HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 2000 MATHEMATICS PAPER 2

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1 000-CE MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 000 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 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 Examinations Authority All Rights Reserved CE-MATH

2 FORMULAS FOR REFERENCE SPHERE Surface area = 4π r Volume = 4 π r CYLINDER Area of curved surface = π rh Volume = π r h CONE Area of curved surface = π rl Volume = π r h PRISM Volume = base area height PYRAMID Volume = base area height 000-CE-MATH 保留版權 All Rights Reserved 000

3 There are 6 questions in Section A and 8 questions in Section B The diagrams in this paper are not necessarily drawn to scale Section A h If A = ( a + b), then b = A A ah B ( A a) h C D A a h A a h A E a h Factorize x x xy + y A ( x y)( x ) B ( x y)( x + ) C ( x + y)( x ) D ( x )( x + y) E ( + x)( y x) 000-CE-MATH Go on to the next page 保留版權 All Rights Reserved 000

4 Simplify A B C D E ( a b ( a b ab a a a a b ) ) 4 6 b b b Let f( x ) = x + ax 7 If f( ) = 0, find f( ) A 7 B C D E 000-CE-MATH 4 保留版權 All Rights Reserved 000

5 y = x 5 If y = x, then y = A 4 B 0 C D 0 or 8 E 4 or 4 6 Find the values of x which satisfy both x + > 0 and x < A x > B C D E x > x > < x < < x < 000-CE-MATH 5 4 Go on to the next page 保留版權 All Rights Reserved 000

6 7 In the figure, a square of side x cm is cut into 9 equal squares If the total perimeter of the 9 small squares is 7 cm more than the perimeter of the original square, then x = x cm x cm A 6 B 8 C 9 D E 8 8 The figure shows a trapezium of area 6 cm Find x A cm B C 4 x cm D 6 E x cm 000-CE-MATH 6 5 保留版權 All Rights Reserved 000

7 9 Let f( x ) = x x 5x + 6 It is known that f( ) = 0 f(x ) can be factorized as A ( x ) ( x + 6) B ( x )( x + )( x + 6) C ( x )( x )( x + ) D ( x )( x + )( x ) E ( x + )( x )( x ) 0 If x + ax + 7 ( x ) + b, then A a =, b = 5 B a =, b = 7 C a = 4, b = D a = 0, b = 5 E a = 0, b = CE-MATH 7 6 Go on to the next page 保留版權 All Rights Reserved 000

8 Which of the following may represent the graph of y = tan x for 0 x 90? A y B y O 90 x O 90 x C y D y O 90 x O 90 x E y O 90 x 000-CE-MATH 8 保留版權 All Rights Reserved 000 7

9 In the figure, ABCD is a rectangle formed by four squares each of area cm DB is a diagonal Find the area of the shaded region A 9 cm 0 A D B 7 cm 8 B C C 5 cm 6 D 4 cm 5 E cm 4 In the figure, the areas of the two triangles are equal Find θ θ 0 cm 8 cm 5 cm 0 4 cm A 7 (correct to the nearest 0 ) B 75 (correct to the nearest 0 ) C 45 (correct to the nearest 0 ) D 5 E CE-MATH 9 8 Go on to the next page 保留版權 All Rights Reserved 000

10 4 A man bought two books at $0 and $70 respectively He sold the first one at a profit of 0% and the second one at a loss of 0% On the whole, he A lost % B lost 0% C gained % D gained 0% E gained % 5 The st and 0th terms of an arithmetic sequence are and 9 respectively The 0th term of the sequence is A 56 B 58 C 59 D 60 E CE-MATH 0 9 保留版權 All Rights Reserved 000

11 6 Which of the following could be a geometric sequence/geometric sequences? I,, 5, 7, II 9, 99, 999, 9 999, III 0, 00, 000, 0 000, A III only B I and II only C I and III only D II and III only E I, II and III 7 In the figure, find the area of ABC A B 5 y C C 6 D 0 E 5 O A B y = x x + y = CE-MATH 0 Go on to the next page 保留版權 All Rights Reserved 000

