M2 Past Paper HKCEE Additional Mathematics (1980 Pl) (1981 Pl) ,. , Find Cho coofll<:leat of ;;' In the explmlon al {! + 2.:r} (I - x) (1983 Pl) (5 marb) -4. Ex i, and O + -ax)'(t - 4x) ) in Mctt1ding p(J'l'i{CT.J of x up- to and including the term containing x l Giw:n.. t?-�� the coe-flicimt of x -ls ttro.. e-v11,ia1� t� -cneffldent of x�. (1984 Pl) J. Li , whet• 11 • ,I, •re .rm. (b) Flrul u.. roal part of f - Integer, (1984 P2) L lo. Ille oxl"""i"" o.f (.r" + � f , whore " r;. 0 , the roefficloot of _.,, is mioled by 8, • Fllld lne muo of a If B,. � 4B, ., (1985 P2) t. fox + ..l )" i:s o:qmded in. <.le�ll®)g powm; o( x , wbm II ili a .,,� l""'lfr 0 • ]t !he lb�rth tern, of tl& ellpamioo is independoot or x and is i:<[1111 w �I , flJ>d th• nfut:• of n mRllsion of (I ... lx)'(t + :t)", '1/hi,n, " it a �live inteser, 11;e .,,,,.m,,;m or :r ,. IO. l'ind !be v1lh>o of " • (5 -, (1993 P2) 3. (Jjv� (I + 4,t: + � 1 :f 111 ] .,. �.:-i + .b,i:!· -t- ¢4:t!ef � .involvlnf msbtt po.,.,. of x • wit,,o • is a pooili"" mlegot. If a� 20,f"md ,r,md b. (1996 P2) 2. II is &n= Iha! (a) (I + x+ai't = l+6.r+t,.-' H:, ... + terms i!MJi.ving higher powersofr, E"JI'= Ai and k2 in ltbe!erm r•, w� n isapo:htwcint¢ :ger Ifthe ,,,,,n;.,;e,,1 .. ;r' is 54. find tho """'8iei<:at of l'. • (1998 P2) t (4 ma:rks) Binomial Theorem (1999 P2) (a) (b) (2000 P2) 2. Expand E xp and (1 + 2x)" in ascending powers of x up to the term x' . where n is a positive integer. In the expansion of (x-�) 2 (1+2x)" • the constant teon is 210. Find the value of n. (6marks) (l. + 2x) 7 (2- x) 2 in ascending powern of x up to the term x 2 (5 c:::�-xs) (2001) Find the constant term in the expansion of (2x 3 +i-)8. (2002) 1. If n :is a positiv,, intege,- and the coefficient of ,? in the expansion of {2003) (l+x)" +(l.+2x)" is 75. find the value(s) of n . l 12. Determlne ffilelherihe e,,pm1$ionof (2" 2 +-)' eonsisls cf (a) a�torm, (b) an,?- mm, I.!! eao!, pan. fm,j tl,e term ifit HiSIS, (4marks) P. 2 M2 Past Paper (2004) (a) (b-) I l Find tb,:,ool!IS!ant it:rm in the ei<;mwion.af (1--+ -HI+ 2x)• . X ,x2 (2005) 2. (a) Exparul (1 + y)'. (4 mark,;) (b) Using(a)�orotherwise,expand O+x+2x 2 ) s inascendingpowersof x up to the term x 2 • (4marks) (2006) 3. It is giv"1l that (1-2x+3x 2 )• =l-10x+h 1 + ttrmsinvolvinglrlgberpowersof :r. whote " is a posili"" ml>:gcr and k is•� Flru! 1he values of • and k. (Smm:s) (2007) 12. If the coefficient of x 2 in the expansion of (1-2x+x 2 f is 66 , find the value of n and the coefficient of x 3 • (2008) 2. (a) Expand (2x+;)' (b) Find.the coefficient of x in the