Mid-year Examination 2015/16 Mathematics Form Three Name : Time allowed: 2 hrs. 15 mins. Class : ( ) Paper Total: 100 marks Instructions 1. Answer ALL questions. 2. All answers in Section A should be marked in pencil on the MC answer sheet provided or marks will be deducted. 3. Answer Sections B and C with blue/black ball pen in the Answer Book. 4. All diagrams and graphs must be drawn neatly in pencil. 5. Unless otherwise specified, numerical answers should either be exact or correct to 3 significant figures. 6. Anything written on the rough work sheet provided will not be marked. Section A Multiple choice (30 marks) 1. Factorize a 2 5a b2 5b . A. (a b)(a b 5) B. (a b)(a b 5) C. (a b)(a b 5) D. (a b)(a 5)(b 5) 2. 3. ( x 1)( x 2 A. B. C. D. x 1) 3 ( x 1) . x3 1 . x3 x 2 x 1. x 3 2 x 2 2 x 1. ( 3x A. B. C. D. y ) 2 ( 3x 0. 2 y2 . 6 xy . 12 xy . y)2 4. The cost of a television was $3500 last year. Suppose that its value depreciates by 40% every year. Find its value next year. A. $2 100 B. $1 400 C. $1 260 D. $560 5. If the price of a book is increased by 60% and then decreased by 40%, find the percentage change in the price of the book. A. 4% B. 20% C. 24% D. 32% F3_1516_ME.docx 6. Factorize ac A. (a b)(c B. (a b)(c C. (a b)(c D. (a b)(c 7. Which of the following values of x satisfies the inequality 4 3x 20 ? A. 5.34 B. 5.333 C. 5.33 D. 5.3 8. Solve x 3 A. B. C. D. 9. x x x x bc ad d e) d e) d e) d e) bd ae be . x . 2 2 2 1 1 If x y and k 0 , which of the following must be true? kx ky I. x k y k II. III. x y 2 k k2 A. I and II only B. I and III only C. II and III only D. I, II and III 1/8 Mid-year Examination 2015/16 Mathematics 10. Which of the following are factors of x3 x2 6x ? x I. II. x 2 III. x 3 A. I and II only B. I and III only C. II and III only D. I, II and III 14. Mary has 18 coins. Each of them is either $2 or $5 coins. The total value of the coins is not greater than $68. What is the greatest number of $5 coins Mary can have? A. 7 B. 8 C. 9 D. 10 11. If x < y < 0, which of the following must be true? x y 0 I. y x 0 II. III. x 2 y 2 15. The total surface area of a hemisphere is 72 cm2. Find its radius, correct to 3 significant figures. A. 2.39 cm B. 2.76 cm C. 3.25 cm D. 3.39 cm A. B. C. D. I and II only I and III only II and III only I, II and III 12. A sum of $60 000 is deposited at an interest rate of 5% p.a. for 4 years, compounded yearly. Find the amount correct to the nearest dollar. A. $72 000 B. $72 930 C. $73 193 D. $73 254 13. Which of the following 2-D pattern(s) can be folded into a solid? I. II. III. A. B. C. D. I only II only III only II and III only F3_1516_ME.docx 16. In the figure, VABC is a regular tetrahedron. The height and volume of VABC are 9 cm and 486 cm3 respectively. Find the total surface area of VABC. A. B. C. D. 54 cm2 162 cm2 216 cm2 648 cm2 17. The running cost of a department store consists of rent, 50%; labour, 30% and electricity, 20%. If the rent decreases by 8%, the labour cost increases by 3% and the electricity cost remains unchanged, what is the percentage change in the running cost? A. 4.9% B. 3.1% C. +3.1% D. +4.9% 18. Mr. Lam wants to let his flat out and earns an annual income of $128 000 after paying property tax. How much rent should he charge each month if the property tax rate is 16%? (Give the answer correct to the nearest dollar.) A. $6667 B. $8333 C. $12 232 D. $12 698 2/8 Mid-year Examination 2015/16 Mathematics 19. The base radius and the slant height of a right circular cone are 6 cm and 10 cm respectively. Find the volume of the cone. A. 60 cm3 B. 80 cm3 C. 96 cm3 D. 120 cm3 20. If x is an even number and 15 2 x find the least possible value of x. A. 6 B. 8 C. 10 D. 12 21. Factorize bc A. (a b)(b B. (b a)(c C. (b a)(b D. (a b)(a ac a a c b 2, (a b) 2 . c) b) a) c) 23. If the angle and radius of a sector are decreased by x% and 50% respectively so that its area is decreased by 85%, find x. A. 35 B. 40 C. 60 D. 65 The sector in the following figure is folded to form a right circular cone. Find the base radius of the cone. A. B. C. D. 2 cm 3 cm 6 cm 12 cm A solid sphere of radius 5 cm is cut into two equal hemispheres. Find the percentage increase in the total surface area. A. 15% B. 20% C. 25% D. 50% For questions 26 30, refer to the cuboid ABCDEFGH shown in the following figure. 26. 22. The volume of a square-based pyramid is 150 cm3. If the height of the pyramid is 6 cm, find the length of a side of the base. A. 5 cm B. 3 5 cm C. 5 3 cm D. 25 cm 24. 25. How many planes of reflection are there in the cuboid? A. 2 B. 3 C. 6 D. 7 27. The projection of D to the plane BCHG is A. C. B. E. C. G. D. H. 28. The angle between planes ABCD and ADEF is A. BAE . B. CDE . C. CAF . D. BDE . 29. The angle between the line FC and plane BCHG is A. CHF . B. CGF . C. FCG . D. FCH . 30. The angle between planes CDEH and CDFG is A. FDG . B. GDE . C. GCE . D. FDE . End of Section A F3_1516_ME.docx 3/8 Mid-year Examination 2015/16 Mathematics Section B Short questions (30 marks) 1. (a) Solve the inequality 1 2 x 3 1 x 9 3 4 and represent the solution graphically. (b) Find the smallest integral value of x satisfying the inequality in (a). 