1. A = (2 ) 5 = (2 5) 2. A a b x y a b x y a 3y b. x y x y 3. D. = (4 + 2x 3 y)(4 2x + 3 y)

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1 HKDSE06 Mathematics (Compulsory Part) Paper Solution. A ( ) ( 5) A a b + x y a b x y a y b x y x y a y b ay x y b. D 6 (x y) 4 (x y) [4 + (x y)][4 (x y)] (4 + x y)(4 x + y) 4. C For A: (correct to significant figures) For B: (correct to decimal places) For C: (correct to 4 significant figures) For D: (correct to 5 decimal places)

2 HKDSE06 Mathematics (Compulsory Part) Paper Solution 5. A 4α + β 7α + β 5 4α + β 5...() 7α + β 5...() () α + β 5...() () () 5α 0 α Put α into () 4() + β 5 β 6. B f ( x ) is divisible by x +, f ( ) 0 4( ) + k( ) + 0 k k k 5 i.e. f ( x) 4x + 5x + When f ( x ) is divided by x +, The remainder f ( ) + + 4( ) (5)( ) 6 7. A 5x > x x > x < 7 and 6x 8 < 0 6x < 8 x < The combined solution is x < 7

3 HKDSE06 Mathematics (Compulsory Part) Paper Solution 8. C The quadratic equation has equal roots 0 k ( ) 4()(8 + 6) 0 k k k 44 0 ( k 6)( k + 4) 0 k 6 or k 4 9. D y ax a ( + ) + a x + ax + + a Direction of opening: a > 0 Therefore, the graph opens upwards y-intercept: put x 0 y + a > 0 Therefore, the graph cuts the positive y-axis 0. C Let $x be the monthly salary of Teresa Then the monthly salary of Peter $ x( 5%) $0.75x The monthly salary of Donald $0.75 x( + 5%) $0.975x 0.975x 60 x 5584 i.e. the monthly salary of Teresa is $5584

4 HKDSE06 Mathematics (Compulsory Part) Paper Solution. D (y 4 x) : ( x + y) 5 : 6 y 4x 5 x + y 6 8y 4x 0x + 5y y 4x 4 x y x : y : 4. D k x Let z, where k 0 y When x is decreased by 6% and y is increased by 60% The new value of z k x( 6%) y( + 60%) k 0.64x.6y 0.8k x.6y k x y original value of z i.e. z is decreased by 50%. A Let $ x / kg be the cost of flour of brand Y Total cost of the mixture 4 + x $(6 + x) Total weight of the mixture + 5 kg 6 + x x 80 x 54 x 7 i.e. the cost of flour of brand Y is $7 / kg 4

5 HKDSE06 Mathematics (Compulsory Part) Paper Solution 4. C The number of dots in the nth pattern 9 + ( n ) 5 5n + 4 The number of dots in the 7th pattern 5(7) B x y y a (alt. s) x + c 80 (int. s) x 80 c x + y + b 60 ( s at a pt.) 80 c + a + b 60 a + b c 80 i.e. II must be true If I is true a + c 80 y + 80 x 80 y x But y and x may not be equal, therefore, I may not be true If III is true b + c 60 b + 80 x 60 b 80 + x But b may not be equal to 80 + x, therefore, III may not be true 5

6 HKDSE06 Mathematics (Compulsory Part) Paper Solution 6. D In AB ABD + BD AD i.e. AB + BD AD therefore, ABD 90 (converse of Pyth. Theorem) i.e. DBC 90 BC CD BD BC 60 cm (Pyth. Theorem) 7. A Let x ABE CEB x (alt. s, AB / / CD) BE CE (given) ECB EBC (base s, isos. ) 80 x ( sum of ) ECB + ADC 80 (int. s, AD / / BC) 80 x x 66 x 48 i.e. ABE 48 6

7 HKDSE06 Mathematics (Compulsory Part) Paper Solution 8. C a cm b cm In the figure, a 5, b 4 c cm c 5 The volume of the prism [4 + (4 + )] cm 9. A Let AOB θ Area of the shaded region θ θ π (9) π () 6πθ 5 6πθ 7π 5 θ 60 i.e. the angle of the sector OAB is 60 i.e. I is true Area of sector OAB i.e. II is not true Perimeter of sector OCD π (9) 60 (78 + π ) cm i.e. III is not true 60 6π π () cm 60 7

8 HKDSE06 Mathematics (Compulsory Part) Paper Solution 0. C Let k be the length of the squares ADP ~ AEQ ~ AHG DP AD HG AH DP k k 9k DP k CP k k k EQ AE HG AH EQ 6k k 9k EQ k Area of DEQP : Area of ABCP ( k + k) k (k + k) k : : 5. B E In the figure, BE AE cos a AB AE AB cos a BF sin c BC BF BC sin c AD AE + ED AE + BF AB cos a + BC sin c F AD and BF CD 8

9 HKDSE06 Mathematics (Compulsory Part) Paper Solution. D BCD 80 ADC (int. s, BC / / AD) BED BCD ( at centre twice at circumference) 6 DFE CDA FED (ext. of ) A 9

10 HKDSE06 Mathematics (Compulsory Part) Paper Solution 4. B ( n ) n 8 n 0 i.e. A is not true Each exterior angle of the polygon i.e. B is true The number of diagonal of the polygon 0 C 0 70 i.e. C is not true Each interior angle of the polygon i.e. D is not true D For hx + ky slope h k 5 x intercept h For 4x + y slope 5 x intercept 4 The lines intersect at a point on the x-axis i.e. they have the same x-intercept 5 5 h 4 h The lines are perpendicular to each other i.e. the product of their slopes is 4 k k 6 0

