HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 1999 MATHEMATICS PAPER 2

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1 99-CE MATHS Paper HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 999 MATHEMATICS PAPER 5 am 45 pm (½ hours) Subject Code 80 Read carefully the instructions on the Answer Sheet and insert the information required (including the Subject Code) in the spaces provided When told to open this book, check that all the questions are there Look for the words END OF PAPER after the last question 3 ANSWER ALL QUESTIONS All the answers should be marked on the Answer Sheet 4 Note that you may only mark ONE answer to each question Two or more answers will score NO MARKS 5 All questions carry equal marks No marks will be deducted for wrong answers 香港考試局保留版權 Hong Kong Eaminations Authority All Rights Reserved CE-MATHS

2 FORMULAS FOR REFERENCE SPHERE Surface area = 4π r Volume = 4 π r 3 3 CYLINDER Area of curved surface = π rh Volume = π r h CONE Area of curved surface = π rl Volume = π r h 3 PRISM Volume = base area height PYRAMID Volume = base area height 3 99-CE-MATHS

3 There are 36 questions in Section A and 8 questions in Section B The diagrams in this paper are not necessarily drawn to scale Section A If f( ) =, then f(a ) = A a a B a 3a C a 3a D a E a y + y = A ( y)( y ) B ( y)( + y ) C ( y)( + y + ) D ( + y)( y ) E ( + y)( y + ) 99-CE-MATHS 3 Go on to the net page

4 3 If + b a = b A B, then b = a a a C D a + a a a + E a a + 4 If 4 = a, then 6 = A 4 a B C D E a 4 a a a 4 99-CE-MATHS 4 3

5 5 In the figure, the graph of y = 6 + k A k 0 y touches the -ais Find k B k 9 C k = 9 D k = 0 E k = 9 O 6 If (3 )( a) 3 + b, then A a =, b = B a =, b = 7 C a =, b = 5 D a =, b = 5 E a =, b = 7 99-CE-MATHS 5 4 Go on to the net page

6 7 Solve > 0 A < or > B < 6 or > 4 C < or > D < < E < < y = If y = + 3, then A = B = or 5 C = or D = 5 or E = 5 or 8 99-CE-MATHS 6 5

7 9 Which of the following may represent the graph of y = 3 8? A y B y O O C y D y O O E y O 99-CE-MATHS 7 6 Go on to the net page

8 0 The n-th term of an arithmetic sequence is + 5n Find the sum of the first 00 terms of the sequence A 50 B 450 C 500 D 5450 E 5700 In a class, students study either History or Geography, but not both If the number of students studying Geography is 50% more than those studying History, what is the percentage of students studying History? A 5% B 33 % 3 C 40% D 60% E 66 % 3 99-CE-MATHS 8 7

9 If : y = 3 : 4 and + 5y = 598, find A 3 B 6 C 69 D 78 E 04 3 If Australian dollar is equivalent to 469 HK dollars and 00 Japanese yen are equivalent to 535 HK dollars, how many Japanese yen are equivalent to Australian dollar? Give your answer correct to the nearest Japanese yen A 4 B 5 C 88 D 4 E CE-MATHS 9 8 Go on to the net page

10 4 Let m be a positive integer Which of the following must be true? I m is even II m ( m +) is even III m ( m + ) is even A I only B II only C III only D I and III only E II and III only 5 In the figure, the bearing of B from C is A A N5 E B N65 E C N85 E D S5 W N N 0 30 E S85 W B C 99-CE-MATHS 0 9

11 6 Which of the following may represent the graph of y = cos for 0 90? A y B y O 90 O 90 C y D y O 90 O 90 E y O CE-MATHS 0 Go on to the net page

12 7 In the figure, find correct to 3 significant figures A 8 B 9 C 35 D 53 E cm cm AC 8 In the figure, = AB A B C C D A B E CE-MATHS

13 9 In the figure, find correct to decimal place A 50 B 84 cm C 5 D 4 cm E 66 cm 0 In the figure, ABCD is a parallelogram Find ABC correct to the nearest degree A 83 D C B 97 C 04 6 cm 4 cm D 4 E 39 A 5 cm B 99-CE-MATHS 3 Go on to the net page