expansion of (3x 2 -x-5\2x+;f (2009} I 1. In the binomial expansion of (;,.2 +;J0 , find {a) thecoefficicntof x 16 , (b) the constairt term, (6marks) (4marks) lb marks) (2011) L It is giv.en that O +x+h 2 ) 3 ::: 1 +ax+b.x: 2 + tenns involving higher powers of .x. (a) Express b in terms of k. (b) If 1, a, b fonnageometricsequence,findthevalueof k. Binomial Theorem (5 marks) P. 3 M2 Past Paper HKDSE Mathematics (Module 2) (PP#0l) 1. Findthecoefficientof x5 intheexpansionof (2-x) 9 • (12#02) 2, It is given that (I +ax)" =1+6x+l6x2 + terms involving higher pmvcrsof x) where n -is a positive :integ� and a is a constant. Find the values of a and n . (13#02) (4 marks) (5 marks) 2. Supposetbecoe:fficientsofx and x 2 intbeexpansionof O+ax} are -20 and 180 respectively. Findthe values of a and n. {14#01) L In.the expansion of· (l -4xj1'(t +xt'" -. the coefficient of x is I . (a) Findthe-vaJm::of n. (b) Find the coefficient of x 2 • (4mm) (16#01) 1. Expand (5+x}'. Henca,findtheconstanttermin1heexpanslonof (5+x)'(1-i)' (17#02) 2- Let (!+ax)'= L"•"" and where a and b are constants. It is given -"""' that J. 2:μ7=7:4 and J. 1 +μ,+6=0. Find a. (5 marks) (18#02) 2. Expand (_"t+3) 5 . Hence,fmdthecoefficientof x 3 intheexpansionof (x+3/(x-.± 1 ) 2 (5marks) ' X Binomial Theorem {19#02) 2. Let P(x)= 0 (x+A.) 2 3 , where AeR. Itisgiventhatthecoefficientof x 3 in the 4 5 (x+JJ' expansion of P(x) is 160. Find (a) ;. ' (20#01) I. (a) Expand (1-x)' . (b) Find the constant k such that the coefficient of x' in the expansion of (I+ kx)'(J - x)' is -3 . (4mru-ks) P. 4 M2 Past Paper HKALE Pure Mathematics (00 Pl#03) 3. Let n be a positive integer. (a) (00 Pl#09) (l+x Y -1 E."q,and ---- in ascending powers of x. X 9. (a) Sh •'--• en en c>rl-1 __.,___ . . __ , ow = , + .-+l = ,+l wm,,.e n • r are pos1t1ve mtegers ow.i n2':r+l. (01 Pl#OS) 5. Let the kill term in the binomial expansion of (1 + :x:) :!n in asceruling powe,s of X be denoted by T; • i.e. T g = ct1:x: M (a) (_b) (OS Pl#02) 1 If x = - , find the range of values of k such that T;+1 ;,, r, . 3 Find the greatest term in the expansion if x = .!. and n = 15 . 3 (5ma,ks) 2. Foranytwopositive!ntegers k and n, lei T, betm:rthttnnmdicexpaosion of {l+x) "' inascending�ofx, i.o, r,�¢:.',x'·'. (a) Suppose x = I . Find, m wnns of It and n , th,, rail!!" of�- of , such that T,.1 s T, , (b) Sttpposo x � l . Using me result of (a). find tlle greatest term in the , cxp.m$il)tl of (l + x) 51 (6marks) (2marks) Binomial Theorem P. 5 M2 Past Paper HKCEE Additional Mathematics (1981 Pl} w F,,ovc,, lby muh,p,aljw �ll<>A, 1hirt 1• * :• � ... � 41 : ¼"{• + 1)(,8 + l) _ �J ,_r..J ht l11Yru I� £.mn • ;pyBmilWb ,oljj! - ""l"""'-"'f - J: :mflR:t iU .l'bawfl ln :flfw>, :!. ll<·i'kl, al Ibo - ilyf th, ,.,_ l�r .,_d!,;dy tl>ol'o l>#.