABCD is a rectangle. E is the mid-point of BC . F is a point lying on CD such that DF = 2CF . If the area of ∆CEF is 1 cm 2 , then the area of ∆ AEF is A. 2 cm 2 . B. 3 cm 2 . C. 4 cm 2 . D. 6 cm 2 . F D E A 18. C B In the figure, AB = 4 cm , BC = CD = DE = 8 cm and FG = 9 cm . Find the perimeter of ∆ AEH . A. 60 cm B. 74 cm C. 150 cm D F B D. E G C 164 cm A PP-DSE-MATH-CP 2−6 6 H 19. In the figure, AB = BC and D is a point lying on BC such that CD = DE . If AB // CE , find ∠CDE . A. 52° B. 58° C. 64° D. 76° E C D 64° B A 20. In the figure, O is the centre of the semi-circle ABCD . AC and BD intersect at E . If AD // OC , then ∠ AED = A. 48° . B. 55° . C. 57° . D. 66° . C D E 24° A 21. B O In the figure, O is the centre of the circle ABCD . If AB = BC = 2 CD , then ∠BCD = A. 64° . B. 87° . C. 93° . D. 116° . B C O 70° A D PP-DSE-MATH-CP 2−7 7 Go on to the next page 22. In the figure, ABCD is a square. F is a point lying on AD such that CF // BE . If AB = AE , find ∠ ABF correct to the nearest degree. E A. 17° B. 18° C. 22° D. 26° F D 112° C 23. 24. For 0° ≤ θ ≤ 90° , the least value of A. 5. B. 6. C. 10 . D. 15 . A 30 2 3 sin θ + 2 sin 2 (90° − θ ) B is Which of the following parallelograms have rotational symmetry and reflectional symmetry? I. II. 6 III. 6 8 A. I and II only B. I and III only C. II and III only D. I , II and III PP-DSE-MATH-CP 2−8 6 6 6 8 25. 26. 27. If the point (− 2 , − 1) is reflected with respect to the straight line y = −5 , then the coordinates of its image are A. (−8 , − 1) . B. (− 2 , − 9) . C. (− 2 , 11) . D. (12 , − 1) . The coordinates of the points A and B are (1, − 3) and (−5 , 7) respectively. If P is a point lying on the straight line y = x + 2 such that AP = PB , then the coordinates of P are A. (− 2 , 0) . B. (− 2 , 2) . C. ( 0 , 2) . D. (3 , 5) . The equation of a circle is 2 x 2 + 2 y 2 + 8 x − 12 y + 3 = 0 . Which of the following are true? I. The coordinates of the centre of the circle are (− 2 , 3) . II. The radius of the circle is 7 . III. The point (2 , 3) lies outside the circle. A. I and II only B. I and III only C. II and III only D. I , II and III PP-DSE-MATH-CP 2−9 9 Go on to the next page Section B y 31. O 6 2 x −2 y = f ( x) The figure above shows the graph of y = f ( x) . If 2 f ( x) = g( x) , which of the following may represent the graph of y = g( x) ? A. B. y O 1 3 x y O 2 y = g ( x) y = g ( x) D. y O 1 3 x y O −4 2 6 x −4 y = g ( x) 32. x −1 −1 C. 6 y = g ( x) B000000002316 = A. 11×1610 + 23 . B. 11×1610 + 35 . C. 12 × 1611 + 23 . D. 12 × 1611 + 35 . PP-DSE-MATH-CP 2−11 11 Go on to the next page 33. 34. 35. 36. If the roots of the quadratic equation x 2 − k x + 3 = 0 are α and β , then α 3 + β 3 = A. k3 . B. k 3 − 3k . C. k 3 − 9k . D. k 3 − 12k . If x is a real number, then the real part of ( x + 3i )( 3 + i ) is A. 3x . B. x+3 . C. 3x + 3 . D. 3x − 3 . The nth term of a sequence is 2n + 3 . If the sum of the first m terms of the sequence is less than 3 000 , then the greatest value of m is A. 52 . B. 53 . C. 56 . D. 57 . Let b > 1 . If a = log 12 b , then 1 . 12 A. log b B. log b 12 . C. log12 D. 1 . log b 12 PP-DSE-MATH-CP 2−12 1 = a 1 . b 12 37. 38. The graph in the figure shows the linear relation between log 3 t and log 3 x . If x = k t a , then k = A. 1 . 