Microsoft Word HKDSE Mathematics Core Paper 2 Sol

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1 DSE-MAT-5-CS 05 HKDSE Mathematics Core Paper Suggested Solutions HKDSE Prepared by Andy Lai Mathematics Teacher (00% M.C. Right!) MC 係分 ABC Grade 既地方, 越出越煩, 越出越深, 同學一定要快又要好小心! Enrollment Hotline: Website: MSN: Address: Rm706, Prosper Commercial Building, 9 Yin Chong Street, Mong Kok, Kowloon, Hong Kong.

) MC 係分 ABC Grade 既地方, 越出越煩, 越出越深, 同學一定要快又要好小心!

2 05 HKDSE Mathematics Paper Suggested Answers. D. D. A. D 5. B 6. A 7. A 8. D 9. B 0. D. C. D. B. C 5. C 6. B 7. D 8. A 9. C 0. C. B. A. C. A 5. B 6. C 7. A 8. B 9. A 0. B. C. A. A. B 5. C 6. D 7. D 8. B 9. D 0. C. B. A. C. D 5. B MC 係分 ABC Grade 既地方, 越出越煩, 越出越難! 轉數快, 概念清! 缺一不可! 同學一定要快又要好小心! Andy s predicted M.C. Grade boundaries: 5**: / 5 5*: 0 / 5 5: 8 / 5 : / 6 : 0 / 5 : 6 / 5

C. B. A. C. D 5. B MC 係分 ABC Grade 既地方, 越出越煩, 越出越難! 轉數快, 概念清! 缺一不可!

3 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS Section A. D ( x + )( x + x + ) x + x + x + x + x + x + x + x +. D 6 (y ) y y y 7y. A. Method : Elimination of Simultaneous equation in two unknowns p + q 5 p + 9q 5p + 5q 0 5p + 9q 6 q 8 q p + () p 5. Method : Program in calculator! p 5 and q. D (correct to 5 decimal places) (correct to 6 decimal places) (correct to 6 significant figures) Therefore, the answer is D 05 A Lai learning Center. ALL RIGHTS RESERVED. P.

Method : Elimination of Simultaneous equation in two unknowns p + q 5 p + 9q 5p + 5q 0 5p + 9q 6 q 8 q p + () p 5.

4 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 5. B x + mx + n ( x + )( x m) + 6 x + mx + n x + ( m) x + 6 m Comparing coefficients and constants on both sides gives m m and 6 m n m and n 6. A 8 + 7x > or 5 x < x > or 7 x > x > or x > Therefore, x > 7. A x 5x 0 and β is the root of the equation. Therefore, β 5β 0 8β 0β β 8β 5 P. 05 A Lai learning Center. ALL RIGHTS RESERVED.

both sides gives m m and 6 m n m and n 6.

5 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 8. D y + a( x b) Vertex form: a[ x ( b)] + 0 y Min Point: (-b, 0) From the graph, b < 0 b > 0. From the graph, the parabola s opening is upward a > 0 Therefore, a > 0 and b > 0 9. B.7 ( 0.6) % Therefore, percentage change -% 0. D 6% 50000( + ) 59, (correct to the nearest dollar). C a : c 5 : b : c : a : b : c 0 : 9 : 6 Let a 0k, b 9k and c 6k ( a + c) : ( b + c) 6k :5k 6: 5 05 A Lai learning Center. ALL RIGHTS RESERVED. P.

From the graph, the parabola s opening is upward a > 0 Therefore, a > 0 and b > 0 9. B.7 ( 0.6) 0. 68 0.

6 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS. D z kx y k() () 7 k 7 z ( ) 89. B st Term: 5 dots, nd Term: 9 dots, rd Term: dots, etc.. th Term + 7 dots, 5 th Term 7 + dots Therefore, 6 th Term + 5 dots.. C Upper limit of the weight of the white sugar in the bad Lower limited of the weight of white sugar in each packet 5500 n The greatest possible value of n kg g 5. C By Pyth. Theorem, AN 5 and CN Since AB BC and AB 0 6 AC + Therefore, Area of ABC ()(6) 96 cm P. 05 A Lai learning Center. ALL RIGHTS RESERVED.

