HKDSE Exam Questions Distribution

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1 HKDSE Eam Questios Distributio Sample Paper Practice Paper DSE 0 Topics A B A B A B. Biomial Theorem. Mathematical Iductio More about Trigoometric Fuctios, 0, Limits 6. Differetiatio 7 6. Applicatios of Differetiatio, 6 9, 6 7. Idefiite Itegratio 3,, 8 8, 0, 9 8. Defiite Itegratio Applicatios of Defiite Itegratio Determiats. Matrices 0. Systems of Liear Equatios Vectors. Applicatios of Vectors 9 7 Remarks:. The followig topics have bee removed from HKDSE sice 06: Applyig mathematical iductio to prove divisibility Usig shell method to fid the volume of a solid of revolutio Questios ivolvig the topics above are uderlied.. Questios requirig eplai your aswer are idicated by (E). II

2 Useful Formulas. Biomial Theorem (a) Summatio Notatio Tk T + T T k (b) Biomial Theorem (i) ( a + b) C0a + Ca b + C a b C a r b r Cb r 0 r r r b r C a (ii) Geeral term Cr a b, where ad r are itegers. - r r. More about Trigoometric Fuctios (a) (i) p rad. 80 (ii) Arc legth rq, where q is measured i radias (iii) Area of a sector r θ rs, where s is the arc legth. (b) Trigoometric Fuctios of Geeral Agles Let P(, y) be a poit o the termial side of a agle of rotatio q. The, siθ y r cscθ r y cosθ r secθ r where r + y taθ y cotθ y, (c) Relatioship betwee Trigoometric Fuctios (i) cscθ (ii) secθ siθ cosθ siθ (iii) cotθ (iv) taθ taθ cosθ cosθ (v) cotθ (vi) si θ + cos θ siθ (vii) + ta θ sec θ (viii) + cot θ csc θ (d) Compoud Agle Formulas (i) si(a + B) si A cos B + cos A si B (ii) si(a - B) si A cos B - cos A si B (iii) cos(a + B) cos A cos B - si A si B (iv) cos(a - B) cos A cos B + si A si B taa + ta B (v) ta( A + B) taata B taa ta B (vi) ta( A B) + taata B (e) Double Agle Formulas (i) si A si A cos A (ii) cosa cos A si A cos A si A ta A (iii) ta A ta A (f) Product-to-sum Formulas: (i) siacos B [si( A + B) + si( A B)] (ii) cosasi B [si( A + B) si( A B)] (iii) cosacos B [cos( A + B) + cos( A B)] (iv) siasi B [cos( A + B) cos( A B)] (g) Sum-to-product Formulas: + y y (i) si + si y si cos + y y (ii) si si y cos si + y y (iii) cos + cos y cos cos + y y (iv) cos cos y si si 3. Limits (a) Limit of a Fuctio Suppose lim f ( ) ad lim g ( ) eist. a a (i) lim k k, k is a costat. a (ii) lim kf ( ) k lim f( ), k is a costat. a a (iii) lim[ f( ) ± g ( )] lim f( ) ± lim g ( ) a a a (iv) lim f( ) g ( ) lim f( ) lim g ( ) a a a f (v) lim ( ) lim f( ) a, where lim g ( ) 0 a g ( ) lim g ( ) a a IV

3 Scorig Target There are basic skills that ca be used to solve the problems i Module. Try to aswer the followig questios to see whether you ca apply such skills. Biomial Theorem Skill To epad a epressio by usig biomial theorem. Epad ( - ) ( + 3) 6 i ascedig powers of up to the term. Try: Mock Q, Mock 3 Q, Mock Q, Mock 6 Q Mathematical Iductio Skill To prove the sum of a fiite sequece by usig mathematical iductio. Prove, by mathematical iductio, that for all positive itegers,. (k 3)(k + ) + k Try: Mock 6 Q6, Mock 7 Q More about Trigoometric Fuctios Skill 3 To apply the compoud agle formulas. Epress cos 3θ i terms of cos θ. Try: Mock Q6, Mock 3 Q3, Mock 6 Q, Mock 7 Q Skill To apply the product-to-sum or sum-to-product formulas. Solve si 3 + si cos for Limits Skill To fid the derivative from first priciples. d Fid d ( 3-3 ) from first priciples Try: Mock Q, Mock 6 Q3, Mock 7 Q3 Differetiatio Skill 6 To fid the derivative by rules of differetiatio. (a) Fid d 3 d +. (b) Fid d ( + ). d XI