81 B. 81 . C. −4 . 5 D. −5 . 4 log 3 x −5 log 3 t −4 Let a be a constant and −90° < θ < 90° . If the figure shows the graph of y = a sin( x° + θ ) , then A. a = −2 and θ = − 45° . y B. a = −2 and θ = 45° . 2 −2 C. a = 2 and θ = − 45° . D. a = 2 and θ = 45° . O 39. O x 135 The figure shows a right prism ABCDEF with a right-angled triangle as the cross-section. A , B , E and F lie on the horizontal ground. G is a point lying on AB such that AG : GB = 5 : 3 . If ∠DAE = a , ∠CBF = b , ∠CGF = c and ∠ DGE = d , which of the following is true? A. a>c>d B. a>d >c C. c>b>d D. c>d >b D F E d a c G A PP-DSE-MATH-CP 2−13 C 13 b B Go on to the next page 40. In the figure, A is the common centre of the two circles. BC is a chord of the larger circle and touches the smaller circle at D . AD produced meets the larger circle at E . F is a point lying on the smaller circle such that E , D , A and F are collinear. If BC = 24 cm and DE = 8 cm , then EF = A. 13 cm . B. 16 cm . C. 18 cm . D. 20 cm . E D C B A F 41. 42. If the straight line x − y = 0 and the circle x 2 + y 2 + 6 x + ky − k = 0 do not intersect with each other, find the range of values of k . A. 2 < k < 18 B. −18 < k < −2 C. k < 2 or k > 18 D. k < −18 or k > −2 Let O be the origin. If the coordinates of the points A and B are (18 , − 24) and (18 , 24) respectively, then the x-coordinate of the orthocentre of ∆ OA B is A. −14 . B. 10 . C. 12 . D. 25 . PP-DSE-MATH-CP 2−14 14 Sample Paper Answer Paper 1 1. 15. 16. 2. 4. 5. 6. 6a a 1 (3m 2n)( m n) (3m 2n 1)( m n) b (a) (b) (a) (b) (a) (b) 7. 8. $400 15% 24 3 54 cm 3 (b) 17. 18. 9.6 (a) (b) (i) (ii) 19. 10 21 11 21 BC 20 2 cm, BD 10 6 cm 46.6 (a) (ii) Yes 2x y 1 0 (b) 19 (a) (b) 9. (a) (b) 10. (a) (b) (i) (ii) A’ = (5, 2), A” =(2, 5) No 60 No 5x + 2 a = 2, b = –1 (a) (b) 12. (a) (b) (c) 13. (a) (b) (i) (ii) 14. (a) (b) (i) (ii) (c) $1 644 2.75 m Part I 8:26 15 m/s 11. (a) x7 y4 3. 21 2 , –3, 1 5 3x 4 y 48 0 Parallel 3x 4 y 32 0 Median = 62%, Mean = 60% 58% a = 63, b = 57 Yes Paper 2 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. C C C A C D D A C A C B D C B 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. B A B A A B C B D A D D B D B 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. B D A A D C C A A C B B D B D Practice Paper Answer Paper 1 1. 16. m 26 n9 2. 3. (a) (b) $585 4. 5. 6. (a) (b) (a) (b) 9. (a) (b) 10. (a) (b) (i) (ii) 12. 13. 14. 15. 17. 18. Yes 196 18 A’ = (3, 4), B’= (5, –2) No 60 No 5x + 2 a = 2, b = –1 68 609 530 (b) 609 1 2 (a) i 5 5 (b) (i) p –4 and q 20 (ii) r 16 (a) (b) (c) 4 L 5 7. 8. 11. 2b 5 3b (3x 7 y ) 2 (3x 7 y)( 3x 7 y 2) a (a) 19. (a) (b) Paper 2 1. A 2. C 3. A 4. D (a) $1 644 5. D (b) 2.75 m 6. C (a) Part I 7. B (b) 8:26 8. D (c) Billy 9. A (a) 8 10. B (b) (i) 45 11. D (ii) No 12. A ∆BCD ~ ∆OAD (a) 13. A 14. B (b) (i) (0 , 4) (ii) ( x 3) 2 ( y 2) 2 13 or x 2 y 2 6 x 415. y C0 (a) (b) 2 , –3, 1 5 3 Decrease 4 19 cm 71.9 CPD increases as P moves from A to Q and decreases as P moves from Q to B . (i) (ii) (iii) Disagree 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. D C A D C C B C D B D B A B C 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. D B C D A B A C A C B A B D C
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