. C Upper limit of the weight of the white sugar in the bad Lower limited of the weight of white sugar in each packet 5500 n 578.

7 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 6. B Similar figures V V l l V V V V 7 Volume of the Biggest circular cone ( π )( 9) ( ) π π Volume of the upper smaller circular cone π 7 Therefore, the volume of the frustum π π π cm 7. D Given DE : EC : Area of ADE : Area of BEC a : a (Same height Base ratio Area ratio) DEF ~ CEB (AAA) 8 a a 6 AEF ~ GEB (AAA) 8 + a a + X X 7 cm Do you know how to calculate the area of ABE? 0 cm Why? 05 A Lai learning Center. ALL RIGHTS RESERVED. P. 5

cone π 7 Therefore, the volume of the frustum π π π cm 7.

8 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 8. A AD AC cosα and AC AB tan β AD AC Therefore, cosα tan β AC AB 9. C cos80 cos sin(90 + θ ) + sin(70 +θ ) + + cosθ cosθ ( cosθ ) + ( + cosθ ) ( + cosθ )( cosθ ) cosθ cos θ cosθ sin θ 0. C 90 ABD ACD (Angle in semi circle) BC CD BAC DAC (Equal chord, Equal angle) 58 BAC 9 BCA (opp angles, cyclic quadrilateral) CBD 9 Therefore, AEC P A Lai learning Center. ALL RIGHTS RESERVED.

C cos80 cos60 + + sin(90 + θ ) + sin(70 +θ ) + + cosθ cosθ ( cosθ ) + ( + cosθ ) ( + cosθ )( cosθ ) cosθ cos θ cosθ sin

9 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS. B Area of AEB ( )( ) sin 90 7 cm. A ( n )(80 ) 60 5( ) n n n Therefore, each interior angle of the polygon ( )(80 ) 50 The number of diagonals of the polygon C 5 The number of folds of rotational symmetry of the polygon is.. C Reflection about y-axis No change in y-coordinate Reflection about y-axis Negative value of the x-coordinate The rectangular coordinates of its image is (, ) By tan θ tan θ θ 0 and r + Therefore, polar coordinates of its image is (, 0 ) 05 A Lai learning Center. ALL RIGHTS RESERVED. P. 7

folds of rotational symmetry of the polygon is.

10 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS. A A moving point P is always equidistant from two points A and B. The locus of P is perpendicular bisector of AB. Locus of P: Perpendicular bisector of AB 5. B ax bx + cy x < 0 a a < < 0 b < a < 0 a b Label the diagram below: a < 0 b b y x + > 0 c c c and > 0 c c > 0 and b < 0 P A Lai learning Center. ALL RIGHTS RESERVED.

The locus of P is perpendicular bisector of AB. Locus of P: Perpendicular bisector of AB 5.

11 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 6. C Standard form of circle: ( x h) + ( y k) r Coordinates of the center of circle: ( h, k) and radius r ( x ( )) + ( y ) ( 0) + ( ) ( x + ) x x + ( y ) + 8x y + 8x + y 6 6y y A 6 0 Expected value $ B A Inter-quartile range 75%th datum 5%th datum A Lai learning Center. ALL RIGHTS RESERVED. P. 9

( )) + ( y ) ( 0) + ( ) ( x + ) x x + ( y ) + 8x + 6 + y + 8x + y 6 6y + 9 6 6y + 9 0 7.

12 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 0. B Total numbers of data 5 The median 8 th datum q m 5 q may be,, 5 Mode r m p m p 5 7 If m q p > q r 5 7 If m q p.9... > q > r 5 75 If m 5 q 5 p 5 q > r. 5 Therefore, Only p > r must be true.. C x x + x + x ( x ) x + ( x ) ( x + )( x ) ( x + )( x ) ( x + )( x ) P A Lai learning Center. ALL RIGHTS RESERVED.

+ 9 + 0 p 5 70 + m p 5 7 If m q p.866... > q r 5 7 If m q p.9... > q > r 5 75 If m 5 q 5 p 5 q > r.

