HKDSE Maths Elective 2 M2 (By-topic 1980-2020) - HKEAA

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M2 Past Paper Binomial Theorem HKCEE Additional Mathematics (1986 P2) (1980 Pl) 2. In the e.xpaus.ion of {x2 + 2}'" in descending po\\."el'.S of x, where n is a positive integer, the coefficient of the third term is 40. Find the value of n and the coefficient of x4 • (S marks) 1. I !Ile eotifici�I <)cf x � hi the exp:in1'loo of f (I +x + x � Y' �- ii �rnd (1981 Pl) I! is a pt,s'ilivi:: iltl�Q.er, foid tile w� of 11 . ,. . . Find Cho coofll<:leat of ;;' In the explmlon al {! + 2.:r} (I - x) , (5 marb) (1988 P2) (1983 Pl) Given (1 t lx} 4 (1 - 2xJ' ,. .I + q::c +- ln: 2 + higJLei: p1lwg;rs □f ,f:. J -4. ) Exi,and O + -ax)'(t - 4x) in Mctt1ding p(J'l'i{CT.J of x up- to and including the term containing x . l fb1d tile w,ru,es of thq �ns!.aml;s. q: a�d b . Giw:n.. t?-�� the coe-flicimt of x -ls ttro.. e-v11,ia1� t� -cneffldent of x�. (7 marks) (1989 P2) (1984 Pl) J. L<t , � ( i - U)' I. {•} U;,ag 111,o bio""1ial thoo-, whet• 11 • ,I, •re .rm. •,:pr,:,• ., In the form " + f>i , (1990 P2) (b) Flrul u.. roal part of f- I. GM11 (I + ZJ; - 3.t' }n ., 1 + ax + f»t1 + �i,m 111:l'OMQI! �r pow,,"' ot :x, whm'<, " lo a posifue int.re Integer, {�) Hlo:Jl'f& a affld b ln let!llilS of Ji. (1984 P2) {t,) iii' � ,. 63. find the 'Qllue- ·af n. (5 marks) L lo. Ille oxl"""i"" o.f (.r" + � f , whore " r;. 0 , the roefficloot of (1991 P2) _.,, is mioled by 8, • Fllld lne muo of a If B,. � 4B, ., I. ·Given tbaot (l • x • .u:1)1 • l ..- ix • lf .., ¥3 -t- temrs in\'olvtoz: hight::r·powen: « x. (1985 P2) (•) Ell.,... � lUld I', ill rem,., ofa. t. fox + .,,..l� )" i:s o:qmded in. <.le�ll®)g powm; o( x , wbm II ili a (I,) If A-1 • 4, find tbe vlll.ue of a_ l""'lfr<e lm!oagor amd " > 0• ]t !he lb�rth tern, of tl& ellpamioo is R�ee find thB vilue of � . (Smarb:J independoot or x and is i:<[1111 w �I , flJ>d th• nfut:• of n m<! ". (S 11»1kl} P. 1 M2 Past Paper Binomial Theorem (1992 P2) (1999 P2) z. In !he HJ>Rllsion of (I ... lx)'(t + :t)", '1/hi,n, " it a �live inteser, (a) Expand (1 + 2x)" in ascending powers of x up to the term x' . 11;e .,,,,.m,,;m or :r ,. IO. where n is a positive integer. l'ind !be v1lh>o of " • (b) In the expansion of (x-�)2 (1+2x)" • the constant teon is 210. (5 -, Find the value of n. (1993 P2) (6marks) 3. (Jjv� (I + 4,t: + �1:f 111 ] .,. �.:-i + .b,i:!· ¢4:t!ef � .involvlnf -t- (2000 P2) msbtt po.,.,. of x • wit,,o • is a pooili"" mlegot. 2. Expand (l. + 2x) 7 (2- x)2 in ascending powern of x up to the term x 2 . (5 c:::�-xs) If a� 20,f"md ,r,md b. (2001) (1996 P2) Find the constant term in the expansion of (2x3 +i-)8. 2. II is &n= Iha! (I + x+ai't = l+6.r+t,.-' H:, ... + terms i!MJi.ving higher powersofr, (2002) (a) E"JI'= Ai and k2 in l<mlS of a. 1. If n :is a positiv,, intege,- and the coefficient of ,? in the expansion of If 6 , t, and k, at¢ m A.P.• :llrul tbe -,alu,:, ,o( • • (l+x)" +(l.+2x)" (1997 P2) 3. is 75. find the value(s) of n . Expand (l+r)"(l-lx)" lnascmiingpawelll;,t' r uptl>tbe!erm r•, w� n isapo:htwcint¢:ger. (4marks) Ifthe ,,,,,n;.,;e,,1 .. ;r' is 54. find tho """'8iei<:at of l'. • {2003) . l 12. Determlne ffilelherihe e,,pm1$ionof (2" 2 +-)' eonsisls cf t (1998 P2) (a) a�torm, (b) an,?- mm, (4 ma:rks) I.!! eao!, pan. fm,j tl,e term ifit HiSIS, P. 2 M2 Past Paper Binomial Theorem (2004) (2011) (a) L It is giv.en that O +x+h2 )3 ::: 1 +ax+b.x:2 + tenns involving higher powers of .x. (a) Express b in terms of k. I l (b-) Find tb,:,ool!IS!ant it:rm in the ei<;mwion.af (1--+ -HI+ 2x)• . X ,x2 (b) If 1, a, b fonnageometricsequence,findthevalueof k. (5 marks) (4 mark,;) (2005) 2. (a) Exparul (1 + y)'. (b) Using(a)�orotherwise,expand O+x+2x2 ) s inascendingpowersof x up to the term x2• (4marks) (2006) 3. It is giv"1l that (1-2x+3x 2 )• =l-10x+h1 + 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 x2 in the expansion of (1-2x+x2 f is 66 , find the value of n and the 3 coefficient of x • (6marks) (2008) 2. (a) Expand (2x+;)' (b) Find.the coefficient of x in the expansion of (3x 2 -x-5\2x+;f (4marks) (2009} I 1. In the binomial expansion of (;,.2 +;J0 , find {a) thecoefficicntof x16 , (b) the constairt term, lb marks) P. 3 M2 Past Paper Binomial Theorem HKDSE Mathematics (Module 2) {19#02) (PP#0l) 1. Findthecoefficientof x5 intheexpansionof (2-x)9 • 2. Let P(x)= 0 (x+A.) 2 3 , where AeR. Itisgiventhatthecoefficientof x3 in the (4 marks) 4 5 (x+JJ' expansion of P(x) is 160. Find (12#02) 2, It is given that (a) ;. ' (I +ax)" =1+6x+l6x2 + terms involving higher pmvcrsof x) (20#01) where n -is a positive :integ� and a is a constant. Find the values of a and n . I. (a) Expand (1-x)' . (5 marks) (b) Find the constant k such that the coefficient of x' in the expansion of (I+ kx)'(J - x)' (13#02) is -3 . (4mru-ks) 2. Supposetbecoe:fficientsofx and x2 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 x2 • (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 1 2. Expand (_"t+3) 5 . Hence,fmdthecoefficientof x3 intheexpansionof (x+3/(x-.± (5marks) ' X) P. 4 M2 Past Paper Binomial Theorem HKALE Pure Mathematics (00 Pl#03) 3. Let n be a positive integer. (l+xY -1 (a) E."q,and ---- in ascending powers of x. X (00 Pl#09) 9. (a) = , Show •'--• en + en.-+l = c,+l >rl-1 __ wm,,.e .. . .,___ n • r are pos1t1ve mtegers __ , ow.i n2':r+l. (2marks) (01 Pl#OS) 5. Let the kill term in the binomial expansion of (1 + :x:) :!n in asceruling powe,s of M X be denoted by T; • i.e. Tg = ct1:x: . 1 (a) If x = - , find the range of values of k such that T;+1 ;,, r, . 3 (_b) Find the greatest term in the expansion if x = .!. and n = 15 . 