�-11,,m,>nd ....... n..r. ill """ (i,)f!}, - m.11 11,;. - of U.. ...-. 11 o!,,QS pat« lmn thll al'. l'Y'- Of ill• - hdght, � "" w --- Mathematical Induction (1983} 9. (a) Prove, by mathcrna:UCI! induction, that iQ-i aJI podtiYf.' integca n. 1 X 2 + 2 :X 3 + J X 4 + . �. + n(n + f) ,.. ½n(n � IXn: + 2). (6 marb) {b) On a battle f"mld, cannon-balls are M.acked u M!own in. Figute 2.. F« 11. stack 1Yitb n laYemafical imlwolioo, that l' + 3> + ... + (2• -1)' = n(2• - !�{2"+ t} for ell p<>,itm i:ntog.t + 1) (5 mub) (5 marks) Mathematical Induction (1992 P2} p B Fig.ur-e I In Figu.rc l .,. l"ABCO, iJJ. a right p-Jlllmid with • squtite bai.c of� of' length 4 cm� L P.A.B - 6(1•. F-ind� ootteoel to- tOO DCP;� 0.1 deg,tt. (a) (b) !he qle 1,,tw.,. 1ho pl,.. PAB .nd I�""'"' ,1/!,;;D , lho_ungi.,-..., lheplaM< PAB md PAD. {1 marl : ,.J,.n(n-l){n-Z}···3·2·L'J (tall positin: um:g,:n ... B marks} (2000 P2) 4. Prove, by mathematical induction, that for all positive integers n. (2001) 12. Pro,·e. by mathematical. induction,, that 1 lx2+2x3+3X4+ ··· +n( n +l)=-n(n+l)(n+2) 3 fur all positive int eger s n. Hence evaluate lx3+2x4+3x5 + ... +50x52. (6marlcs) (8marks) (2002) 12. (2003) 1. (2005) (a) (b) Prove, by mathematical induction, that for all positive integers n. Show that for all �l'e � n. 8. Prove, by mathematical induction. that: nx2" lx2 2x2 2 3x2 ;. -+--+--+ �-· +---- 2x3 3x4 4x5 (n+l)(n+2) for a:11 positive integers n. (2007) 5. Let a ;< 0 and a;, I • Prove by mathematical mduction that I l I I I a-I a ,l- a" a"(a-1) for all positive integers n . Mathematical Induction .{8 mmks) (5 lllaTD) (5 marks) (5marics) P. 3 M2 Past Paper (2008) S. Prove, by mathematical induction, that 13 +23 +33 +···+n3 =..!..n2(n+ 1)2 4 fur all positive inregers n . (2009) 5. Prove-by mathematical induction that for all positive integers n � I 1x4+2x5+3x 6+· •· + n{n+3)= -n{n + l)(n + 5) Mathematical Induction (Smmiintegers n • "' (b) Using(a),evaluate 2)-tf+'k' k=3 (18#06) (a) Using mathematical induction, prove that I. k(k + 4) ,., integers n. (b) llsing (a). evaluate IC; 2 J(�; 3 4) . h<;JJ (-l)"n(n+I) 2 n(n+1X2n+13) 6 (5 marks) (5 marks) for alt positive (8 marks) for an positive (6 marks) for all positive (7 marks) (19#05) 5. (a) (b) (20#05) s. (a) (b) Mathematical Induction • • • • 2n l n+ I Usmg mathematical induction, prove that L -, _c..c_;;_ for all positive integers n . 