4 FORMULAS FOR REFERENCE A + B A B si (A ± B) si A cos B ± cos A si B sia + si B si cos cos (A ± B) cos A cos B ± A + B A B si A si B sia si B cos si ta (A ± B) ta A ± ta B A + B A B cosa + cos B cos cos taata B A + B A B si A cos B si (A + B) + si (A - B) cosa cos B si si cos A cos B cos (A + B) + cos (A - B) Aswers writte i the margis will ot be marked. si A si B cos (A - B) - cos (A + B) ********************************************** SECTION A (0 marks). Epad ( - ) 3 3. Hece, fid the costat term i the epasio of 6 ( ) +. ( marks) Aswers writte i the margis will ot be marked. Aswers writte i the margis will ot be marked. EP(M) MOCK - Hog Kog Educatioal Publishig Compay

5 SECTION B (0 marks) 9. Defie f () 3 + p + q + for all real umbers, where p ad q are costats. Deote the curve y f () by C. It is give that P(-, 9) is a statioary poit of C. (a) Fid p ad q. (b) Is P a miimum poit of C? Eplai your aswer. (c) Fid the relative etreme poits(s) of C. (d) Fid the poit(s) of ifleio of C. (3 marks) ( marks) ( marks) ( marks) (e) Let L be the taget to C at P. Fid the area of the regio bouded by C ad L. ( marks) Aswers writte i the margis will ot be marked. Aswers writte i the margis will ot be marked. Aswers writte i the margis will ot be marked. EP(M) MOCK - Hog Kog Educatioal Publishig Compay

6 Mock Eam 3 Mock Eam 3 Sectio A. Referece: HKDSE Math M0 Q (a) ( + ) ( - ) ( ) ( + + ) + + Coefficiet of - \ - -7 M A (b) Coefficiet of ( - ) - + M (- ) - () + A (). Referece: HKDSE Math M PP Q6 d + h ( ) lim d h 0 h lim h 0 + h + h + h + h + + h lim h 0 h ( + h + ) lim h 0 + h + M M A A () 3. Referece: HKDSE Math M PP Q (a) + cot θ cot θ + cos θ si θ cos θ + si θ cos θ si θ cosθ M (b) ( + )( ) ( ) + + M Sice is real, we let cot θ for some θ. \ ( + )( ) cos θ (by (a)) + M - cos θ - cos θ ( + )( ) \ The least value of + is -. A () Hog Kog Educatioal Publishig Compay

7 Mock Eam Mock Eam Sectio A. Referece: HKDSE Math M0 Q y cos y d ( + si y) dy d d dy \ d + si y d y cos y d ( + si y) cos y ( + si y) d y Whe 0, d cosy 0 y cos 3 dy d M M A (3) Alteratively, you may fid dy dy by usig. d d d dy d y d d dy. Referece: HKDSE Math M PP Q8 (a) e cosd cos d( e ) e cos ed(cos ) e cos + e si d e cos + si d( e ) e + cos e si e cosd e cos + e si e cosd \ e cos d e cos + e si e cosd 0 0 (b) e cos d e (si + cos ) + C, where C is a costat e (+ cos ) d e d + e cosd 0 0 e + 0 e (si + cos ) e + e ( ) [ () ()] 7e 3 0 M M M A () cos θ + cos θ Hog Kog Educatioal Publishig Compay

8 Mathematics: Mock Eam Papers Module (Eteded Part) Secod Editio Solutio Guide (b) P + y y + y y + y 3 y P MP + y y 3 + y 3 3 y y ( 3 y) y( 3 y) 3( + y) 0 + y 0 ( + y)( 3+ + y) y times P M P ( P MP)( P MP)...( P MP) ( P MP) ( 3+ + y) A A A M A () 3+ ( ) ( 3) (c) Note that A ad (-)(-3) > 0. ( ) 3+ ( 3) Puttig M A i (b), P A 3 0 P 0 ( 3 3) A P 3 0 P , where P 3 M M M M A () Cosider the values of ad y i order to apply the result of (b). 6 Hog Kog Educatioal Publishig Compay

Mock Exam 4. 1 Hong Kong Educational Publishing Company. Since y = 6 is a horizontal. tangent, dy dx = 0 when y = 6. Section A

Mock Exam 4. 1 Hong Kong Educational Publishing Company. Since y = 6 is a horizontal. tangent, dy dx = 0 when y = 6. Section A Mock Eam Mock Eam Sectio A. Referece: HKDSE Math M Q d [( + h) + ( + h)] ( + ) ( + ) lim h h + h + h + h + + h lim h h h+ h + h lim h h + h lim ( + h + h + ) h + (). Whe, e + l y + 7()(y) l y y e + l y