3 (5ma,ks) (OS Pl#02) 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) P. 5 M2 Past Paper Mathematical Induction HKCEE Additional Mathematics (1983} (1981 Pl} 9. (a) Prove, by mathcrna:UCI! induction, that iQ-i aJI podtiYf.' integca n. w F,,ovc,, lby muh,p,aljw �ll<>A, 1hirt 1 X 2 + 2 :X 3 + J X 4 + . �. + n(n + f) ,.. ½n(n � IXn: + 2). * :• � ... � (6 marb) 1• 4 1 : ¼"{• + 1)(,8 + l) _ {b) On a battle f"mld, cannon-balls are M.acked u M!own in. Figute 2.. F« 11. stack 1Yitb n laY<C#. the 'balls 1n the bQttom fayer au, ammged u shown fn MglUe 3 with JJ balts on a:ch side. For du, �d bottom la}'ff'. th� .arrangemwf ls- .similar bu, cadi Sde co1Js:1$b of (11 - I) balls; for the tl:urd bottom b.yer� eadt side has {,r - 2) balls.. ,and so on. � top layer coasists of 011ly one hall. �J ,_r..J<Ul,f,;albdm 111'11 plkif VJ> ht l11Yru I� £.mn • ;pyBmilWb ,oljj! - ""l"""'-"'f - J: :mflR:t iU .l'bawfl / / ln :flfw>, :!. ll<·i'kl, al / Ibo - ily<c ..,_ i \ ' of n bl!kb� M ; \ ..,. r>f th, ,.,_ l�r .,_d!,;dy tl>ol'o ; l>#.�-11,,m,>nd ....... n..r. ill<!!!ly - b<kk In i'.boO,,p �------� r.y,.,. ro flrul 11M ""- al 11M ,.1,, llry<r """"'!ill tn lll,o ill\t'­ --- -., fu,i iho MJ[UlDO tit/JO � (i) find the number of b2Us in the r-lh layer counting from the top. (ii) thing the mult of {a), 01 otherwise. find the total numl,ct of Cfflfl.oo-h2ils in. .a stack Oil tlmu: !'ht RRllb <!f (-> """ (i,)f!}, - m.11 11,;. - of U.. ...-. 11 oonsistin_g. of n layer.s. o!,,QS pat« lmn thll al'. l'Y'- Of ill• - hdght, � "" w {IJ!} J.f tlu: ume rcqtltr-cd to deliver and fire .11 cmne.n-.ball � ftQJn the r�th l.ye,r It f minutes, rilld the time requi� to deliver JOO fire .11U 1he cann.on-baJh: In 1M Mh layer. liDru:e find ffi': tot=l time- rn-:1:!dcd to � up all tile -cannon�balts. in a std of to byttt. (J4 :mark.} (1985 P2) 2 Let T =i. .r.rfo + - ) ., wli,ete n i5; -a pc,um'c int:egt::r. f'rme by 12 {f'l' + l)' mafb�'l-/e2l bliductlo.n that Tc -� T-= X ... X T.ir ""2{:: 1) for all n, P. 1 M2 Past Paper Mathematical Induction (1986 P2) (1992 P2} 1. Prove, by mathematical induction: that for any positive integer n, p 1 1 1 n --+--+···+�-� lx2 2x3 n(n+l) n+l (5 marks) B (1987 P2) 1. i'rov<', by TMC"'1i!llrticol illidu<ct;oo, lh!,t Fig.ur-e I _l_+ .J.... + + --L... + I = __,,_, In Figu.rc l.,. l"ABCO, iJJ. a right p-Jlllmid with • squtite bai.c of� of' lx4 4x7 Ts10 ... (:'ln-2)(3,n+ l) 3n1 I length 4 cm� L P.A.B - 6(1•. F-ind� ootteoel to- tOO DCP;� 0.1 deg,tt. (a) !he qle 1,,tw.,. 1ho pl,.. PAB .nd I�""'"' ,1/!,;;D , (b) lho_ungi.,-..., lheplaM< PAB md PAD. (1988 P2) {1 marl<s) 5. Provo, by ""1.ll>emafical imlwolioo, that 7. I•) Prove. by matbeIDllieat mdu;:;tioo� that l' + 3> + ... + (2• -1)' = n(2• - !�{2"+ t} for ell p<>,itm i:ntog.<ro " . ('o) U••�E lb• romwl• Ut (�). fmd t]lo sum (1989 P2} 1 x :2 + 2 :i.: 3 -+ • • • "'" t:(11 + l). i. Prov�� by mniu:f111ilit111l tr1d,m·1ii;m.1 llttt I� 1 • l•l • 3 "4½ ... +•(1'+ ])(n•2J = •(Jl+ l){�+l )fn +ll (1992 P2} t. Prove:, by �tbcm:s�i�t imhtclirutt.. that 1x2 +2.x5+3x$+ .... +11(3,tJ-l)=iJ-;i(n-t-l) (1990 P2) 1;. L,t .;, " ,i' + " Ior a;iy tpoSitl"" !Jllepr 11 • (5mub) (1993 P2) r,.,..., by motoomu.lG1I iTidootllon, di,t T, + T, +, •• + T,, w 3l ,r(n + !) (>t + 1) (5 mub) P. 2 M2 Past Paper Mathematical Induction {1997 P2) (2002) 1. Let T ; .. (n' +l}(nl) fat llllyposilive � ,, • 12. (a) Prove, by mathematical induction, that P,-,1,y�mdwtion, that 1i +T, + ...+T• =n[(n+l)lJ tbr"l!YJ)Q6111,,,inkgol' n. for all positive integers n. (N'.llll> : ,.J,.n(n-l){n-Z}···3·2·L'J (b) Show that (<imam) (1998 P2) Pro;,,, by:malbemalical blt!uo&ll, that .{8 mmks) (2003) 1. fi>tall positin: um:g,:n ... B marks} (2000 P2) 4. Prove, by mathematical induction, that for all �l'e � n. (5 lllaTD) (2005) for all positive integers n. (6marlcs) 8. Prove, by mathematical induction. that: (2001) lx2 2x22 3x2;. nx2" -+--+--+ �-· +---- 12. Pro,·e. by mathematical. induction,, that 2x3 3x4 4x5 (n+l)(n+2) 1 for a:11 positive integers n. lx2+2x3+3X4+ ··· +n(n +l)=-n(n+l)(n+2) (5 marks) 3 (2007) fur all positive integers n. 5. Let a ;< 0 and a;, I • Prove by mathematical mduction that Hence evaluate I l I I I a-I a ,l- a" a"(a-1) lx3+2x4+3x5 + ... +50x52. for all positive integers n . (8marks) (5marics) P. 3 M2 Past Paper Mathematical Induction (2008) S. Prove, by mathematical induction, that 13 +23 +33 +···+n3 =..!..n2(n+ 1)2 4 fur all positive inregers n . (Smmi<s) (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) (5 marks) P. 4 M2 Past Paper Mathematical Induction HKDSE Mathematics (Module 2) (19#05) (12#03) 3. Prove, by mathematical induction, that for all positive integm n , 5. (a) • • • • Usmg mathematical induction, prove that L2n l k•nk(k+l) -, n+ I _c..c_;;_ for all positive integers n . n(2n+I) 200 l (5 marks) (b) Using (a), evaluate I; --- . ;.,o k(k+ 1) (13#03) (7 marks) (20#05) 3. Prove, by mathematical induction, fuat for all positive integers n. l+-1-+_l_+_I_+····+---''-- 4n+1 n(n+3) • s. (a) Using mathematical induction, prove that for all positive l lx4 4x7 7x!O (3n-2)(3n+l) 3n+l Bk(k+l)(k+2) 4(n+l)(n+2) (5 marks) integers n. (15#08) 123 50 (b) Using (a), evaluate - prove ·•- ,_,.mductton, • 2x� . nx (n+l)x ,... = sm2cos-- Bk(k+J)(k+2) 8. Using mathemati"'"" Llli1t sm. L.i cos/L,I; - for alt positive (a) integers n. ,., 2 (7 marks) Using{a)� evaluate I;cos ....!. . 561 k (b) k=I 'l (8 marks) (16#05) (-l)"n(n+I) (a) Using mathematical induction, prove that I,(-1/ k 2 for an positive 2 5. kd >integers n • "' (b) Using(a),evaluate 2)-tf+'k' k=3 (6 marks) (18#06) (a) Using mathematical induction, prove that I.,., k(k + 4) n(n+1X2n+13) 6 for all positive integers n. (b) llsing (a). evaluate IC;2J(�;3 h<;JJ 4 ) . (7 marks) P. 5 M2 Past Paper Mathematical Induction HKALE Pure Mathematics (12 Pl#0S) (94 Pl#0S) 5. Let ai =� • a2 = !_ and 6 a11 +i = 5a11 +1 - a,1 for all posittve integers ,r. 2 12 5. Let {11.} be a ,gequeoce ofi-(tive mmabets Sll<:h tbat \TfJ (1+11 (a) Using mathematical induction, prove that = 1- + _!_ for any Jl<lsitive integer n . 1!' 311-1 (111 a� + ",z•�•. -+-a" • for 1t • l,2,3, .... Does there exist a positive integer m such that 'I,a > 3 ? Explain your answer. ,., (b) k (7marks) �"-1�::l•-···"''--� (94 P1#08) •I «. ]" · L iwmben. (b) • Let M • = "" "• 11. . :b: my posmve ill� n • show dnl: b. ,; "• "• , "• and .:0 ate-�"" ml a11o11beo � ... +b._ +c0 • (ft+b+1if (01 P1#02) 2_ (a) Showthat Lr=ln r�, =-n(n+ 1 6 1)(2n+l) . (06 Pl#0l) 1. A sequence {a.} is defined by a, = 1, a1 = 3 and a.,2 = 3a.+1 + a,, for all n = 1, 2, 3 .... Using mathematical induction, prove that 1 a. =--[( 3+ ✓11 ✓17·. 