200 l Using (a), evaluate I; --- . ;.,o k(k+ 1) Using mathematical induction, prove that integers n. Using (a), evaluate 123 50 Bk(k+J)(k+2) k•nk(k+l) n(2n+I) (7 marks) • l Bk(k+l)(k+2) n(n+3) for all positive 4(n+l)(n+2) (7 marks) P. 5 M2 Past Paper HKALE Pure Mathematics (94 Pl#0S) 5. Let {11.} be a ,gequeoce ofi-(tive mmabets Sll<:h tbat (1+11 f a� + ",z•�•. -+-a" • \TJ for 1t • l,2,3, .... (94 P1#08) L �"-1�::l•-···"''--� iwmben. (b) (01 P1#02) 2_ (a) (06 Pl#0l) I «. •• "·] Let M • = "" "• 11. . :b: my posmve ill� n • show dnl: b. ,; "• "• , "• and .: 0 ate-�"" ml a11o11beo � ... +b ._ +c 0 • (ft+b+1if L n , 1 Showthat r� =-n(n+1)(2n+l) . r=l 6 1. A sequence {a.} is defined by a, = 1, a 1 = 3 and a., 2 = 3a.+ 1 + a ,, for all n = 1, 2, 3 .... Using mathematical induction, prove that a. =-1-[(3+ ✓11 f -( 3- ✓11 )"]·. for any posi t ive integer n. ✓17·. 2 ) 2 (6 marks) Mathematical Induction (12 Pl#0S) 5. Let ai =� • a 2 = !_ and 6 a 11 +i = 5a 11 + 1 - a, 1 for all posittve integers ,r. 2 12 (a) Using mathematical induction, prove that (1 11 = 1- + _!_ 1 for any Jl 3 ? Explain your answer. ,., (7marks) P. 6 M2 Past Paper HKCEE Additional Mathematics (1980 Pl) 2. Find Jim ..!..( 2x + 3 + 2h h->0 h x+4+h {1986 Pl) L Find, from first principles, ! (..-r 3 }. (1988 Pl) 2x+3)· x+4 {5 marks) (4marks) {2000 Pl) 3. (b) {2003) 1 l Show that Jx+Ax -[; F. d(l ,,__ . 1 md - r) :from.w.stpnnc,p es. dx ,JX [. (11-) Simplify {-.J;x + 1 + t:u = ,,/Jt' + 0(-.J.:r +.I+ i!,x + ..JiTIJ. 2. (l>) Ln y = .,/x+I Fmd ! (x') ftmn fimprincipl1;S. (1993 Pl) {1996 Pl) 2.. d nm! dx (x 2 ) bin Jim pn.nciplies. {1998 Pl) L Find -j; { ./;:) from ftt$1 principles. (2005) (S mllW} 3. Find!(;) fromfirstprincipJes. {2007) 4. Find ! (x2 + 1) from first principles. {2009) 8. Find ! (..r;:;i) from first principles. (4 :111.mS) {4 marks) Limits and first principle -Ax (5mru:ks) (4marb) (4marl-2. Find f'(2) fromfirstprinciples. ,t2+x (4 marks) (4 marlcs) P. 2 M2 Past Paper HKALE Pure Mathematics {89 P1#03) {90 P1#02) • I - (h,) .Bvt\'!lltl .. �.!,}(ii;+ f)(t t 2) {99 P1#07) 7. A seqnenc e {a.} isdelinedasfollows: (a) l 1 l a1 =- and ---=2n+5 for n=l.2,3•.... 5 amI an 1 Show that a. , for n = l, 2. 3, .... n-+4n (b.\ n ..., r+2 • Ir_ • 'J .I.U:SU1'Ve -, ') mto partta: aact:mns. (x·+4x)· Hence or otherwise.. evaluate fun '\' n (k + 2)a/ . 7.1--¥-> k-1:=1 {03 P2#01) 1. Eva!