13 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS. A By slope-intercept form: log y ( )log x + log y ( )log x + log y log x + log y log x log x y log ( yx ) yx y 9 x y 79 x. A B x + kx 0 and roots are α and β α + β k and αβ α + β ( α + β) αβ ( k) ( ) k + 5. C z 6 7 ( a + 5) i + ( a ) i z n ( a + 5) i + ( a ) i ( i for all integer n) z ( a + 5)( ) + ( a )( i) a + 5 ( a ) i Since z is a real number ( a ) 0 a 05 A Lai learning Center. ALL RIGHTS RESERVED. P.

A + + + + + + + + 00 0000 6 0 6 0. B x + kx 0 and roots are α and β α + β k and αβ α + β ( α + β) αβ ( k) ( ) k + 5.

14 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 6. D To solve graphical inequalities: Substitute some points co-ordinates in the shaded region. To coordinates of the vertex of shaded region: (,9), (,) and (0, 0/) Therefore, for convenience, substitute (a, b) be (, 7) into equations. The results follows. 7. D x 6 and x 96 ar 5 6 and ar r and 656 a or r and 656 a 656 ( ) x 79 or x 656 ( ) 79 x5 ar 9 x5 ar 9 > 6 or > 6 x7 ar r x7 ar r x + x + x x n ar + ar + ar ar n a( r + r + r r n ) ( ) a( r) < 968. < 05 r 5 9 P. 05 A Lai learning Center. ALL RIGHTS RESERVED.

To coordinates of the vertex of shaded region: (,9), (,) and (0, 0/) Therefore, for convenience, substitute (a, b) be (, 7) into equations.

15 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 8. B cos x sin x for 0 x < 60 sin x sin x sin x ( sin x) 0 sin x 0 or sin x 0 x or x 80 or x 90 for 0 x < 60 Therefore, there are only solutions. 9. D 0.5T T 80 k The sine curve is shift to the left by 5 o k( x ο θ θ + ) 5 θ 0 k k 0. C By Pyth. Theorem, BE cm AC AB (Tangent properties) CE 0 6 and EE 8 r where r is the radius of the circle By Pyth. Theorem, r (8 r) 6 6 6r + r r 6 r 8 8 r r 6 Therefore, the diameter BD 6 cm 05 A Lai learning Center. ALL RIGHTS RESERVED. P.

solutions. 9. D 0.5T 65 75 90 60 T 80 k The sine curve is shift to the left by 5 o k( x ο θ θ + ) 5 θ 0 k k 0. C By Pyth.

16 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS. B Method : Discriminant 0 x + y + x 6y + k 0 x + y + 0 x + ( x ) + x 6( x ) + k 0 x + 6x k 0 0 (Only repeated solution!) 6 ()(0 + k ) k 0 k 8 Method : Calculator program (Problem solved within minute!). A Method : Draw diagram in scale (Problem solved within minute!) Method : Calculator program (Problem solved within 0 seconds!!!). C 8 6 P 7 P D By drawing a tree diagram P. 05 A Lai learning Center. ALL RIGHTS RESERVED.

) 6 ()(0 + k ) 0 56 0 8k 0 k 8 Method : Calculator program (Problem solved within minute!). A Method : Draw diagram in scale (Problem solved within minute!

17 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 5. B a + a + a a 50 x 50 a + a y 5 6 and a + a + a a 9 ( a z 9 a 5 y and z ( a x 50x a + a + a a 50 + ( a + ( a x) 50 9x a + a + a a 9 + ( a + ( a x) ( a ( a 9 50 If x a50 50x 9x + x x x If a 5 a6 y y but if a 5 < a6 y < y y y If x a50 ( a z ( a z + ( a + ( a + ( a 50 + ( a x) ( x ( a 9 However, z ( a z ( a + ( a + ( a + ( a x ) 9 + ( a x ) ( a ( a 9 Comparing ( a x ) + ( a x) + ( a x) ( a9 x) z and 50 ( a x ) + ( a x) + ( a x ) ( a9 x ) z Numerator is the 9 same but denominator is smaller z > z The end. 05 A Lai learning Center. ALL RIGHTS RESERVED. P. 5

.. + ( x +... + ( a 9 However, z ( a z ( a + ( a + ( a + ( a x ) 9 + ( a x ) 9 +... + ( a 9 +... + ( a 9 Comparing ( a x ) + ( a x) + ( a x) +.