2 ) f -( 3- ✓11 2 )"].· for any positive integer n. (6 marks) P. 6 M2 Past Paper Limits and first principle HKCEE Additional Mathematics {2000 Pl) ..!..( (1980 Pl) 1 l -Ax 2x+3) 3. Show that 2 x + 3 + 2h · Jx+Ax -[; 2. Find Jim {5 marks) h->0 h x+4+h x+4 {1986 Pl) . d(l ) :from.w.stpnnc,p ,,__ . . 1es. ! (b) Fmd - r dx ,JX L Find, from first principles, (..-r3 }. (4marks) (5mru:ks) (1988 Pl) {2003) [. (11-) Simplify {-.J;x + 1 + t:u = ,,/Jt' + 0(-.J.:r +.I+ i!,x + ..JiTIJ. 2. Fmd ! (x') ftmn fimprincipl1;S. (l>) Ln y = .,/x+I . (4marb) (2005) Find!(;) fromfirstprincipJes. (S mllW} 3. (1993 Pl) {2007) 4. Find ! (x2 + 1) from first principles. (4marl<s) {2009) 8. Find ! (..r;:;i) from first principles. {1996 Pl) (4marks) d 2 2.. nm! (x ) bin Jim pn.nciplies. dx (4 :111.mS) {1998 Pl) L Find -j; { ./;:) from ftt$1 principles. {4 marks) P. 1 M2 Past Paper Limits and first principle HKDSE Mathematics (Module 2) (18#01} (SP#0l} L Let f(x) == (x2 -l)e·� . Express f(l + h) in terms of h . Hence, find f'(l) from first principles. (4 marks) 1. Find �(.fi;) from first principles. dx (4 marks) (19#01) (PP#06} 4 3 6. Find !(�) from first principles. 1. Let f(x)=� . Prove that f(l+h)-f(l}= principles. 7+3x� h- h' 10+6h+3h2 . Hence, find f'(l) from first (4 marks) (4marks) (12#01} (20#02} L Let f(x) = e'' . Find f'(O) from first priodples. 2. Define f(x)=� forall x>-2. Find f'(2) fromfirstprinciples. (4 marlcs) (3marks) ,t2+x (13#01} l. Find ! (sin2x) from first principles. (4 marks) (14#03) 2. Consider the curve C : y = x' - 3x (a} Find dy from :first principles. dx (15#01) I. Find _<!_ (x' + 4) from first principles. (4marks) dx (16#02} 2. Prove drat -,=x I 'VX I ,----;- - --tx+n (x+h) x+x x+h -,/-=-- ,5h ..j;-:;: . Hence, find dx-.x dl3 from first principles. (5 marks) (17#01} I. Find ddB sec60 from fmst principles. (5m,u-ks) P. 2 M2 Past Paper Limits and first principle HKALE Pure Mathematics {09 P2#01) {89 P1#03) 1. Evaluate lim {e'-l)sinx , (a) z--)(1I-cos� {90 P1#02) • I - (h,) .Bvt\'!lltl ..�.!,}(ii;+ f)(t t 2) {99 P1#07) 7. A seqnence {a.} isdelinedasfollows: l 1 l a1 =- and ---=2n+5 for n=l.2,3•.... 5 amI an 1 (a) Show that a. , for n = l, 2. 3, .... n-+4n (b.'J\ n ..., .I.U:SU1'Ve -,r+2 ') mto . partta: • Ir_ • aact:mns. (x·+4x)· n Hence or otherwise.. evaluate fun '\' (k + 2)a/ . 7.1--¥-> k-1:=1 (]marks) {03 P2#01) 1. Eva!- (a) . ( --- hm I ) I mx .z-+os-I:nx P. 3 M2 Past Paper Differentiation HKCEE Additional Mathematics ,l lllllll l, lo 1111b • d of �y y <U!i� ""lf<I !,om (1980 Pl} lillll! """' ""'"' 1-h< - :I, "' - <i!. ,i&fm -.. ey!War iH ""' �•, .. - o, Plpro l. n.. A IOlid' ri.g:ht .:il'CWl:f «1a1t .uf v-nlW1ll! If ¢ubM· �,t11es .m:nl: bn:� OOW!', r ;mt:trtt two a �l!4.:1t :$.Urfa� ma: .,-,110{"1... h ...,,,.mdw.iiio,, nt•i>.,._ or w squaK �t-.:s. l '� ,.,r-,,,.,r.o, (I,) � - ""' ........... ,if lh< � {c) �IJ:Hit lhe- en,;'Ph. of 1..s -"" r. Ii k wt,lll, "'1t 0£ 111• -� lo lt. il.o!illo"""!Nnw:lr\Jlh•,.._�t,c C. r (1980 P2} . -•::z::. J. f.inil, 11,e shJJ'I! ,r tht t.am.g,i!D!I. 'l.0 lltl!l ttrv>t ro Flpml .ii Llu: point (!. o, , !t � • 0 , BOIi • to. ..,,.. al v. ,i;, Sb.,.. 11w! 1111< "111< al r p,m • ,n1n1m..,, - •I C . •• (1981 Pl} It" :, ,. - ..L:;::.J. , �•1 �dz ,I-;, . {� !Dlllla} (1982 Pl) ht 11'.�J • r" � n ".I!', � � IIDII b m •••.I. ncOJ...,. J• .,_ �- JG:•) ;;,, /(-f) foull of .>: • -.. .. ·•chlnW. fi!,.-1 ib - - .,r ;,,• - ,/f!J;r; + s. (• lllllm} ,. l.ct :r - J(>) • � - � tl'taJ: thEi �Ji\ 11} [h � "'""'- 'JI :. ;,::T - 9 ;- l + ,3{):!€ + 4 """""' b< �I to ·t!bo i)} - - f(-•) • �Jb). (ij mub} ,,i,,,., x -,I, i i1J1<11 " ;, • P<"'""' "'"'I"'• $-,, <i!l'f•=<l<ti1>s i>oth !idol ot � - vimJi1Y l'llol Iha - - ., y • {(:,} ud ·- _ .. t!btJ ffl ,._, o, 'With �et 00 � :! ;fl,md! the :sum -polm. p.,,.1a1 I • '1;11!! + 3�:1: .. -�� +- (.ri - E)_.x -1 ¼ .!1.X i'!l �·I � --�,i,, ..... ..: Sbt.ck.m � 7 • f(,a '}- � -·• < .r < •, Ht� 1:1.ru! tho ..iu., .,/ I " ·2(1} ♦ :!�' • . , . + 9(l)' + iO(l:)' -�(Iii"" - l(r - !) J> -�-ll • 1,-1;>+1 -16 m..:a) P. 1 M2 Past Paper (ill """"I Differentiation (1983 Pl) Figure. 1 ah= .U1 b0$C.l'� tim,g!e ABC with 1l>e gap!,. ,ot • futwti<m. y �-4 "' 3. BC• 2X and AB • AC. lhe J)fflmetet of A lnii,,.;,,llrffll l < X < Ii -� ·i➔ the triangle b 2 tnetm.. The uiangle b =lVOd about BC so ,; tu fonn a tolid consii:ting or _,..�tho poinb (Z, I�� two e011es with a common li� of r.1dius AD. (3. 15). wl {S. 0). tit, ilJ'phs Expre,s the -rolume of fliis solid 111 tem,s of in·"""'--�;.�­ of sJ.J:: "'dof ..,,; ,..,.,..,.. x. Hence find the VdU<: 9f X' for which lhli; IIX ol4· volume fs a ma.ximum. B � X D X C (6 mttks) -� fll<<qutim, .Of '1>0 Fl- l """" oi y ' IO. IA Figure 4, PQR ii an isoscdes triangle with F base QR = 2r. N is the tuid-point of QR. @ .. -1aemumwm...i l and M ate variable point,: on l'Q and PR, respeett:ldy, such that LM I QR. Lei -painmofiMs,,FI, lM '= x. �f y � (a) fintl x such that the area of t:.LMN is amuimum. (to nw:kliJ (8 marks) (b) Jf the figuu. is n:m,lved ;il,out lW, find x so d12t the volume of the com: Q R generated by lilMN 1$ a maximum. (6 maoo) {c) Show that the volume of 1he eoae generated by revohling the l).LMN specified in (a) about PN is only .,2 ! 7 of the -Y<>hime generated QI; (b}, {6 marb) JL FIJllRI 5 shows a r.a.ll POQ 'With 12 LPOQ = 120 . A rod AB of length ° y7 m Is free to -slide on Uu: 1'-il with it;, ef!dAC1nOP:ara<let1db<m:OQ, 1.ct OA ., .t: mettu :and 08 .., y melRS, (a) (l} Find a tebtion be1ween x and y �d hence fmd the *. '1!1UC of y when :x "' 2. {D) Find ·•·· Given that x and y a� fllnctions of 11me t (ill seconds), lhQ.w th1t '' ' ' . . .. ·f'. . (ID mub) (b) Tha end A is pushed tl)W,Jrds O with a uniform �peed or ½m/s. When A bat :a distanee of 2 ntelfe! ftom Q, fiOO the speed of the end B. (c) Suppci�e the perpendfcol:ir distinct rmm O to !he rod i& p mctrt:s. Sh01'f thtl p = u ,,r; fJ. ' Hence find A.