14 In the figure, a square is inscribed in a circle with radius cm Find the area of the shaded region A ( π ) cm B ( π ) cm C ( π ) cm D ( π ) cm cm E ( π ) cm The figure shows a right prism Find its total surface area A 04 cm B 08 cm 5 cm C 4 cm 4 cm D 0 cm E 40 cm 8 cm 4 cm 99-CE-MATHS 4 3

15 3 In the figure, a cylindrical vessel of internal diameter 6 cm contains some water A steel ball of radius cm is completely submerged in the water Find the rise in the water level 6 cm 6 cm A 3 cm 7 B 8 cm 7 C 6 cm 9 D 4 cm 9 E 8 cm 3 99-CE-MATHS 5 4 Go on to the net page

16 4 In the figure, the solid consists of a right circular cone and a hemisphere with a common base Find the volume of the solid A 30π cm 3 B 33π cm 3 C 48π cm 3 D 54π cm 3 E 7π cm 3 4 cm 3 cm 5 In the figure, ABCD is a semicircle Find the area of the shaded region correct to the nearest 00 cm A 533 cm D C B 87 cm C 67 cm D 33 cm A 5 0 cm B E 07 cm 6 In the figure, O is the centre of the circle Find A B 0 C 4 D 40 E 60 O CE-MATHS 6 5

17 7 In the figure, AB is a diameter of the circle Find A 6 B 3 D C C 38 A 6 B D 5 E 64 8 In the figure, ACD and ECB are straight lines If EAC = CAB and EA = EB, find A E D B C 44 D 46 C E 68 A B 99-CE-MATHS 7 6 Go on to the net page

18 9 In the figure, ABCDE is a regular pentagon and ABFG is a square Find A 8 D B 7 C 30 E G F C D 36 E 45 A B 30 In the figure, AEB and ADC are straight lines Find ED A 0 cm 3 A B 40 cm 3 8 cm C 3 cm D D 40 cm E 5 cm E 80 cm B 5 cm C 99-CE-MATHS 8 7

19 3 A( 4, ) and B(, 3) are two points C is a point on the y-ais such that CB AC = Find the coordinates of C 3 A (, ) B (, 0) C (, 0) D (0, ) E (0, ) 3 In the figure, OABC is a parallelogram If the equation of OC is y = 0 and the length of CB is 3, find the equation of AB A y 3 = 0 y B y 3 = 0 C y + 3 = 0 C B D y 6 = 0 E y + 6 = 0 O A 99-CE-MATHS 9 8 Go on to the net page

20 33 Find the median and mode of the ten numbers 6, 8, 3, 3, 5, 5, 5, 7, 7, A median = 5, mode = 5 B median = 5, mode = 55 C median = 55, mode = 5 D median = 55, mode = 6 E median = 6, mode = 5 34 A student scored 50 marks in a test and the corresponding standard score is 05 If the mean of the test scores is 60 marks, find the standard deviation of the scores A 0 marks B 5 marks C 95 marks D 0 marks E 0 marks 99-CE-MATHS 0 9

21 35 Two cards are drawn randomly from four cards numbered,, 3 and 4 respectively Find the probability that the sum of the numbers drawn is odd A B C D E Tom and Mary each throws a dart The probability of Tom s dart hitting the target is while that of Mary s is Find the probability of only one 3 5 dart hitting the target A B C D E CE-MATHS 0 Go on to the net page

22 Section B 37 In the figure, a right circular cone is divided into two parts X and Y by a plane parallel to the base such that the lengths of their slant edges are 4 cm and 3 cm respectively Find the ratio of the curved surface areas of X and Y X 4 cm Y 3 cm A 6 : 9 B 6 : 33 C 6 : 49 D 64 : 7 E 64 : It is given that F( ) = 4 + a + b F() is divisible by When it is divided by +, the remainder is Find a and b A a = 5, b = 0 B a =, b = C a = 3, b = 6 D a = 4, b = 7 E a = 7, b = 0 99-CE-MATHS

23 39 If log y = + log, then A y = 0 B C D E y = 00 + y = ( 0 + ) y = 0 y = = 3 A ( )( + )( + 3) B ( )( + )( + 3) C 5 ( 3)( )( + ) D 5 ( 3)( )( + ) E ( 3)( )( + ) 99-CE-MATHS 3 Go on to the net page