- (a) . ( I I ) hm --- .z-+o s-I:nx mx (]marks) {09 P2#01) 1. Evaluate (a) lim {e'-l)sinx , z--)(1 I-cos� Limits and first principle P. 3 M2 Past Paper HKCEE Additional Mathematics (1980 Pl} A IOlid' ri.g:ht .:il'CWl:f «1a1t .uf v-nlW1ll! I f ¢ubM· �,t11es .m:nl: bn:� OOW!', r ;mt:trtt two a �l!4.:1t :$.Urfa� ma: or w squaK �t-.:s. {c) �IJ:Hit lhe- en,;'Ph. of 1 ..s -"" r. (1980 P2} J. f.inil, 11,e sh JJ'I! ,r tht t.am.g,i!D!I. 'l.0 lltl!l ttrv>t •• .ii Llu: point (!. o, , (1981 Pl} ,l lllllll l, lo 1111b • d of �y y .,._ (I,) � - ""' ........... ,if lh< � ,.,r.o, Ii k wt,lll, "'1t 0£ 111• -� ,.,r-,, lo lt. il.o!illo"""!Nnw:lr\Jlh•,.._�t,c C. ro !t � • 0 , BOIi • to. ..,,.. al v. ,i;, Sb.,.. 11w! 1111< "111< al r p,m • ,n1n1m..,, - •I C . Differentiation '� r -•::z::. l Flpml It" :, ,. - ..L:;::.J. , � dz �•1 ,I-;, {� !Dlllla} (1982 Pl) ,. ht 11'.�J • r" � n ".I!', � � IIDII b m •••.I. .,_ �- JG:•) ;;,, /(-f) foull ncOJ...,. of .>: • -.. .. ·•chlnW. fi!,.-1 ib - - .,r ;,,• - ,/f!J;r; + s. l.ct :r - J(>) • � - i)} - - f(-•) • �Jb). l'llol Iha - - ., y • {(:,} ud ·- _ .. t!btJ ffl ,._, o, -polm. p.,,.1a1 Sbt.ck.m � 7 • f(,a'}- � -·• < .r < •, l(r - !) -�(Iii"" - --�,i,, ..... J> -�-ll • 1,-1;>+1 -16 m..:a) J • � tl'taJ: thEi �Ji\ 11} [h � 'JI :. ;,::T - 9 ;- l + ,3{):!€ + 4 "'""'- (• lllllm} """""' b< �I to ·t!bo (ij mub} ,,i,,,., x -,I, i i1J1<11 " ;, • P<"'""' "'"'I"'• $-,, s i>oth !idol ot � - vimJi1Y 'With �et 00 � :! ;fl,md! the :sum I • '1;11!! + 3�:1: .. -�� +- (.ri - E)_.x ..: -1 ¼ .!1.X i'!l �·I � Ht� 1:1.ru! tho ..iu., .,/ I " ·2(1} ♦ :!�' • . , . + 9(l)' + iO(l:)' P. 1 M2 Past Paper 1l>e gap!,. ,ot • futwti0 """" oi y ' @ .. -1aemumwm...i -painmofiMs,,FI, �f y � (to nw:kliJ ' -� ·i➔ ·•·· .. .. . '' ' ' ·f' (ill """"I "' (1983 Pl) 3. Figure. 1 ah= .U1 b0$C.l'� tim,g!e ABC with BC• 2X and AB • AC. lhe J)fflmetet of the triangle b 2 tnetm.. The uiangle b =lVOd about BC so ,; tu fonn a tolid consii:ting or two e011es with a common li� of r.1dius AD. Expre,s the -rolume of fliis solid 111 tem,s of x. Hence find the VdU<: 9f X' for which lhli; volume fs a ma.ximum. A (6 mttks) � B X D X C IO. IA Figure 4, PQR ii an isoscdes triangle with base QR = 2r. N is the tuid-point of QR. l and M ate variable point,: on l'Q and PR, respeett:ldy, such that LM I QR. Lei lM '= x. (a) fintl x such that the area of t:.LMN is amuimum. (8 marks) (b) Jf the figuu. is n:m,lved ;il,out lW, find x so d12t the volume of the com: generated by lilMN 1$ a maximum. (6 maoo) Fl- l F Q R {c) Show that the volume of 1he eoae generated by revohling the l).LMN specified in (a) about PN is only ! 