£. when :x: = 2 • di (6 marks} P. 2 M2 Past Paper (1983 P2) Differentiation 11. In Figuri;,, J , Ab is. :a :r.ail:w$11y SO krn lottg. C b; ·s factory h. fdl:o-matre:s s. Find tbi. eqntiom of the two llnet which 11:l"e. both. p:1.r.tlM to the li11¢ Jx -2y= 0 and frolll B .,t<h ,11,,1 LA1'C = 9f:t . Good,; ,,. "" b• ,,_,.d from C tangent te> the ellipse lo A • Thi, trnnspw(•lian •""1 per toM< of goods ""'""' 111" oo,mtl}' 4x2 + y 2 " 16, (6 t11Mlcs) by i,uek i, $ 2 J>'' km, whu,e.. by ,.nvny ii is $ J J>CI km. (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 C fast fa � voh.1mc-. oi' che 1phcre i�,uBlng wm:ff the rurf.;n:;e ilfe;t is: .3-6ffcm l·? (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<ildoo to ronn 11. rlg.M pyramid Wl(h ti<'!5e PQl(S . Let PQ = lx metr,u :awi k.l the irolomc .of tltC" pyr.ttnld .,. V -cubU. meu-es. A A (•) t.,t P be • pooot on iii,: rail..-.y , LPCB = a q , .. l<t $N be th<> 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>) (b) H h = 5-0- • fillow tll!tt tlkC lent tr�rtatlon cost for lQllt)e: -0f good,: from C w A i. $50(,n 1). + (c) (I} Supi""" It > SO.fj . Show !Jut ton B < -A , ,nd <l«luce lit>! �1 < O for llll po,sil,lc ,a!u,,o of 6 , (u") If It = Wll , what roots ,hould i,. takon so th>I the lrl!mp<lt!alioo _, iS tho l<a8t 7 (1985 Pl) {111 Fit1d the �tati,omiry poin!:5- af the _guph or -V , l. Let Find ll-11 cqw.tioM ,of tbe t.me,enf.!I to the. graph �t the :sb;��ri2ry p,;:,i.td!:tnti&t xe]. (5 maib) (12 m:rrks) P. 3 M2 Past Paper 5. Differentiation 9. In Flga'11 4, P(• , l>) i> • point In tile tlr•• quadrant, A \t&tlnblo {�) lino ""'Pitl QR v•= t!uou!jll P wltb 11,� �"" Q oo Ill• ;,.a1<i< a""1 R oo tm _,,,,,:<1,_ Lot L RQO = ti and QR = •. r y R l tl 4 (0 fucpm,s; 1 .!I' term< of "• /I and O. (il) Sh,;., llrat • will he i-t l1lhlm. m 8 = '¥ .�'i , (11 mark!) FiSJU>!' 2 !ihowi; � �sac! in th� .JIAp,i of "' iighl ci.lcular �on� with .base (o)' Hgure S ,h,.- two �c,.,:ri.do.u mu(illg 21 <wit IIJ!� The wld!h 11r tadttii 4 cm ""d hJ!:lght 12 ""'• W.t�r ill l')l)llred lnto· tile 'J,elal!!!] t'1rm1gh ooe oomd<>t ll: 0.8 m !l!!dl that of CM o-tll,er fa i:1 JII • A P!P" !s t!I ilie a,peoc. Fimd tbe volume of the wai�r ill th� V!!Mel wb� the depd! im JIID...Jr fttlffi ono rot!ldor into llie otllcr. <'f Um water .ii; h cent�ttes. If \TIiier 1$ poured int9 th,, ,se!Ml 8l a fl[ts or 1r am� fs, how f,u;I ls the waler le<rol rising wh� the d�plh ad' ti;,;., waler ill 6 i;m 1 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 ,:orJid<l< I< 3 m, •!l;,4 tl,c t.ngci1 of � l□n@OSI: pill" ht an l)c =t...i "'""" tl:le =nor. {9 marks) P. 4 MZ Past Paper Differentiation IL .In l'f&,lro 6, Jjl) Is •� <<im'll,,.11.,nt ,rnl11ted o� • vc<tk•i w•ll BOC (1986 Pl) of a boil<lld:j!. 8D = 20 m , IX = 10 m . a\O <ll>inn,c, at A , x metro, 4. Find the equation of the tangent to the curve x2 + xy + y2 = 7 at the point {2� 1 ). from tho woll. ,"'1ls ll,e •11i:I• �di:d by the >olmtl,.,m,mt a1 h.i ey¢ '" b• it • (6 marks) B 11. Figure 2 shov."S two rods OP and PR in the xy-plane. The rods,. each 10 cm long, are hinged at P. " The -end O is fixed while the end R can mo,,e along the positive x-axis. OL = 20 cm, OR= s cm and LPOR = 8, whe.re O � 0 S 2· y 20m D p !Oen !Oan A C • '"" R L Show tha.t t3lll 9 "" :20x .'{� + 300 Figure 2 (a) Express s 111 terms of 8. By dlffer,ot!,tirng botb. sfoo o, tho """'' ia (•) vritb ,espool lo If R moves from the point Oto the point Lat a speed of 10 cm/s, find the rate of change of 0 "' ,h""" th>( !!! = 20{300 - x') v.ith respect to time when s = 10. dx x• + 1000x1 + 90 000 (5 marks] Berm:. find the 'lt:illtie of l!' f,co: whlch t ls a maxtmum. {6 omle) (c) A square of side£ cm is inscribed in �QPR such that one side of the square lies on OR. Show that 20sinBcosfJ fiad tho ..lt., of l!.! �x . at x = $0 , t-OH•<t W 4 4•cl""'J place,;. £-----. 5in0-+2cos0 Hence .find fJ when the area of the. square is a maximum. H<!ru:::1?· utimart@ the Jnefl?':a:!:li! 11:1 thee di!t.tnct bctwe�n the -obscl:'l,�f attd tho wall lf tho •oglo rubteodod ls t<> ho d«:roa...S by I• '""' (10 marks) that oo,om,d at x = 50 (yom aru,w,or mould b• oo,m:t .,o I�� nearest ...L m}. rn (6 maria) (di) Sketch (l,e graph. of Ii ago,ra,t X for x >0, (1985 P2) Flnd 111,e �qQ;.li<l"" of the lW<J t�m!a"<iU <lrawll to tc!,,,, parab,:,la y 1 .. 4x. (6 marks) P. 5 M2 Past Paper Differentiation 4. 12. Figure 3 .shov."S a rectangular picture of area A cm2 mounted on a rectangular piece of cardboard of B area 3600 crn2 with sides of length :r. cm and y cm. The top, bottom and side margins are 12 cm. 13 cm and 8 cm wide respectively. A C F;guu, J in figure l, AB= 3;::rn�AC= 6cm, BC ,.. ;reui !•) :Exptus x� :i,i. tcmli of D . (ib} lf B laru�;es .at the �-�of½ Iadli:an PH second. iind the r.ru: Qf ctJ311.g,e of :,; \14th respect t-o time ,.,,hem 8 "" i• (6 :macb) 1----x ,m------l A Figure 3 [a) Find .4. in terms of x. (2 marks) (b) Show that the largest value of .4. is 1600. (5 marks) (c) (i) ·Find the range of values of .x for which A decreases as x increases. (ii) 1f x � 50, find the largest -value of A.. FlS?O-� 4(:;i.) lilOW$ a circle- of -eCJ1t:r.:: 0 1uid t2diuJ -II' m�ribed fo (G marks) no Jrosoetei tri:n�-t: ABC wm,, AB,.,. AC, Lit L OAB-= O. 4 X 9 (d) If g$yS 16, find the range of values of x and the 1arges.t value of A. (I) Find). in ltmt� 1!1f a artd 61 � ·th� ™'igbi AH of l!. ABC, Jknct :sliow th�l the :Stea 'Cir t!:.ABC � (7 marks) ��- I + ,fll-B}t sS:n6 o!i;l0 (1987 Pl) (ii) f'"or Wl1.1t i'alue �r fj i!; til� MC:!: of !!.ABC :;: tnlldm:um? ('t�ifog for ma.:drnumfmir.ii�wm h not required.)- (10 m,arb) l I' qt� lb) Li:t f{x) = oog:c I;!;,1 . find 2. Lei :t. = ,. -t 3111)'. 'l!�y l -d;,c Q fi,gu.H' 4(b) �i,rvw1. 11 d,,;Je .of C("t1lrc O :,ind 1:»diU1 b circ1111«riblng :Ut &.au:eJes tri:3ng.le P.QR wJlfo .PQ ""l'.R. U:t l 0QR "" r/J, U) Sbow th::11 th<? 1m:m· of l!, PQR ls b� ci:ti; ,P( l + B.ln?)- (ti) Wf!ffl ll.i'QR fa e,q�illlt,:.,riL �fao,w lfurt its an:a is ai mm)lifflllm. (JO mules) P. 6 M2 Past Paper Differentiation (1988 Pl) 2. find the eqiu:600:s o{ ttte t,vo �oQ:l!!!i:iU: to tlb8 tur'fe y2 = x2 y + i at th; point< wl,,te x = l . (S imrlij RIVER 4. Ll::i y = �ill.X - l:,;o:SX wh.