18 05 HKDSE Mathematics Core Paper II Suggested Solutions DSE-MAT-5-CS 黎 sir 教室將於 05 年 9 月推出中六數學 / 物理 / 化學 / 經濟科精讀班同學想奪星? 梗係要上由黎 sir 教室 5** 導師團隊教授既課堂啦! 集齊最少位同學報名, 可以即時開班, 課題任選, 內容為你度身訂做! 詳情請致電查詢. 或瀏覽本教室網址 : P A Lai learning Center. ALL RIGHTS RESERVED.

集齊最少位同學報名, 可以即時開班, 課題任選, 內容為你度身訂做! 詳情請致電 677 00 查詢.

19 黎 sir 簡介 Andy Lai BEng CUHK, MIEEE 畢業於香港中文大學電子工程學系, 黎 sir 教室創辦人之一. 超過 5 年教授中學文憑 / IB Diploma / GCE / HSC / SAT / AP / GCSE / IGCSE / IB MYP 課程經驗. 與學生面對新制度的中學文憑試, 黎 sir 親身上陣, 以實力於物理科和經濟科奪取 5**, 證明寶刀未老. 熟悉出題趨勢, 教授考試取分技巧 ; 鼓勵同學獨立思考, 增強同學理解能力. 善用生活化例子講解, 教法生動, 增加學習趣味 ; 深入淺出, 明白學生學習上的困難和需要. 精心編制筆記, 適合中文和英文中學學生就讀 ; 精心編制練習和試題, 協助同學盡快掌握答題技巧. 黎 sir 在中學和大學時代已是一名傑出學生, 曾獲取的多項學業上和運動上的獎學金及獎項. 曾代表香港參加國際性運動比賽, 取得優異成績, 又讀得又玩得, 絕不是死讀書的書呆子. 任教科目 : 所有數學科, 物理科, 化學科, 生物科, 經濟科, 商業科. 黎 sir 教室學生佳績 : 首屆香港中學文憑 (HKDSE), 多位學生取得 5/5*/5** 級以上佳績. 更有學生考獲 5 科 5** 級科 5* 級科 5 級優異成績, 在全港 760 考生中, 排名 8, 入讀港大醫學院. 英國高考 (GCE AS/AL), 多位學生取得 A*/A 最高級別, 更有學生考獲 5 科 A*. 國際文憑 (IB Diploma), 多位學生取得 6 / 7 級別, 更有學生取得 /5 總分. 英國會考 (IGCSE / GCSE), 多位學生取得 A / A* 成績, 更有學生取得 8 科 A* 加拿大大學預科 (CESI) 數學課程 MCVU, 取得 98 / 00, 99 / 00 成績學生成功拔尖 (EAS), 提早入讀港大理學院和中大法律學院. 香港中學會考 (HKCEE), 多位學生取得 0 分以上佳績. 保加利亞國際數學競賽 (BIMC 0) 隊際賽金牌. 奧數華夏杯 / 港澳杯 / 華杯, 多位學生取得特等獎 / 金獎 / 一等獎 / 全港第二名. 還有更多, 怒不能盡錄, 詳情請瀏覽以下網址 : 黎 sir 教室課程特色 : 小組教學 ( 6 人 ), 導師親身教學 ; 照顧每位學生需要, 事半功倍. 精心編制筆記, 練習以近 0 年本地和外國公開試題為藍本. 概念理解, 取分技巧並重 ; 協助同學盡快掌握答題技巧. 歡迎自由組合小組上課, 時間及課程內容編排更有彈性. 詳情請瀏覽以下網址 :

善用生活化例子講解, 教法生動, 增加學習趣味 ; 深入淺出, 明白學生學習上的困難和需要. 精心編制筆記, 適合中文和英文中學學生就讀 ; 精心編制練習和試題, 協助同學盡快掌握答題技巧.