2. 3. (1 mark) The population of a city increases by 10% every year. (a) Find the percentage increase in the population over the next 3 years. (3 marks) (b) The population will be 322 102 five years later. Find the population this year. (3 marks) (a) Factorize 27 x 3 (2 marks) (b) Hence simplify 4. (3 marks) y3 . 27 x 3 y 3 9 x 2 y 3xy 2 . 3x y (4 marks) Figure 1(a) shows a net which can be folded into a right pyramid with a square base as shown in Figure 1(b), where AB = 12cm. A 12cm B 10cm Figure 1(b) Figure 1(a) (a) Find the total surface area of the pyramid. (b) Find the height of the pyramid. (c) Find the volume of the pyramid. 5. See page 7. 6. See page 8. F3_1516_ME.docx (2 marks) (2 marks) (2 marks) 4/8 Mid-year Examination 2015/16 Mathematics Section C Long questions (40 marks) 7. Figure 6 shows an inverted right circular conical container. The height and the base radius of the container are 10 cm and 4 cm respectively. It is filled with hot water to a height of 9 cm. 4 cm Figure 6 9 cm 10 cm (a) Find the capacity of the circular conical container in terms of . (2 marks) (b) Find the curved surface area of the circular conical container. (3 marks) (c) By finding the radius of the water surface, find the volume of water in terms of . (3 marks) (d) A sugar block is in a shape formed by a cylinder and two hemispheres of common bases. Its cross-section is shown in Figure 7. The length of the block is 2.5 cm and the diameter of the hemisphere is 1 cm. Figure 7 1 cm 2.5 cm (i) Find the volume of a sugar block in terms of . (ii) At most how many sugar blocks can be added to the conical container with hot water as shown in Figure 6 so that it will not overflow? (6 marks) 8. Mr. Tsui is planning to rent a car for a trip of 5 days. He has checked online the rates offered by two car rental companies, A and B, and the rates are shown in the following table. Company Basic fee ($/day) Free distance (km/day) Extra distance ($/km) A 300 200 1.9 B 500 300 1.6 use up the free distance provided every day. (a) Which company should he choose if the total distance travelled is 1500 km? Explain your answer. (b) Let x km be the total distance travelled by the rental car. (i) Show that the total cost of renting from Company A is given by $( 1.9 x 400 ). (3 marks) (ii) Find, correct to the nearest km, the maximum total distance travelled such that the total cost of renting from Company A is cheaper than that from Company B. (4 marks) (c) Mr Tsui has estimated that his total distance travelled is 2000 km. Company C then offers him a car rental package of $5400 including fuel. If the cost of the fuel is $1.2/km, Mr Tsui claims that his total cost spent on fuel and renting a car from Company B is 5% higher than that in renting from Company C. Do you agree? Explain your answer. (3 marks) F3_1516_ME.docx 5/8 Mid-year Examination 2015/16 Mathematics For questions 9 10, refer to the allowances and tax rates in the current financial year of a city shown in the following tables. Allowances: ($) Basic Allowance 130 000 260 000 Child Allowances: For each of the 1st to 9th child 80 000 For each child born during the year, the Child Allowance 80 000 will be increased by Dependent Parent Allowance Aged 60 or above 30 000 Aged 55 or above but below 60 15 000 Additional Dependent Parent Allowance Aged 60 or above 30 000 Aged 55 or above but below 60 15 000 Net Chargeable Income ($) On the first 35 000 On the next 35 000 On the next 35 000 Remainder Standard tax rate Rate 5% 10% 16% 23% 20% 9. Mr. and Mrs. Lee are citizens in the city and have a daughter born in this financial year. Mr. Lee is a policeman while Mrs Lee is a housewife. They live with Mrs. Lee -year-old father and 54year-old mother. They have to pay $8 450 for the salaries tax this year when assessed on the net chargeable income. (a) Find the total allowances of the Lee family. (b) Find their net chargeable income. (c) Find Mr Lee . 10. Let $P be the total income of a citizen, who is single, in the city. (a) If the citizen has to pay salaries tax at the standard rate, (i) express the amount of his salaries tax in terms of P; (ii) show that the total income of the citizen is at least $1 440 000. (3 marks) (2 marks) (2 marks) (3 marks) (b) Andy is a citizen in the city and is single. His total income is $1 480 000 this year and thus the standard rate of tax is used to assess his salaries tax. In order to pay his salaries tax, he begins to save money 12 months before the due day of paying salaries tax. A deposit of $145 000 is saved in a bank on the first day of the 1st and the 7th month at an interest rate of 4% per annum, compounded quarterly. (i) Find the amount in his account after 6 months before his second deposit. (ii) Find the amount in his account after 12 months, correct to the nearest dollar. (iii) Will he have enough money to pay his salaries tax on the due day? Explain your answer. (6 marks) End of Paper F3_1516_ME.docx 6/8