12 8 Consider the three straight lines L : 6 x + 4y = 0, L : y = x + 4 and L : 6 x 4y + = 0 Which of the following is/are true? I L // L II L // L III L L A I only B II only C III only D I and III only E II and III only 000-CE-MATH 保留版權 All Rights Reserved 000

13 9 In the figure, ABCD is a parallelogram Find BDE A 0 B 5 C 40 D 50 D 40 C E 55 A E B 0 In the figure, O is the centre of the circle EAOB and EDC are straight lines Find x A 40 C B 46 C 57 E 0 A D x O 4 B D 66 E CE-MATH Go on to the next page 保留版權 All Rights Reserved 000

14 Two fair dice are thrown Find the probability that at least one 6 occurs A B C D E A bag contains six balls which are marked with the numbers,,,, and respectively Two balls are drawn randomly from the bag Find the probability that the sum of the numbers drawn is zero A B C D E CE-MATH 4 保留版權 All Rights Reserved 000

15 {x, x +, x + 4, x + 6, x + 8} and {x +, x +, x + 5, x + 7, x + 9} are two groups of numbers Which of the following is/are true? I The two groups of numbers have the same range II The two groups of numbers have the same standard deviation III The two groups of numbers have the same mean A I only B II only C III only D I and II only E I and III only 4 In the figure, AB = CD, CAB = ECD and ABC = CDE Which of the following must be true? I ABC CDE II ABC ~ EAC III EAC is an isosceles triangle A I only E B III only A C I and II only D I and III only D E I, II and III B C 000-CE-MATH 5 4 Go on to the next page 保留版權 All Rights Reserved 000

16 5 In the figure, PXQ, QYR and RZP are semicircles with areas A cm, A cm and A cm respectively If A = and A = 5, find A A Z B 7 C 69 D π A cm R A cm Y 69 E π 8 P A cm Q X 6 In the figure, find the area of the triangle correct to the nearest 0 cm A 7 cm B 07 cm C 7 cm D 50 cm E 9 cm 50 6 cm CE-MATH 6 5 保留版權 All Rights Reserved 000

17 7 In the figure, find x correct to significant figures A 68 7 cm B 785 C 845 D 87 E 89 6 cm x 5 cm 66 7 cm 8 In the figure, ABCD is a rectangle Find CF C b cm D a cm B A θ F A ( a + b) sinθ cm B ( a + b) cosθ cm C ( a sinθ + b cosθ ) cm D ( a cosθ + b sinθ ) cm E a + b sin θ cm 000-CE-MATH 7 6 Go on to the next page 保留版權 All Rights Reserved 000

18 9 In the figure, DAB is a straight line tanθ = C D A tan 0 0 θ A B B tan 0 C tan 0 D tan 0 E tan 40 0 According to the figure, the bearing of B from C is A 050 B 0 N 40 B C 40 D 0 E 0 A C 000-CE-MATH 8 保留版權 All Rights Reserved 000 7

19 In the figure, CAB is a semicircle and ABCD is a parallelogram Find the area of ABCD A 65 cm B 60 cm C 5 cm D 5 cm A D 5 cm B C cm E 0 cm The figure shows a square, a triangle and a sector with areas a cm, b cm and c cm respectively cm cm cm 4 cm 4 cm 4 cm 4 cm a cm b cm c cm cm 4 cm 4 cm Which of the following is true? A a > b > c B a > c > b C b > a > c D b > c > a E c > a > b 000-CE-MATH 9 8 Go on to the next page 保留版權 All Rights Reserved 000

20 In the figure, a solid wooden sphere of radius r cm is to be cut into a cube of side cm Find the smallest possible value of r r cm cm A B C D E a b 4 If 9a b = 0 and ab < 0, then = a + b A B C 0 D E 000-CE-MATH 0 保留版權 All Rights Reserved 000 9

21 5 y varies directly as x and inversely as z = 9, find y when x = and z = 4 z If y = when x = and A B C D E Tea A and tea B are mixed in the ratio x : y by weight A costs $80/kg and B costs $00/kg If the cost of A is increased by 0% and that of B is decreased by %, the cost of the mixture per kg remains unchanged Find x : y A : B : C : D 5 : 6 E 6 : CE-MATH 0 Go on to the next page 保留版權 All Rights Reserved 000

