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

Size: px

Start display at page:

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

Maximilian O’Brien’
5 years ago
Views:

1 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 + 7y Differetiate both sides with respect to, dy e + dy y y dy ey y 7 + 7y Whe ad y, dy e () 7() + 7( )() 8 (). y ( + ) Let u +. The du. \ y u du + C u C + +, where C is a costat. Whe dy, ( + ) Sice y is a horizotal taget, dy whe y. Hog Kog Educatioal Publishig Compay

2 Mathematics: Mock Eam Papers Module (Eteded Part) Secod Editio Solutio Guide So, the horizotal taget y touches the curve at. \ + C C \ The equatio of the curve is y + +. (). Referece: HKCEE A. Math Q ( ) + ( + ) [ + C ( ) + C ( ) +...] + [ + C ( ) The coefficiet of ( ) \ ( 8)( + ) 8 or - (rejected) ( ) + C ( ) +...] Sice the coefficiet of is give, we should epad the epressio as far as the term i. The terms i higher order are omitted ad deoted by the symbol.... (). Referece: HKDSE Math M PP Q9 dy Slope of L Whe dy, 8 + ( )( ) or Whe, y -; whe, y 7. \ The equatios of the two tagets are y + ( - ) ad y + 7, i.e., - y - ad 8-7y - 9. (). (a) si cos si cos cos 8. siq siq cosθ. si - cos q q Hog Kog Educatioal Publishig Compay

3 Mock Eam (b) (si + cos ) [(si + cos ) si cos ] cos 8 (by (a)) ( + cos ) si + + C, where C is a costat. 7. Referece: HKDSE Math M SP Q7 The augmeted matri is 9 9 ~ 8 9 ~ 9 ~ (R - R R, R - R R ) () R R R (R R ) y + z 9 We have. y z Let z t, where t is ay real umber, the we have y t -, t +. The required solutio is t +, y t -, z t, where t is ay real umber. () 8. Referece: HKDSE Math M Q (a) Area [ ( )] + [( ) ] ( ) + ( ) + 8 (b) Volume π [( ) ] π ( ) π ( + ) π + 8π + () Hog Kog Educatioal Publishig Compay

4 Mathematics: Mock Eam Papers Module (Eteded Part) Secod Editio Solutio Guide 9. Referece: HKDSE Math M SP Q9 (a) The area of the parallelogram ABEC AB AC ( i + j) ( i k) i + j 9k (b) The volume of the tetrahedro ABCD AB AC i AD ( ) ( i + j 9 k) i (i j+ k) (c) D ca be ay poit o the plae FDGH ecept D. Take D F. AD AF AC + CF AC + AD (i k) + (i j + k) 7i j k (or other reasoable aswers) (). (a) For, L.H.S. M cos si si cos R.H.S. \ The propositio is true for. Net, assume the propositio is true for k, where k is a positive iteger, i.e., k k M k cos si k k. si cos Whe k +, k+ L.H.S. M k MM cos si cosk si k si cos si k cos k coscos k sisi k cossi k sicos k sicos k + cossi k sisi k + coscos k cos( k + ) si( k + ) R.H.S. si( k + ) cos( k + ) \ The propositio is also true for k +. By the priciple of mathematical iductio, the propositio is true for all positive itegers. Hog Kog Educatioal Publishig Compay

5 Mock Eam cos si (b) (i) Sice A r, si cos we have r cos...( ) rsi...( ) () (), ta Substitutig ito (), r cos r (ii) A cos si si cos cos( ) si( ) si( ) cos( ) A is a diagoal matri whe si(). Whe si(), 8m, where m is a iteger. m 7 For the least positive iteger, ( 7 7 ) Whe, we have A cos cos (8) Sice cos < ad si <, should be i quadrat III. If m is a multiple of 7, the is a iteger. To make the least positive iteger, we take m 7. Sectio B. Referece: HKALE P. Math Paper Q8 (a) P y ( + y) Sice y >, ad y are both positive or both egative, so + y ad P. Therefore, P is ivertible. () A matri A is ivertible if ad oly if A. Hog Kog Educatioal Publishig Compay