�F.! 0 � X :(: 211'. UNI) Find. '•)· JU'. ood l!:.l!. , ' dX d.x' B Figure S •.n Figure :5., A J5. a fk:(,ed pol.nt ln -water a km from a straight rr.•er b.ilak. (!; marks) B is: a ti:t-ed pt1i,i, on '1a11d h klrt fonti the fivu. M and N are tfo;- po:i□ts Dll 1le. b.iink 1te:uc,.i to A ;md B 1<:5-fH!c-li-vely� P Is. a point bctwf!'il:ln M :md N, Let l Jf.JA� ""' 8 lLtid l.. NOP = tJr- • A 1mn <::m swlm at a s. ·n.� ,.,...c 4 c , y., "'' + "' -.2 cues !he line y " l •t P. sp�cd of u km/h a:1ul nm at a s.pcOO of v km/h .. where u < v. x2 +4 (a) 11,c m.t.n svrl� ft-01'lt A to P- :and llL-cn nms to• B. (i) hp"'"' MN ·in '"""' .,r • , b • 8 ,oo f. lfonc-e show that � = - -are-c·:i:,e d9 bsee,: f, (U) Lei t ho1us 00 11n: t�mit' tabn to tra•�l frorn A to fj 'Ila �-. (\F) For d>0 """"" C , firnl Show tbat J' Ui! .!. sm: a + .(i .! -c: .c. V_,,., 'V· (i) tlm X• ai,d y. m;torcapls; If l Is; 3 mi11mttli1n, :sf!O'W Oi.a1 Iii. '"'· � l-' slur ,#t ' · ; 00 1M l'allflC of val.,.,. of " for which tM okq,B. h nepli\iB (1't:stie1;t fo:r n1a'X.t'IHW1t/mtuimtim i:s: tiot f-fQ.-uJr-ed.) {J2 ma.1'ks) (Iii'> tbs � poifit.!; Dd f« "'""" ptml, - ,,.bocther ii ls a {bJ Ld MN ·= h ki1•. SoppOse ti� man swhms ft0am A mo P and ;maximwn point Of a mini;mlm po.in!. llii!n run, I<> N. (T•lllm! for � ii; 11<1t m11.wil'Od.) (i) Expro» th• llme lakrn in t<m� or a , h • " • u a,,d Q • (ll n,a,:b) (fill Using 01-� r:esul1 fiu �b)(O,. r11.a MP- ln t-crms of wb1m fil<' 1imc- !:ikt!lll ll II mit.limtllct�. (Tit:sth1g for Tilla!l:ITTnimfmio:imum i:. 11ot requiredr} (4 mea:b) (c) Supp<r.,-e C' is a poin1 in W-di!CT ,: km '1'0111 N ;ind CN l MN, If lhl;!: 11m1 �wim� from A �'° C ,,ta r In 11� 'fflklilTIUm time. {d) (i) Exprm: tl1'l ,quorum of tho "'"'" C ln ti,,; foon find MP; JW. (3 marks) y • a + ltr +" (41, li •" are C911Stmm), (1987 P2) ..,. + 4 . 7. H11.d llu! "'ll'otlo,u of tile lw<> Onm,,ntt to 1� c•= x' - y1 = 3 ..,b,ch fl"""" show "1al if x > 1½ &n y >1• �r� pqrallet to Lile Jin• .v "' '.Ix . (ii) In Figuro 2(b), ,t,,;c:h the oun". C for ;r; ;:,. S . P. 7 M2 Past Paper Differentiation 12, (b) C )Q p l'1Q 0 I Flgure 4 Fis.ttl\!, 4 .$hoWJ 3 clreubr paffl oi :rullus. , m� ci:ntrt:d at O , Two nfflJI, X .tnd Y • holding the- ,sruJs or a. loog rod, arc 'Wfflkirig :In lh.e dir«:tkm sbollVll :at a speed of 11 metre$ per xoond. At .a i:.Crl.m:11 lDStant. lhe P1)!rilM .!JD of the. rod subtttl'.ld$ tfl tingle cf () 20-m:Jims lrt O imd jt; 1li 11, �,;,t; i!.'lffllttl!! fi'cm. too mid­ point C of th, rim 11D of 11>, pOOl. {i) Express. t in U!rms. of r end fJ • :figme 3(a) � 11 pi� af pap<!:( !n � sllapi: of a �e®r of ra,diiw Let .A square metres � d1e Bl"(!a -of Ute :alm>t!ed �n. 2 0n l!lld L POQ ., ,pt�. It m made tG (MIil 11 �I �I (U) E.:e.� A in t-enm: of ,- IPJd fJ • of !'lltEll!I rtl\'l whl:ff (IP � w/.'!.h OQ (- FlgJ,r� ;;(!,)t (lo) L;t � (fu m'•-·) ,. !ho ,.,. of ....... of ti...,.. of tho l:'hadcd :r.eg:km wilh: mpcct lO GJne. IE:qirm: d.Ad, ' En t-erms of r , ,8 and u • (Jiml� .!!! == u ) 41 Hence -deduce chat d.A - ni w'ften O • J! , dt 6 (1989 Pl) Jk:n,:.e fimt q ,d,;< .. + 25y , (4 marls) By finding � , detemtiric tlte 'l'lliu� or Fi!id tlte ooo�iJ:iat,e:i: of the 1,wo jXlill1$ -cm ,be cru;rn1 .y "" ,;,;- 3 r,t wbitih. W d¢ J. "•l"'C!IY o.f lb<: � fa • �-- tlll'l�l'.IU, 1(1• tite curvt blM! ii dOpl! ,of ¾. '[NO'le, You lllllY � th" fa,;t that V Is a ,;i:µitoum "IM11 f)t!IX1' f1111d the �uatiora ,cf: tht! twtl ta�I.$ to the c:uni11 Jf' � ;e:l wbidt �t-a p&rau.;·1 io U'li, Li� 3-x - 4y · = 4' .. �, ii; a mam,imi,] (ID 1,•uks) P, 8 M2 Past Paper Differentiation {1989 P2) (a.) .Ex:pr= y ln 1em:d of t11d�. Il- (.rp ,y0) is. a poJ1u. an d:U! p.tn'ttola y :1 ,,:r S.r wlline x ri �Qi_ (b) � � .. O , tind th� 'Yalua -of 11 • 10. A.U1Ut!rlpt<iteub.fey�hobgth: :R.mwt:tu'ld�adtu• rmetrts. (S 't!Ylb:) (,) Floil ¼; . Thi!! wm ,of iu \!rig(h llf!d die- drcumlcrenee o!' .U. Cf01$<�Dn is 2 Jl'lrire!_ H1?:t1:C:t1 sfuow tha1 the: cqmt:ian uf the nmgtatt tu tlw: �ln:lla at (?-'+. Y+) h Ytl!�,., 4x + 4X+ • (4 m::arb) (b) Ut tlJr, total � imi. of tltt eyllrltkr &e S :aqoo-e �i!!lnd. (b) U$lmg. tbe .r---iJJ.lt in (a}. M\.QW 1ha1t. tbf! eqi,llJtioo of the tan&fflt of ;tlope {I) Expum S in ttn:111 od" , • (U) FilllCl the "w.11:n::- ,;,£ r suet. 1b111t • S b- -1 �- m W· 11he puabul11 Ui p • mx • l. . ,. (!i.) S\l{I� O.lS <:r<o.t:. D!itmmiD� (h� UIIIIP, of � -H of r {Qr wbich $ ii, ,.,, 1h, slopes oif 1ho two 1""'-""" from Ille p,int {~-4, ~2) 10 !IV! pu,1>-ob, (!J - {1 n,,rlo,) {l) d!ert!Wlll- {1990 Pl) l½fflec. ..,.. �. liM4 !JUI � -wiluo. or S , (9 nw'ks) 12- I( !'(OJ• l, lied ti>< v3luo of k. 0 F • lkne¢ fipd the e{�tmtions (If the two ,�ngents to C whid1 a� -pan1Uet to l 1t1 Fipl� 2, B , dWl< ust of O • Q tlti! tfflnklm of a nilway OB tit' tne: m1:e .t.· = - --x . 2 (7 lUatl<s) let!gth bbn .and A b.21own ,ir:kzn oorthof O. A road di' is l& bl- b:llilt � _,i to i!m pi{'ll'll.y It. P W (Dt � l:all be 10. (aJ ,tmnsported from A to. B 'YI.I P • n,� .cxist at WilSportiq, 1 to1:ri:o:, d �&- pM .km by red a $ t (.t > I) ma 51 pa mt by 1'1ilfty. l.e1 OP"" xkm ,. whffc O <: 1t < b , mil let UM! -0ffl ol 1r.m59ottlf:l1- 1 toM:I of' pm &om. A to- Jt "1a. P be $ T , Find (1) (Ji) l.1Le (um:ins p-0lnts of C1 ; and fo, ea,ch point. stO:I,!!· whe-th.er it fii) U k"":l_.nnd,.lnUlm18� ir.tbfi�mmuC"·of T, i!',: a ma�iknurc or a minimum point. (lasting ftir maximum/ minimum is �1ot require,.cL} (lll) Fiud tlle n:ng,e oi Wim of t ror- .,.hi.cit .,) s; > ;J� • (t mat!«) k -I He'J:I� de.ll.lmibie tiff! tllll!f.l of wluf::1 af 1; for ..ti5ch it wo'JH (b) w,t � w u:ampmt � from A to B via. P thlllill (4 "'"'"') dfn,et'ty from .A ta 1J 11141�1 uil.t!a ultw;lly. O'l llWi:s) (c) C� UtheClilrvc: y ,. 1 (b) w .t = 2 • F.lttd. in tacw ot ,r • thi. Plin[m!Jm '1:illllc of r !Ol u,ing the ri:,ull of (b) or othoiwi><!, Jko;ch !ho OLITT'e C, {i) ,r,-.1... fo, -3 .;_ ,r < 3 in Flgu,. 3. (li} b "" ½ir. P. 9 M2 Past Paper Differentiation 10- 'The equation of a parabola S k y = x?