22 Section B 7 Simplify a b + + a + b b a a ab b A a + b a b B a b a + b C a + b a b + 4ab D a a + b b E 8 If log( x a) =, then x = A B +a 0 a C 000 a D a E 0 + a 000-CE-MATH 保留版權 All Rights Reserved 000

23 9 Which of the following may represent the graph of y = x + x? A y B y O x O x C D y y O x O x E y O x 000-CE-MATH Go on to the next page 保留版權 All Rights Reserved 000

24 5 40 If + = x, then x = A 0 B 0 4 C D 0 E In the figure, the graph of y = f(x) intersects the x-axis at P and Q only In order to find a root of f( x ) = 0 using the method of bisection, which of the following intervals can you start with? I < x < 0 II < x < III < x < y = f(x) y A I only B III only C I and II only D I and III only E I, II and III P Q O x 000-CE-MATH 4 保留版權 All Rights Reserved 000

25 4 According to the figure, which of the following represents the solution of 0 x 4 y x y? 0 y 4 V VI A Region I IV B Region II I C Regions I and VI II III D Regions II and III E Regions II, III and IV O 4 x 4 In the figure, ABCDEF is a right triangular prism It is cut into two parts along the plane PQRS, which is parallel to the face BCDF, and volume of the prism APQRES AP : PB = : 5 Find volume of the prism ABCDEF A 7 E B C D B P A Q F C S R D E CE-MATH 5 4 Go on to the next page 保留版權 All Rights Reserved 000

26 44 π degrees = A B C π π π 80 radian radians radian D 80 radians E radian 45 In the figure, AB is tangent to the circle at B Find DCE A 70 B 75 C 90 C D D 95 E 05 A E 0 B 000-CE-MATH 6 5 保留版權 All Rights Reserved 000

27 46 In the figure, AB : BC : CD = :: Find ADC A 56 D B 60 C 6 D 7 E 84 A 96 C B 47 The bar chart below shows the distribution of scores of a test Find the mean deviation of the scores of the test A 0 mark 0 B C 8 mark 9 mark frequency D marks score (marks) E 6 marks CE-MATH 7 6 Go on to the next page 保留版權 All Rights Reserved 000

28 48 If the centre of the circle x + y + kx + ( k + ) y = 0 lies on x + y + = 0, find k A B C 0 D E 49 If the straight line y = mx + is tangent to the circle ( x ) + y =, then m = A 4 B 0 C 4 D 0 or 4 E 0 or CE-MATH 8 保留版權 All Rights Reserved 000 7

29 50 A(, 4) and B(, 4) are two points The line x y = 0 cuts AB at P so that AP : PB = r : Find r A B C D E 5 If cos θ = and 0 < θ < 90, then tan(θ 70 ) = k A k k B k C k D k E k 000-CE-MATH 9 8 Go on to the next page 保留版權 All Rights Reserved 000

30 5 The figure shows a right triangular prism Find its volume m α β A sin α cosα sin β cos β m B sin α cos α sin β cos β C sinα cosα sin β cos β m m D sin α cosα sin β cos β m E sinα cos α sin β cos β m 000-CE-MATH 0 保留版權 All Rights Reserved 000 9

31 5 In the figure, the area of ABC is A π y C B π y = cos x C π D 4π E π O A π B π x 54 In the figure, AEC and BED are straight lines If the area of ABE = 4 cm and the area of BCE = 5 cm, find the area of CDE A 45 cm D B 5 cm A C 6 cm E D 65 cm E 9 cm B C END OF PAPER 000-CE-MATH 保留版權 All Rights Reserved 000 0

32 000 Mathematics (Paper ) Question No Key Question No Key E B A B C A 4 E 4 A 5 B 5 D 6 B 6 C 7 C 7 E 8 B 8 D 9 D 9 E 0 A 40 B E 4 C B 4 D A 4 C 4 A 44 A 5 C 45 E 6 C 46 C 7 E 47 B 8 A 48 B 9 A 49 D 0 E 50 A E 5 B C 5 E D 5 C 4 D 54 D 5 B 6 C 7 D 8 D 9 A 0 D 保留版權 All Rights Reserved 000