Mock Eam Mock Eam Sction A. Rfrnc: HKDSE Math M Q ( + a) n n n n + C ( a) + C( a) + C ( a) + nn ( ) a nn ( )( n ) a + na + + + 6 na 6... () \ nn ( ) a n( n )( n ) a + 6... () 6 6 From (): a... () n Substituting () into (), 8 nn ( ) 48 nn ( )( n ) 6 n n 84 n ( n ) 48 nn ( )( n ) 8n 66n + 76n 496n 6 nn ( 8)(4n ) Sinc n is a positiv intgr, n 8. Substituting n 8 into (), 6 a 8 (4). Rfrnc: HKDSE Math M 6 Q ( + h) ( + h) ( + h) ( + h) ( + h) ( + h) + ( + h) + ( + h) + ( + h) + ( + h) ( + h) [( + h) + ( + h) + ] h ( + h) + ( + h) + ( + h) (a - b)(a + ab + b ) a - b Hong Kong Educational Publishing Company
Mathmatics: Mock Eam Paprs Modul (Etndd Part) Scond Edition Solution Guid d 6 lim 6 6 d h h ( + h) 6h lim h [ ( + h) + ( + h) + ( + h) ] lim 6 h h 4 ( + h) + ( + h) + ( + h) 6 + + 4. (a) By long division, ) 7 + 7 + 4 4 + + (by (a)) () \ 7 ( + )( + ) + + + + Dividnd Quotint Divisor + Rmaindr (b) ln( + ) d + d 6 ln( ) 6 6 ln( + ) 6 d[ln( + )] 6 6 6 ln( + ) 6 d 6 6 + 6 7 ln( + ) d 6 + 6 ln( + ) + + d 6 (by (a)) 6 ln( + ) + d + d 6 ( ) + 6 ln( + ) d + + d + 6 ( ) ( ) 6 + 6 6 4 ln( + ) ln( + ) + + 6 6 8 6 + C, whr C is a constant (6) Hong Kong Educational Publishing Company
Mock Eam 4. (a) Lt u -. Thn du -d. Whn, u ; whn, u. f( ) d f( u) du f( u) du f( ) d cos (b) Lt f( ) +. By (a), w hav + sin cos + + + d cos ( ) d sin + sin( ) + + cos d cos sin + sin d + + + + + cos d cos d cos + d sin sin sin cos + d + d sin cos + + sin d (6). Rfrnc: HKALE P. Math 997 Papr Q7 (a) A (A - - I) [A(A - - I)] (I - A) [-(A - - I)] (-) (A - - I) - (A - - I) 4 (b) (i) ( A I ) 4 7 (4 )( + ) + 88 4 8(4 ) + + 4...(*) Whn, (A - I) - + () + () - 4. \ - is a factor of (*). \ is a root of (A - I). (A - I) ( - )(- + ) For n n matri P and ral numbr λ, (λp) λ n P. \ Othr roots ± ± (ii) A 88 - -4 By (a), (A - - I) if and only if (A - - I). By (b)(i), - or ± \ or ± 6 (7) Hong Kong Educational Publishing Company
Mathmatics: Mock Eam Paprs Modul (Etndd Part) Scond Edition Solution Guid 6. Rfrnc: HKDSE Math M 6 Q6 (a) cos θ sin θ cos θ cos(9 - θ) θ 9 θ θ 9 θ 8 (b) cos θ sin θ cos(θ + θ) sin θ cos θ cos θ - sin θ sin θ sin θ cos θ ( cos θ - )cos θ - sin θ cos θ sin θ cos θ ( cos θ - ) - sin θ sin θ (cos θ for < θ < 4 ) cos θ - - ( - cos θ) sin θ 4 cos θ - sin θ 6 cos 4 θ 4 cos θ + 9 4 sin θ 6 cos 4 θ 4 cos θ + 9 4-4 cos θ 6 cos 4 θ - cos θ + (c) By (a), θ 8. By (b), w hav 6 cos 4 8 - cos 8 + sin4 cos 6 cos(8 ) cos 8 + 8 + 4 cos 8 ( ) ± ( ) 4(6)() (6) cos 8 + 8 or - 8 (rjctd) (7) 7. Rfrnc: HKDSE Math M 6 Q4 (a) Not that - is th vrtical asymptot. f( ) + + + ( + ) + + + ( + ) + ( + ) + + + Not that whn ±. + \ y + is th obliqu asymptot. 4 Hong Kong Educational Publishing Company