7 of the -Y<>hime generated QI; (b}, .,2 J L FIJllRI 5 shows a r.a.ll POQ 'With LPOQ = 120 ° A rod AB of length y7 m Is free to -slide on Uu: 1'-il with it;, ef!dAC1nOP:ara the ellipse Jx -2y= 0 and 4x 2 + y 2 " 16, (1984 Pl) 2. The sut"fne a.1-ea: or .a $_F1iu,a fa tn�f!:Uhng :al a r.:rte of S cm 1 / s. Ho'iW fast fa � voh.1mc-. oi' che 1phcre i�,uBlng wm:ff the rurf.;n:;e ilfe;t is: .3-6ffcml·? (B ma11t:s.) 6. ABC is tJ. trJllrsfe :Iii whJ,clt AD "" AC ani, .lBAC r 2.9' _ The median AD• Ir . Find :a point P on AD � th;iit IM product of tM: d.i:s:bnoes fram P to the dtrce sfd.es af 6..tBC 13 3 maximum. (6 marks) to. Jn FiSUn 2, .A.BCD E3. a 5';!_UAM tb\ pla� or:!:� 2...,fft, m. l'QRS &- iii � who1t1 -ccn!�C � with 1hit tif A8CD. Tbe :;f1lld�d p�rt:s _ uc (:tit off :and the rcnu:[.:id11g put is f'' km, whu,e.. by ,.nvny ii is $ J J>CI km. (•) (b) C A t.,t P be • pooot on iii,: rail..-.y , LPCB = q , .. a l tot al trao�O{btion -coot �r S tonne of goo-ds: fro1n C to P a:n:d then to A • Fmd N in tCl!11> of fl Md h (4 mark>) H h = 5-0- • fillow tll!tt tlkC lent tr�rtatlon cost for lQllt)e: -0f good,: from C w A i. $50(,n + 1). (c) (I} S up i""" It > SO.fj . Show !Jut ton B < -A , ,nd ! �1 < O for llll po,sil,lc ,a!u,,o of 6 , (u") If It = Wll , what roots ,hould i,. takon so th>I the lrl!mp!' 2 !ihowi; � �sac! in th� .JIAp,i of "' iighl ci.lcular �on� with .base tadttii 4 cm ""d hJ!:lght 12 ""'• W.t�r ill l')l)llred lnto· tile 'J,elal!!!] t'1rm1gh ilie a,peoc. Fimd tbe volume of the wai�r ill th� V!!Mel wb� the depd! <'f Um water .ii; h cent�ttes. If \TIiier 1$ poured int9 th,, ,se!Ml 8l a fl[ts or 1r am� fs, how f,u;I ls the waler le) i> • point In tile tlr•• quadrant, A \t&tlnblo lino ""'Pitl QR v•= t!uou!jll P wltb 11,� �"" Q oo Ill• ;,.a1t ll: 0.8 m !l!!dl that of CM o-tll,er fa i:1 JII • A P!P" !s t!I im JIID...Jr fttlffi ono rot!ldor into llie otllcr. 2.7m (i) lf tloo pipe i> ·11> Ile •roit11'foboly oo !h• bi:iru:oota1 floor 'll!mm i! "' l>e,o g rnD'IW w"'1<1 lhe eornex, w®I ls tba gr.al� pombie looEl]I of tlte pJf>1i 1 (ll) If the he!.e;flt of tho ��"'i of eaeh ,:orJidinn,c, at A , x metro, from tho woll. ,"'1ls ll,e •11i:I• �di:d by the >olmtl,.,m,mt a1 h.i ey¢ '" b• it • B 20m D A C Show tha.t t3lll 9 "" :20x .'{� + 300 By dlffer,ot!,tirng bo t b. sfoo o, tho """'' ia (•) vritb ,espool lo "' ,h""" th>( !!! = 20{300 - x') dx x• + 1000x 1 + 90 000 Berm:. find the 'lt:illtie of l!' f,co: whlch t ls a maxtmum. {6 omle) fiad tho ..lt., of l!.! at x = $0 , t-OH• ho d«:roa...S by I• '""' that oo,om,d at x = 50 (yom aru,w,or mould b• oo,m:t .,o I�� nearest ...L m}. rn (di) Sketch (l,e graph. of Ii ago,ra,t X for x > 0 , (1985 P2) Flnd 111,e �qQ;.li

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