24 4 The method of bisection is used to find the root of sin + = 0 starting with the interval [0, ] After the first approimation, the interval which contains the root becomes [0, ] Find the interval which contains the root after the third approimation A [0, 05] B [05, 075] C [05, 075] D [05, ] E [075, ] 4 John goes to school and returns home at speeds km/h and ( + ) km/h respectively The school is km from John s home and the total time for the two journeys is 54 minutes Which of the following equations can be used to find? A B C D E + + = + = + [ [ ( + )] = 4 4 = + ( + )] + ( + ) = CE-MATHS 4 3

25 43 In the figure, find the point (, y) in the shaded region (including the boundary) at which b ay + 3 attains its greatest value A (0, 0) B ( a, b) ( a, b) y (a, b) (b, a) C (a, b) D (b, a) O E (b, a) (b, a) 44 The sum of the first two terms of a geometric sequence is 3 and the sum to infinity of the sequence is 4 Find the common ratio of the sequence A B C D E or 99-CE-MATHS 5 4 Go on to the net page

26 45 It is given that y varies inversely as then y is 3 If is increased by 00%, A increased by 800% B increased by 700% C decreased by 300% D decreased by 875% E decreased by 5% cos(90 A) cos( A) 46 = sin(360 A) A cos A B cos A C sin A D E cos A sin A cos A sin A 47 If 0 θ π, solve (cos θ 3)(3sinθ ) = 0 correct to 3 significant figures A 0730 or 3 B 0730 or 4 C 0730 or 387 D 0730 or 6 E 0734 or 4 99-CE-MATHS 6 5 Ç All Rights Reserved 999

27 48 The figure shows five sectors Which of the marked angles measures radians? A a B b C c D d E e 99-CE-MATHS 7 6 Ç All Rights Reserved 999 Go on to the net page

28 49 In the figure, ABCDEFGH is a rectangular block Find the inclination of EM to the plane ABCD correct to the nearest degree A 3 B 5 E 4 cm F G H C 65 D 7 E 75 D 36 cm C 9 cm A M B 50 In the figure, AT is tangent to the circle at T and ABC is a straight line Find AT A 9 cm C B cm 6 cm C 5 cm D 6 cm B 9 cm E 0 cm T A 99-CE-MATHS 8 7 Ç All Rights Reserved 999

29 5 In the figure, find the equation of the circle with AB as a diameter A + y + y 3 = 0 y B + y + y 3 = 0 A(, 3) C + y + y 3 = 0 D + y + y 3 = 0 O E + y 5 = 0 B(4, 5) 5 The figure shows a circle centred at C and passing through O(0, 0), A(6, 0) and B(0, 8) Which of the following must be true? I C lies on the line + y = 6 8 II The radius of the circle is 0 III OC is perpendicular to AB y B A I only B II only C C I and II only D I and III only O A E I, II and III 99-CE-MATHS 9 8 Ç All Rights Reserved 999 Go on to the net page

30 53 Two circles with equations ( + ) + ( y + ) = 5 and ( ) + ( y 8) = 00 touch each other eternally at a point P Find the coordinates of P A ( 3, ) 7 4 B (, ) 5 5 C (3, ) 7 D (5, ) E (7, 5) 54 In the figure, ABCD is a rectangle M is the midpoint of BC and AC intersects MD at N Area of NCD : area of ABMN = A : B : 3 A D C : 3 D : 5 N E 4 : 7 B M C END OF PAPER 99-CE-MATHS 30 9 Ç All Rights Reserved 999

31 999 Mathematics (Paper ) Question No Key Question No Key A 3 E B 3 D 3 D 33 C 4 B 34 E 5 E 35 E 6 C 36 C 7 A 37 B 8 D 38 E 9 A 39 E 0 D 40 E C 4 C C 4 B 3 C 43 D 4 B 44 E 5 E 45 D 6 A 46 A 7 B 47 B 8 D 48 E 9 B 49 C 0 B 50 C A 5 A E 5 A 3 A 53 C 4 A 54 D 5 D 6 D 7 C 8 B 9 B 30 A