Mock Eam (b) f () f ( ) + ( + ) 9 f () + ( + ) 4 Slop of th normal - 4 9 4 Th quation of th normal is y + 9. (7) 8. Rfrnc: HKDSE Math M Q8 (a) Whn n, L.H.S. sin cos () sin cos R.H.S. sin cos ( + ) sin cos \ Th proposition is tru for n. Nt, assum th proposition is tru for n m, whr m is a positiv intgr, that is, m sin cos k sin m cos( m + ), k whn n m +, m + sin cos k k m sin cos k + sin cos ( m + ) k sinm cos( m + ) + sin cos( m + ) (by th assumption) sin(m + ) + sin( ) sin(m + ) + sin( m ) + sin(m + ) sin m + m + cos 4 sin sin( m + ) cos[( m + ) + ] \ Th proposition is tru for n m +. By th principl of mathmatical induction, th proposition is tru for all positiv intgrs n. Hong Kong Educational Publishing Company
Mathmatics: Mock Eam Paprs Modul (Etndd Part) Scond Edition Solution Guid (b) Lt 8 By (a), k and n. k sin cos sin cos 6 8 9 8 8 k + + sin k cos 9 cos 9 sin 8 sin cos + 8 sin 8 sin 8 sin 8 (8) Sction B 9. Rfrnc: HKDSE Math M 6 Q9 (a) Sinc (, -) is a point on C, f () a + + b() + a a a f ( ) + b + + b Sinc (, -) is a stationary point of C, f () + + b b () (b) By (a), f ( ) + f ( ) f () < \ P is a maimum point of C. \ Th claim is agrd. () 6 Hong Kong Educational Publishing Company
Mock Eam (c) Whn f (), + - ( ) - + ( - )( - ) or or ln f (ln) > \ C has a minimum point at ln. f (ln) ln ln \ Th coordinats of Q ar (ln, - ln ). () (d) Whn f (), ln < ln ln > f () - + f ln ln ln \, ln ln is th point of inflion. () Not that th quation of L is y - ln. ln () Ara ln ( ln ) d + ( ln) (ln ) ln ln () 7 Hong Kong Educational Publishing Company
Mathmatics: Mock Eam Paprs Modul (Etndd Part) Scond Edition Solution Guid. Rfrnc: HKCEE A. Math 99 Papr Q (a) PA + PB ( + 4) + ( y ) + ( 4) + ( y ) + 8 + 6 + y + 8 + 6 + y + 8 + 6 + y 8 + 6 + y + 8 + 6 + y 8 + 6 + y 8 + 6 + y 6 8 + 6 + y 4 + 8 + 6 + y ( 8 + 6 + y ) 6 + 6 + 4 + y 6 + 6 9 + y y + 9 (b) (i) Lt V cubic units b th volum of watr. + h y V dy 9 + h (9 y ) dy 9 y 9y 9 + h + + ( h) 9( h) + 8 9 h h 7 (9 ) dv dh (ii) h h 7 (8 ) dh (6 h h ) 9 dh (6 h h ) 9 dh 9 (6 h h ) Whn dh is minimum, 6h - h is maimum. Considr 6h h h + 6h ( h 6 h) + + 6 6 h 6h ( h ) + 9 \ Whn h, 6h - h is th maimum. \ Whn th rat of chang of th dpth of watr is minimum, th dpth of watr is units. () Not that whn, y ±. Thrfor th lowr limit is -. Furthrmor, sinc h is th dpth, th uppr limit is - + h. Th rat of chang of volum of watr Th rat of chang of volum of watr pourd into th containr - Th rat of chang of watr vaporatd. 8 Hong Kong Educational Publishing Company