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

7 Mock Eam. (a) Sice C passes through (, ), i.e., q. Whe + r, r. p( ) + ( ) + The equatio of the vertical asymptote of C is q r, r. Sice it is give that the vertical asymptote of C is, r \ r p y + p p ( + ) + p p + \ The horizotal asymptote is y p. \ p p The equatio of C is y +. () + (b) The equatio C is y Whe, y. \ The coordiates of the poit of itersectio are,. () 7 Hog Kog Educatioal Publishig Compay

8 Mathematics: Mock Eam Papers Module (Eteded Part) Secod Editio Solutio Guide (c) d ( + )( ) ( )( ) + ( + ) ( + ) < for d + ( )( ) ( + )( ) ( ) > for () + (d) Sice the equatio of C is y, + y The vertical asymptote of C is. The horizotal asymptote of C is y -. () (e) for the correct shapes of C ad C for the correct asymptotes, itercepts ad poits of itersectio for all correct () 8 Hog Kog Educatioal Publishig Compay

9 Mock Eam. Referece: HKALE P. Math Paper Q8 (a) Let ta ta q. The sec sec q dq. + cos sec sec + sec ta + sec θdθ ta θ + sec θdθ sec θ (b) Let y p -. The dy -. Whe π, y. Whe π π, y. dθ θ + C + C ta ta, where C is a costat π si si( π y) π si y f( y)l( + e ) dy π si y + e f( y)l dy π si y e si f( )[l( + e ) l e si ] π f( )l( + e ) f( π y)l[ + e ]( ) dy π si π Therefore, f( )l( + e ) f( )si. si \ f( )l( + e ) f( )si π π () () The upper ad lower limits should be chaged accordigly whe substitutio takes place. 9 Hog Kog Educatioal Publishig Compay

10 Mathematics: Mock Eam Papers Module (Eteded Part) Secod Editio Solutio Guide si (c) Let g ( ). + cos si( π ) si g( π ) + cos ( π ) + cos π si sil( + e ) + cos π π π si + cos π π si + cos g( ) l( + e ) cos + cos ( + cos ) + cos + cos ta ta π si π π π ta ta.7 (cor. to sig. fig.) (by (b)) (by (a)) () The result of (b) is based o f (p - ) -f (). Therefore, we must check that g (p - ) -g () to apply the result.. Referece: HKCEE A. Math 9 Q (a) PV a + b () (b) PR a + b r + k a b + ( a) PT r + r + k r a + b r + ( r + ) Sice PT // PV, r + k r r + ( r + ) ( r + k) r r k () Hog Kog Educatioal Publishig Compay

11 Mock Eam (c) (i) RQ k a + k a b a ( ) b Sice RQ PV, RQ PV. ( k ) a b ( k ) a a k k r (by (b)) + PT a + + ( a + b) ( ) b ( + ) Sice V is the orthocetre, PVT is the altitude with respect to the base RQ. (ii) QV ( a + b) a a + b RU Let s. UP sqp + QR QU s + s a + + a b s + s + a + b ( s + ) ( s + ) Sice Q, V ad U are colliear, we have s + ( s + ) 9 s s + RU \ UP ( s + ) (7) Hog Kog Educatioal Publishig Compay

Mock Exam 1. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q1 (4 - x) 3

Mock Exam 1. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q1 (4 - x) 3 Moc Exam Moc Exam Sectio A. Referece: HKDSE Math M 06 Q ( - x) + C () (-x) + C ()(-x) + (-x) 6 - x + x - x 6 ( x) + x (6 x + x x ) + C 6 6 6 6 C C x + x + x + x 6 6 96 (6 x + x x ) + + + + x x x x \ Costat