- - ·2x + 3. u. {a) Let t Oc 1he x-ooordlastfl of st1y polnt P on S. Mad. the p equation of the tallgmt ta S st P. (b} y B x-m 2 In Flgu,:1, 4, 1'0Q i• 1 ,.U w&•re OQ i. horl2ontlll ond Li'OQ � . :," . AB is • rod of Joogtft 3 m "11.eh I, fr,,e t<> >lid• nn tile rel! with imd A ilft OP and md B <>n. OQ . Eod ,..j is iradlllly •t polnt O ond <:ru! B is pu,lled l<>IYO!do (J at a c""5tll>l ,p«d <>r "'! ,..-1 , Afte, , -00<!� ! Jo Ftfttr-e J:, the x-eoo.rdimtc of a po.Int A tm. S :Is. T. /J is � 01'1!1!:.:ms from O end ,the: wd m:i:ik!$ 2n al:'.lgle< 8 with the horizontal .. {2 o,arks} (li) Tht llnc x = l cuts S md T1 at pmnn C -and D :te$�nJy. Ffud the coo«dlaates of C atJ.d D. {1,) .Lot $ m• l>e ·tho am of llAOB. (iii) 'Find the iX!Ord'IrulkS Df anotbet po,int B on ,s S\lCh tftal the :tangent T!! to S cat B pa!!Se:S tmciugb D. Slimv that£• d6 3'1/'!M(l!.. - 29). 3 (1991 Pl) !km:<, Jlml Iii< !l!Ulll'l\lOl ,.aJue <>f S. (c) (i) (JI) Find lh• """A" ..r tho po'5il>t. ""1uc, <>f oo; (; - Q) • (b) di-emax:imum"1td �nlmum vs[uegofy. 9 &,;ic,i .,•.,.nn1m, tho g,,cato.,t .,,,i Jew ..iue, of d . dt (o, mark<) •• (11) Find ,e,qua_i:i:on!I of die ta:ngoot and non:msl. to Cat P. (&) Show Utait the- latlgoot to C at Q � through lb,; �l A(O, '), (1 mad:s) P. 10 M2 Past Paper Differentiation u. (d) To ,.,..., 2(1,), ,k..,,b the gnpb. «I p ,..inst h fur h > 5. (ii) Hence. write ,dl)'l,VO the nnge CJf vmte6-o£ p lot which two­ different� triJff.,a:IM whose- i� d� wi: of radii :3 -cm (:Q'l; tiav,c the _ssmc:, perimct>:r p· cm-. <•.....bl "" ' o s; t----- 21�m ____,. � , ";, FW!U:r-t 2(11) , -:� . .ABC is .a variable � tti:a.np wlt'b AB ,. .AC Sllth lhld the r.adius of i1::5 inscribed circle is J cm. 1'hc b:::igbt.AD and tho lHl&c IJ.C ,of AA.BC 11te ;i�i;l:il;r�1ii>:l-----1----------:i:- � §li;itr 1, em •nil 2., cm n:spectn'�y. � /!I > 6. (Sec, Figmc l{m).) Let fJ cm be tltc: peritrtewr ·of AASC. ·-· • . . 21,• (b) Sbow tlist p. --- •. (h-6)' {c) Fmd (�) ili,!s :r.am,ge. ofviltUCS of b for which � is porilive. {li) the � va1oe of p • P. 11 M2 Past Paper Differentiation J' L FigqRi 3 5bows a� ofrm:lius 2 eeotc«l -A-!- �- point C(O, I). A varwito straighi. 'lina L wi1b po&it:nlc dope� 100'.Qush the-origin O .md makes- an Figun:, 4 (a) shows lill $0Ud oomi:Uing: of a right ;pyrmrld and. a ccboid :mgle e 'llith the. �.:HXis. L � the cirt:.!e at pOmtS A and B·. with a t:DIJUilOO £a.ea whi-.:b b a !Np;urre of s:i:de x cm, Tbei sl.anl. edge of Let S 'OQ. the Area of the SMdi:d.�. Pis the point OD L &ucb that CP the pymnid is � em and ihe height of the cuooi.4 -k; (to - 2.:J)cm. is �iettl.at toAB. Ld: il'CA -=- 4'. wben!:O < x< 5. (a) (i) Find: the lcag:lh of CP in. tmru of 8. Heace $.bow that cos 6 ,. 2 O;)S ,41,, (a) Leth i:::'to be the height of Jhe- solid'. Sbmw tlm,1 h = 10 "' .t. (3 muk,J {ii) soowtb$: s •4t-2 smu. (5 ma"'9) (b) (,) Fimi : i1t tenm of 0 and ,¢1. (o) Hecce find dS intmos of EL Fmd 1'11:1 tattge of '\llli'OOB o.f -11: for whldl. V is (<) !., -rotaw llb!,ut.-0 Ill die. .i;;�i,,. .Jir�on such lhat O &c:roascs i..,reasing. :steadilyat.ar.flltc,of � mdiA��- Findtherate.o.f� Henoo write &own the rzngi:: of vat�es. of .i. fur whtt-h Y '5 dtueasini;g. of S with re&peet lo-ti.me� 8 .. �- 3 {4 motb) (1992 Pl) FJJ?t:1(le, 2. dwwa a 'l.'csscl :iti •ttc dJ::.pe of .11. right cirwliilr � Mith serni� d ver1kal mgle 30". W:a.tcr is flio\lirm.a: oot oftheceooc dirou.g:;h :its:� 'Ill a OOflSl.aDt rate of ;r- �t • {a) Lot V =' ho ll,e vo�..,nh- in Ille ._I wb"' lhe depth ,of Mter is ft em, E:xpcess V in tcro1s or h . "'- g) (b) H¢W fllit is. tl:te v,a� � &Uir:ig, � the cteptlt cf waler i.s 4 em? " ""'""J P. 12 M2 Past Paper Differentiation (c) {1993 Pl) (a) (h} Fiod !ho equation of lb< ""Pl lo C at 11,o point (2, -1). (I mule;} JScnt A,�, The snltd � pb.eed COMPLETELY- imlde .a. 11"&:tangular !box u :shown in fi.e:t1te.i 4 {b). The. base of 1� 00,;, �s: a. S<l'IW'>C: of -stde ' C 3.5 -cm sml the imtltt of lhe box is 7 cm. B (i) Show that 3 :$ .x S ).5. E,___ '1'a----t-- JC:nt --1 Ht!:RCl?l find. <:O<JTI:Ct �o one: decimal piaoe, die �est vo,1ul'1e of the :so1id:. Flgute 1 ™"''" •"""'gbt rod All of longtll 8 m -..,, °" • ,,.,.;..i �u CD of height 1 m. 11t& end. lJ is fTClCI to .slide .along & :OOri:mm.tl (<l) Thi: :Side of the- &Q\l•tu·� b.rse of the l,o,- � {c) is now i::hnn,eed to mii1 � � AB ls. ·v�icaU,y it� tbe mH. � .E b¢ tbc projcctlOll 4.1 cro a.nil the- height .S.5 cm� Fi�d. ootw:t to one decimal of A: oot:lwnU. f:J!t = sm � IJD ... xm� wbe�O < x < 3,rf, place. 11'¢ Jmil:est "olume ,of Hu: $0-]id lbat ,::m be- pl� COM.l'LETfil..V Wt-de the bo"t.. (3 tmrlc,) {•J )2. c, is i.he,; eur'\fit. y .. 2cos2x - 4siJ1i.r -+ 1. w.h1:re osx.�'it". () Ollllb) (i} M Fmd l&:i;; muimum valu of s ... (I mad<s) (ii) (e) {iii) (l2 ......, (i} C, is ,•• w,w )' - ( 'Whcn::OS.x>11'- =· - 4sinx + I l, (ii) Does I' - • .,...;,,,um 'Mlm - 1 ll:<plalA)'Ollt-. s - ;,. USU1$ -thee reiSWl eif (a} (iii), d;etch Uil!: CUf',,'c ¼ in f;gu,. (bJ. s (ii) Hence wri.ce dloWR di& greaecs,; Gd lest values iaf I 12cos�-45:in.:: + 1 !.or O � � S l'l"- (4 m�) P. 13 M2 Past Paper Differentiation 6. 10. (a) :Find the v&Wle of � ..i,t P. dx l( C ..... the .x,axi,, at two poinls 1' ;m,I Q ...,i /'Q = I, (l>) Fino tho _,;.., of tl,o normal to t!io cnrv• at P. f"IIWI tho vsillo(•) 6f k. (7mmb) 9. +o [ (h) Find a,,, •""'8l' ..t: -.aim.. of ,I; .-d> thal C ..,.,. '"' OIJI ' I ' , I Show dJat C _,.. - ,h,oogh lW<> i....r points fur all val- of k not <q1D! to -l. What.,.,, the�� .,f mo 1 I - � (lffl11ts'?' hm ' i1m ... ----- ... II. J.. A B Figure 2 A -II bunp O is pl..,,.! h m •oov• tli• gro...,d, wh.r,, I < h s 5, Vcrticat!y below the lamp is tho -centre of a round table of raCUw 2 m �rid heigM 1 m, The lamp �s .a shadow ABC of the table on t.ht!: S:t'cUttd (c) find � � point(•) of Ibo """'" Y - :fi(x) • • • (�- Figtm! Z), � Sm:: be the .area o!tfu:. .shadow. f.o:r- l.M:b. po,in.l, tes;1, �b�r j1 _is I maximum: or • :tntmm1um poi.ot, 41"11' Show th.at S :::: --- • {h - I)' (4 ""'rb) (b) If!he.lmp i• lowered vertically at• coo,•••O•O: of ¼ m •-\ find Ebe: rate of change of S with n!spc.ct to- time when lt ·= 2. (1995 Pl) {S m&!ks) 3. u...g Ibo .,..,,..,..,. in the !ollowine lable; - Ibo graph of Let V d ·be the volume of the coott OABC. y - Iµ), ....... Iµ) is•Pol}'DOmial. 