Mock Eam dv (iii) t + 8 ( ) V ( t + ) 8 t t + C, whr C is a constant 6 8 By (b)(ii), whn t, h. V [9() ] 7 \ C t t V + 6 8 Whn V, t t + 6 8 t + t 8 ( t )( t + 4) t or -4 (rjctd) \ Th tim rquird is minuts. (). (a) Th augmntd matri is a b c a ( R R R ; b a R R R c a ) a b a c b a ( R R R ) \ a If c a + b, th augmntd matri is b a. \ (E) is consistnt. Lt z t, whr t is any ral numbr. Thn b - t, y a - b + t. (4) (b) Sinc (F) is consistnt, th systm of linar quations formd by th first quations of (F) is also consistnt, which is quivalnt to (E) for a, b and c α. By (a), α () + 6, and - t, y t, z t, whr t is any ral numbr. Substituting - t, y t and z t into + y - z β, ( - t) + t - t β β 4 () 9 Hong Kong Educational Publishing Company
Mathmatics: Mock Eam Paprs Modul (Etndd Part) Scond Edition Solution Guid (c) Sinc (G) is consistnt, by (a), w hav p + q 7 Lt f + y + z. q 7 - p... (*) f (q - t) + (p - q + t) + t (7 - p - t) + (p + t - 7) + t (from (*)) df (7 p t) + (p + t 7) + d f 6(7 p t) + p 6t + 44 df Sinc f attains its local minimum at z (i.., t ),. t -(7 - p - ) + (p + - 7) + -( - p) + (p - ) + -p + 66p - 84-6(p - )(p - 7) p or 7 d f Whn p, () 6() + 44 8 >, which givs a local minimum. t Whn p 7, d f 7 6() + 44 <, which givs a local maimum. t \ p q 7 - () (). Rfrnc: HKDSE Math M 6 Q (a) Sinc OC is th angl bisctor of AOB, O, A, B and C li on th sam plan. 4 4 6 m n 4m 8n 6 4m n... () Whn O, A, B and C li on th sam plan, th volum of th paralllpipd formd by OA, OB and OC is, i.., OA ( OB OC). Hong Kong Educational Publishing Company
Mock Eam Also, AOC BOC. cos AOC cos BOC OA OC OB OC OA OC OB OC 4 + + + + i j 6 k ( mi + nj + k) (i 4) j ( mi nj k) m + n + m + n + m 4n + 9 m + 4n m + 4n m 4n + 4 m n... () Substituting () into (), 4(n - ) - n - Substituting n 8 into (), m (8) (b) (i) OA OB n n 8 (4) i j k 4 4 6 4i 8j 4k Not that DE // OA OB. \ DE 4ti 8tj 4 tk, whr t is a non-zro ral numbr. OE OD + DE + + 7 4ti 8t j 4t k 8 Sinc DE OE, DE OE 7 (4 t) 8t 8t 4t 4t 8 t t t or (rjctd) 48 \ DE 4 i 8 j 4 k 48 48 48 i j k 8 6 Hong Kong Educational Publishing Company
Mathmatics: Mock Eam Paprs Modul (Etndd Part) Scond Edition Solution Guid (ii) 7 OE 4 i + 8 j + 4 k 48 8 48 48 i + k From (a), OC i + 8j + k. Not that whn E is th incntr of DOAB, E lis on th angl bisctor of AOB, i.., E lis on OC. Sinc th j componnt of OE is whil that of OC is not, it is impossibl to find a ral numbr λ such that OE λoc. \ E dos not li on OC. \ Th claim is disagrd. Whn E is th incntr, E is th point of intrsction of th thr angl bisctors of th triangl. (iii) + OE 4 EA OA OE i + 4 j k + + EA 4 74 Sinc EA OE, E is not th circumcntr of DOAB. \ Th claim is disagrd. (7) Whn E is th circumcntr of DOAB, a circl with E as th cntr and passing through O, A and B can b drawn, i.., EO EA EB. Hong Kong Educational Publishing Company