11 HKDSE06 Mathematics (Compulsory Part) Paper Solution 6. B Let C ( a, b) C lies on the line x y 0 a b 0 a b AC BC ( a 9) + ( b + ) ( a + ) + ( b 8) (b 9) + ( b + ) (b + ) + ( b 8) b b + + b + b + b + b + + b b b a i.e. the x-coordinate of C is 7. C For the equation of the circle C: x + y x + y x y x y Centre i.e. III is true Radius 4 0 (, ) (, 5) 65 () + ( 5) i.e. I is not true Distance between origin and centre of C ( 0) + ( 5 0) > 9 i.e. distance between origin and centre of C > radius of C i.e. the origin lies outside C i.e. II is true

12 HKDSE06 Mathematics (Compulsory Part) Paper Solution 8. C P(at least $) P(less than $) P(she takes out $,$,$5 coins) C C B The expected number of tokens B As the mode of the data is 68, there should be at least two among a, b and c are 68 Without loss of generality, assume a b 68 mean c c 770 c 85 Arrange the data in ascending order:, 68, 68, 68, 79, 85, 86, 88, 98, Median 8. C 9 a b a b a b a b 4 4 5a 5 a 6 6 L.C.M. 5 a b 80a b 6 6

13 HKDSE06 Mathematics (Compulsory Part) Paper Solution. D y ab x log y log ( ab ) x 9 9 log a 9 log a + x log b 9 9 Put x 0, log9 y log a + (0) log b 9 9 log 9 log 9 a Put x 4, log9 y log b log9 b log 9 9 log 9 b A BC000DE ( 6 + ) 6 + ( 6 + 4)

14 HKDSE06 Mathematics (Compulsory Part) Paper Solution 4. B 7 7 a i 7( a i) 7a 7 u i a + i a + i a i a i a + a a + i 7( a + i) 7a 7 v + i a i a i a + i a i a + a + 7a Real part of u real part of v a + i.e. II is true a + i a + i u a i a i v Imaginary part of u 7 Imaginary part of v 7 i.e. III is not true uv a + i a i a + If a +, then uv is not rational i.e. I may not be true 5. D To achieve the greatest value of 7y 5x + The greater the value of y and the smaller the value of x is better i.e. the greatest value is attained at the point at the top left corner i.e. the greatest value is attained at point S 4

15 HKDSE06 Mathematics (Compulsory Part) Paper Solution 6. B Let a and r be the first term and the common ratio of the geometric sequence a ar...() a 6 7 ar 89...() () () r r ± i.e. I is not true a 4 ±, which is irrational number i.e. II is true If r, a 7 The sum of the first 99 terms If r, a 7 The sum of the first 99 terms 99 7[( ) ] >.96 0 > [ ( ) ] < 0 ( ) 4 4 i.e. III may not be true 7. A When x 0, y a cos (0) a When x b, y cos b cos b b 80 b 90 5

16 HKDSE06 Mathematics (Compulsory Part) Paper Solution 8. B θ θ 5sin + sin 4 0 (sinθ + )(5sin θ 4) 0 4 sinθ or sinθ 5 ( root) ( roots) 9. A In ABC AC cm AC AP PC 0 cm In PCQ PQ cm In APF PF cm In FHQ FQ cm In FPQ FP + FQ PQ cos PFQ ( FP)( FQ) (6) + (5) ( 8) (6)(5) sin PFQ

17 HKDSE06 Mathematics (Compulsory Part) Paper Solution 40. D Join BD PB PD (tangent property) PBD PDB (base s, isos. ) ( sum of ) 56 DAB PBD ( in alt. segment) 56 ABC 90 ( in semi circle) In ABQ AQB ( sum of ) 4 4. C Let ( a, b ) be the mid-point of PQ x + y y () x y () From () y x 6...() Sub. () into () x x x x x x x 5x 40x x + ( 6) 8( 6) x Let x and x be the roots of the above equation 8 x + x 8 + x 8 x a 4 b a 6 (4) 6 i.e. the y-coordinate of the mid-point of PQ is 7

18 HKDSE06 Mathematics (Compulsory Part) Paper Solution 4. A P(at least cans of tea) P( tea coffee OR tea coffee) C C + C C C4 4. D The number of different committees n( girls 4 boys OR girl 5 boys OR 0 girls 6 boys) C C + C C + C C B There are 0 students in the group Median Upper quartile i.e. I is not true Mean Standard deviation.58 < i.e. III is not true Standard score of Ada.879 < i.e. II is true 45. C The variance of the new set of numbers

1. B (27 9 ) = [3 3 ] = (3 ) = 3 2. D. = c d dy d = cy + c dy cy = d + c. y( d c) 3. D 4. C

1. B (27 9 ) = [3 3 ] = (3 ) = 3 2. D. = c d dy d = cy + c dy cy = d + c. y( d c) 3. D 4. C HKDSE03 Mathematics (Compulsory Part) Paper Full Solution. B (7 9 ) [3 3 ] (3 ) 3 n + 3 3 ( n + ) 3 n + 5 3 6 n + 5. D y y + c d dy d cy + c dy cy d + c y( d c) c + d c + d y d c 3. D hl kl + hm km hn