20 黎 sir 簡介黎 sir 教室 A Lai Learning Center HKDSE / IB Diploma / GCE AS AL / AP / SAT / HSC IGCSE / GCSE / IB MYP / KS / MO / F. F.6 / Y9 Y 資深中學補習導師小組補習事半功倍!!! 簡介 Andy Lai BEng CUHK, MIEEE 畢業於香港中文大學, 黎 sir 教室創辦人之一. 超過 6 年教授中學文憑 / IB Diploma / GCE / HSC / SAT / AP / GCSE / IGCSE / IB MYP 課程經驗. 與學生面對新中學文憑試, 黎 sir 親身上陣, 於數學科, 物理科和經濟科奪取 5**, 證明寶刀未老. 現於黎 sir 教室任教補習班, 學生就讀於英文中學, 中文中學, 國際學校及英國留學生. 熟悉近年出題趨勢, 教授考試取分技巧 ; 鼓勵同學獨立思考, 增強同學理解能力善用生活化例子講解, 教法生動, 增加學習趣味 ; 深入淺出, 明白學生學習上的困難和需要. 中英對照筆記, 適合中文和英文中學學生就讀 ; 精心編制練習和試題, 協助同學盡快掌握答題技巧. 黎 sir 在中學和大學時代已是一名傑出學生, 曾獲取多項學業上和運動上的獎學金及獎項 ; 曾代表香港參加國際性運動比賽, 取得優異成績, 又讀得又玩得, 絕不是死讀書的書呆子. 黎 sir 在就讀大學時曾於全球最大美資電腦公司任實習生超過一年, 大學畢業後旋即於全港大型英資電腦公司, 負責主理該公司所代理的全球大型美資電腦公司儲存系統銷售業務. 於短短半年內將該產品線銷售業績提升超過 50%. 同時更被公司評選為 " 傑出表現員工 Outstanding Performer", 成功將書本上的知識靈活運用於工作上. 黎 sir 為了教學理想, 毅然辭去工作, 全身投入教學事業, 希望將自己的一套學習方法教授學生黎 sir 教室課程特色小組教學 ( 6 人 ), 導師親身教學 ; 照顧每位學生需要, 事半功倍. 精心編制筆記, 練習以近 0 年本地和外國公開試題為藍本. 概念理解, 取分技巧並重 ; 協助同學盡快掌握答題技巧. 歡迎自由組合小組上課, 時間及課程內容編排更有彈性. 時間及課程請瀏覽以下網址 : 地鐵 : 旺角 E 出口, 油麻地 A 出口小巴 :, A,, C, 6, 6C, 6F, 9, 0X, 5A, A, A, 60X, 6X, 68X, 69X, 8S, 87D, 9K, 95, 0, 7, 0,, 0X, P, X, 8P, 8S, 59B, 70P, 8A 小巴 : K, 7, 7S 上課地址 : 查詢熱線 : 電郵地址 : 網址 : 香港九龍旺角煙廠街 9 號興發商業大廈 706 室 enquiry@andylai.hk

與學生面對新中學文憑試, 黎 sir 親身上陣, 於數學科, 物理科和經濟科奪取 5**, 證明寶刀未老. 現於黎 sir 教室任教補習班, 學生就讀於英文中學, 中文中學, 國際學校及英國留學生.

所 3 學分課程, 及兩門跨領域課程共 6 學分以上課程學生在修課前, 必須填寫課程修課認定表, 經班主任或指導教授簽名後始認定此課程學分 ) 10. 本規章未盡事宜, 悉依學位

所 3 學分課程, 及兩門跨領域課程共 6 學分以上課程學生在修課前, 必須填寫課程修課認定表, 經班主任或指導教授簽名後始認定此課程學分 ) 10. 本規章未盡事宜, 悉依學位 95 年訂定 96 年 11 月修正 97 年 10 月修正 100 年 2 月修正 101 年 4 月修正 102 年 1 月修正 103 年 4 月修正 103 學年度入學新生適用, 舊生可比照適用 1. 研究生須於入學後第二學期開學前選定指導教授, 經課程委員會認定後方得繼續