4 (i) Show that V • .-h' 3(h - I)' (ii) Find the micrimom value of Vas b. varfCS. Does $ ilfflin a minimum wbm V :a.flams its minimum'? Explain yoor -.-. (8 ,..rio,) (Smaw) P. 14 Differentiation M2 Past Pa er 6 12. E 6. Find the oquatians of the two tan� tn - the curv.c C : y= - x+l whii:h are parallel 10 lhc line x+6y+10- o_ ' (7 marl;,;) G C'i-1 ---',,;--,B F . 4x-3 Ci u:thc CUI'\,"(: y = -,-. X +.l {•) Fmd (� tile z. and y. inle""pt, oftliecurve C,; 4 3 (ii) the ran� ofvalues. oC r for which ;- .is decreasing; X +I C """"""° la Fipn:; 4, OAB is the positio.ct :of a .square of si.d e. l. '.'!'he square is (llij the tt.tmin,g: poi.nl(s:) of Ct � staling wbcthcr cscb rolaled mliclod.vru.e about:. 0 to ;a llCW PQA.tion ODEF•.BC culS DE at .fJ pomt is a maximum ot a ttllllltQW1l potnL ('fcstulg for and.OC�roti:EF�H. I.a .l.COC -0,- � ..!.. < Q < .!... '"3Ximum/mln!mum is irot =· 8 4 (a) - • lriangl,, whicl, ,. <ODSNtnl IO (b) In F,gm,, l(a). skdch tllcauvc c, for -lOS .,,au Hoooc, - � the area of fWFH is _('_ . 21ao.2a Omarn) (<) (b) Let S be lhe """ of lhc .,_ ot AOFH ond lhc �I ODGC. Uw,g the rerutt of (b). ·- tho <Ur:VC c, for -1os.. s10 m Flgm,:l(b). (i) 14r-31 llalce write dowD. !be gJeOkSI and ·- vat... of-,-- foe -g. "'+1 (u) -!Osx:ao. (l) mcreuing, (Z} Hcnoo fllld Ilic- minimum vale of S. (ll im:rb) (c) Ficd the ...,.;mum ..i.. ofllle an:a of tho 4""drib!eral CGEH. (2:mad:o) {1996 Pl) 1. Let f(x)=Sll' x. Find f'(r) aod f"(x}. P. 15 M2 Past Pa er Differentiation IL (1997 Pl) I, (3 -) atP. Fig:u,rcJ(;IJ (3matks) f1gn,c 3(;) .i,,,,,. a >=1 ..U, a "'1'2cill' of 24 cd)ic "'"1S. Tho lellglh of !he vessel .. t and its ...,;oa] - .... equlla..-.1 lrnlllgle of si<lo x. Tho """'1 is made ofthin metal pWes aodhauo lid. Let S be □ !he lotal -of """"1 platosmcd ta mal<c lhe vcs:.:L D .□ 64 (a) Soowthat S = .fj x' + .fi, (4marks) 2 • D "' □ (b) F'md !hevoil... of • and t - ""' lhe..,. of mdal plall:S 0 " used tomakethewsse1j$minimwn. . �-) {c) At time tc0, 1hcvcsscl.�Ulpan- �) :iscompldclyfl!lcd lrilh "'31Cr. 5"ppo,;o1he waiu.�at a rate-ni<ioal to IJ km '' " □ ." 1hc.an::a ofwaterswfaceat that :instant such that wbi:<c V and A an: gspe<ti"'1y lhe voiume or water .arid -Ox:: amt or D waler surface at time t , ,.._._ 3(1 lll'--'--t (i) Leth bclhedepthof.....,.,,..,¥CSSclatomc t. (See Figure3(b,),) Sllowthat A:"4/, and V=2Jr2 _ ffenoc. or -•· dh . find -. uwi.rwue, Flgll�1 dt (ii) Fmd the time required for tlle water in thl:. vessel to. Al!W1 � at a'horlzoolal d.ista:Jo<>e,of 30 mm,masight� el""'®!' _. """pl<tdy. of a lll!Thllllgas mmm in Flgme I. TIie � is tisitig- mtleally wnh a unifoo:n. speed,:,£ l.S m -� -l . When 1Jio ,:lmllor is at a ltQghl hm .ab<m, (1996 P2) !he .grl)1:illd. it,; angle cl devation fuJm. lb,;. - i.s e, Find lie rntc oi x" E islhedllpsc - +-'- �1 . 2 7 2 ,, °"""' ebang,,of e wllllrespc,;;tto�wben!beebalorisalalteigbt ,oJim the _.-i. IN""- : Yoo may """"""' tJw U.o .,.,.. of 1J>c clov31<)<" and lbe llllm are negligible.] Let the.line y:c:-m.r.f.c be a tan.gent to E. Show that (.S.1n11rb) 1 c1 =2.m +1. (b) \Ising (a), find 1he <quafi.ollS oftlJe two lang<IJIS from the point (!), S) 10 E. (7mamJ P. 16 Differentiation M2 Past Paper l 12. ) x +kx+-9 10. Th== :function f(.r wbcrc k :is-a. WOllant. attaim a x:i.+l stabmlaly value, 01 X : ). (a) l'im f'(x) in terms or k and r. He!= sboW that k=-6. (4 mm,;) (b) {i) Fmdtlle·x- and )'-inlen:epl&oftlie®t>C y=f(r}; 0 H Fmd th<> .,..;.,... and mmlmwn point, <>f .th<> """"' y=fM. (7malb) (c) Sl=oll1llt gnp11_ar y=f(•l liot -6;;-,:s� in F;p,,,, 3. In Figui-c: 4, OA.B is a or'unit :radius .and LA08�28,wbere l!em:e adi:h. tile graph $CC(Ol' sameliguri,. • a[ yad'-f(•) -• 1. fM ·_.,;;-;r,;;5 . . ut tile 0<8<�- 2 Ci is :an. . inscribed . . . radius s jn the-�- '2. is circle or (5m:ub) :anothet-cin:lc ofr.!dius r 'klUd,.mg 0A. OB :md Ci. I.et E.anii F be the,: ce:mrcs at C, and C,: n:spcctivdy. OA t.ouobes C1 311d C:a: at G and H respectively, (a) (4marl<s) (b) By ccasidcring AOFH mid MJEG. tJq)l'ess r in terms of s. dr·= 0S8(1-3s:m8). Hcnccsbowthat d8 (l+sin,9}" (c) By comidcmg!he ,..g,:,rnf..iue, of 9· fur Which r Is (ij �£.and (ll) dccn:asing. :Ond1ht-trra\"im,rmatea·or� C2: , {No(c.:Ywmay gi,-cyour answ<n C<lffOd to-,lgnificant fig=s.J (Smam) (d} Does the area of circle C1 .attam. a_� when the area of cin:le C, almlnsilS_lllllXimum7 Explainyoorl!IISWU. P. 17 M2 Past Paper Differentiation ll. In 1111t _,.,..,. - - rn11 be _...., COl'1"e<t IO lh..., lO. Let f(x)=2ais2x+4sinx-3 � whcr:c -nsx:S:;rr .. sii,tilk:llrit figures. (a) (0 Find.the x- and y-.interceptsofdte:cmve y•'f{x). (ii) Find the tn8Ximum .arid minimum pomt!, of the -c�· ' c.u,,tryX y= f{x). '41t (!Omaxk>J m (b) h,figurel,sl<etclithecurve yd(x). i· 15km Q Yigure5 p I He11Q; wrik do-wn the � and 1Ctil valu,es ol j2cos2x+4-sinx] f-or -;:�x,s,:r. �2bn Count,yY g In Fw,re s, u.., L � I.lie - of (WO =mes X and r. Amy 13. lives at .place A In Colllltty X while Billy lives at place B in Omnlcy f. l' and Qare respedivtly the fe<t of perpend1cular from A atul B·ro the li<>tder and AP - 4 km, PQ- lS km. QB= 2 km. Amy •anc1 Billy want to mcoi cadI other as early-as possibk at a -certain poi-nt on che- lfflrder. They st.art -i ws:1kissgfrom home to tha.1 poia:it al the same time. !f one arri"VCS , earlier, he/she' Im. [01_1'ai[ fot�otbc,:. I --'Bi"-"; p A____ • L!-L------ X 0 (a) Let R·bea_polntonthebordersuchlhai AR=RB. · Jnfigm,,5,Pllg isamland LPOQ•i· All inrodofleo&'h 1 m (Q P"m<! the distance of R from Q . ftich -is. iree to slide on the tail with cad A on OP =d CDl1 B on DQ. 4i: &dB.Ii [Ii)· Snpp,s,,A:myondlli!lywalkatequahpecdsof 4kmli1 . Iniwlly,cod A isat:lbcpolnton OP Ak:b.1hat LO..f8=9· Elq,laln bridly wl,y 1l,oy ,""'11d waJlc 10 R In omer lo meet - - wilhln .the - ll,nc; !;ind !hi, �-Oot•-,p«d. ADert-..i.. 0,1�,m, """"'51 t!o,e, (Ii nw:b) 4,r OB=ym mid ,£0AB=-6'. where 0s6!!:: g:-, (b) S1jppo,e Billy nm, at a SjlCOd of 8 km h4 ins"'8<l anc! Amy Sill! walks ata lfl"'4 of 4 Im, h-l. To whici, point (a) &pms. :r. .and y in terms of O. on the l>crdcr slu>uld they 80 in order lo meet each otlu,, >rithintheshodcsttimc? (b) Let Sm 2 t,t1he.areaof tJ.OAB. (ii) Suppose !lilly tides on a bicycl,, at � spe,:,! of 16 km h_, --�=,in(�-29). instoad and Amy otill - at a ,peal of 4km h_, . To d9 6 winch J>Oi!'I on theborder should they 80 iu order lo.mcol <ach <lll!er witliln the-.,. limo 7 (10 mm:.,) Hence find tl:,,e'valll'eof o fflCh that s is:a.rnaximum. (6 marks)- {1998 Pl) SK dx -=(6-9) dy $. P(0,2) isapoiotoodlec..,,, :i'-xy+3Y =12. (c) Using(a.)..ffiOwtbat di - (4 ...,1:s) ca,s9 dt (a ) Find lhe nlueof : ar P. A stvdclrt ,..... u,, ful!owmg pn,lie!ion o:g,rding ""'mc,/jon of (d) elld A ofthe rod; (b) Filldlheeqltlfumoflhenorm.,l !Olbecw-Y<" p" As end B maves fiom W. initial position to po.&u. O. end A wil nm move aw&)' ftom Oimd then it will change its. dl.rectioo and mi»"C � O · ls the-srodont's. -ptedi.CUQD correct 1 &.�ain :yo11r answer. (3 .,.,.,) P. 18 M2 Past Paper Differentiation {1999 Pl) Let f(x)=asin.2.x+bcosx� where O"Sx:5:z and a, barecom-tam:s. Firui d . -sm(x 2 +I), L (a) .Figum 2{a) showstbepaphof y= f'(x). dx 2 � sin(x + 1\ (b) 1 dx x (4marks) 6. Thepoint P(a,a) isonthecurve 3x 2 -xy-y2 -a 2 =0, 't'.i:J.ei:e a isa ncn--zero constant. dy (a) Find the value of - at P. dx (b) Find the equation ofthe tangent to the curve at p_ (6marks) 8. ,,•ball / I T / I hm 1 / ' l Figo:re 2(a) / □ □ (a) (,.) Find f'(x) in iEm>, of a, b and x. / □ 40m (u) Usingfigum 2(a), sbow&rt a=-2 and b=-4. (4marks) 0 (b) (i) Firui tbe,:-and ;wmtercepts ofthe cun-e y =f(r}. J-:55m find the maximum :and mi?JimJtID _points of the cun-e. y=f(x). Figure 1 (o) (3 matks) A ball is 1hrown vertically upwar& :from the roof of a building 40 metres in height. After t seconds, the height of the ball above the roof is (d) Let g(x}=iasin2x+boosx-61, 'l'rll.era O ;f;r,;n. Using tbe h metres, where h =20t-St1. At this instant;. the angle of elevation of the result of (e), 'Write doom tber.mg,, ofpossil,l,,-ralues of g(x). ball :6:om a point 0, '11inch is at a horizontal distance of 55 metres from the (2marks) building, is 6. (See Figure 1.) (a) Find (i) tan6 in tenns of I. (.tl.1 the value ◊f 8 when t = 3. (b) Find the:rate-of change of 8 with :respect to time when t= 3. (7 marks) P. 19 M2 Past Paper Differentiation 12 13_ T ------ 1 /J-cm SOUP A mad oowpanyprodnces ,cans ofinstmtsoop_ Each can is in fue form.of a right C}lmda- wiJh • base radim of x cm and a height of h cm (see Figure 6) and its capacity is Vern', wiere Vis com;tmt_ The ems are made oflmn mettl shee!s. The cost oftlre =-ed. surfuoe oflhe can is 1 rent per cm' and the - of fue plane suda<:es is k <le!lts per an'. I.et C """' be figu:rc 5 fue � rosl of me can. Fm- eronomir :reascais, fue wlue of C is Figu,e 5 ·s1u:,ws a RCtal1gle ABCD wilh AB:2an mi AD=21ran, minimised. ,me,, 1r is a posilire mrmh<t- E :md F are -variable poims m 1be sides BC mi CD � mdi that CF :xan mi BE:2xan, (a) E,quss h intem,s of :r,x and J?: ,me,, x is a�-- I.et Scm'deno!e 1beareaof l!.AEF. 2V Haire show that C = -+ mk:r2. (3mmh) (3) Showthat S=i'-2x+21r. (3mm) 3 (b) lf � = 0 , e:,:p,,,,s i' inta:ms of ,r, k and v: {b) Suppose .. =1 · lieooe·showthat C is�miirinmrovdten �-..!__ (6mai:ks) (i) Byronsiilerlngihatpoims E mi F lie<m1besides BC h-2k mi a> re;pecti..iy. show that Osxs¾- (c) Suppose k = 2 and r = 25&<. (ji) Fmd 1he least value of S and 1he � ,-.Jt.. of (i) Fmd 1be'ralues of r ml h. x. (in) Find1be g,ealest value of s. Jflhe v.due of k inaeases, how woold fue dime,,sioos of fue cm be affected ? Explainyour """'"'· (5mmh) (c) Suppose k=¾- Astlldaltsaystbat S isleastwben r = L (d,) The company imends to prodm:e a bigger can of caparily 2V cm', Mlicll is alro·. in fue form of a right Cjfulda-. Suppose fue coots of fue ,cun,-.d samce and plane rumre; of the bigger can are (i) Explainmot!Je:lheSl!ldattisoonect. maintllmed at 1 calf and k cems per om' respecti,,"1y. A v.,:d:er Find1heleastvalueof S. ""gg,,sts fuat fue ratio ofoose radius 1o heigh! of fue bigger can (ii) (4marks) should be twice 1hat of the s:n:ialla: -can in order to mfuimize the prootldion ro;t Explain M!elher fueworl:eris comrt (2 mam) P. 20 M2 Past Paper Differentiation (2000 Pl) d . 2 13. 2. Fmd (a) -sm x, dx (b) (4marks) 4. P(-L2) isapomtoo.1hecw:ve (x+2)(y+3)=5. find dy (a) the value of - at P� dx (b) the equation ofthetangent to the Clln,·e at P_ (5marks) 7 4x 1<1--lOOm-- 10. Let f(x) = _;- . , -� x"" +2 Two boats A and B ace initially located at points P and Q in a lalre respectively, where Q is at a distance 100 m due nocth of P. R is a point (a) (i) Findthe x-andy-imerceptsofthecurve y=f(x). on the lakeside which is at a distance 100 m due west of Q. (See Figure 5.) Starting :from time (in seconds) t = 0, boats A and B sail northwards. At (ii) Find the r.mge of values of x fur whicli f(x) is time t, let the distances travelled by A and B be xm and y m decreasing. respectn-ely, where O $ x $ I 00 _ Let LARB = 0. (iii) Show that the maxrnmm and minimum values of f(x) (a) Express tan LARQ in terms of x. l . Iy. are 4 a:nd -- respecilve 100(100-x+y) Hence show that tan 0 (9marl:s) 10000-lOOy+ xy In Figure 3, sketch the =ve y=f(x) fur -2 S x S 5. 1 (b) (b) Suppose boat A sails uith a constant speed of 2 .m s- and B (3 marks) adjusts its speed continuously so as to keep the value of LA.RB unchanged. (c) T •• _ 7-4sin0 , where v � is real_ J..Gt p- 2 lOOx sin 0+2 (i) Using (a), show that y=---. 200-x from the graph in (b), a student concludes that the greatest and 1 (ii) Find the speed of boat B at t = 40_ least values of p are 4 and -- cespeci:J.vely. Explain whether 2 1 (iii) Suppose the maximum speed of boat B is 3 m :s- • the student is correct. If not, what should be the greatest and least Explain vroetberitis poss,ole to keep the value of LARB values of p? unchanged before boat A reaches Q. (4 marl:s) (12 marks) P. 21 M2 Past Paper Differentiation (2001) (2002) 2 ). _i_( r I. Find Find fbe equation of the tangent to the curve C:y=(x-1)' +4 which is dx 2r+l (3 marlcs) parallel to the line y = 4x+8. 7. P(2, 0) is a po.im on the curve x-(l+siny) 5 = L Find the equation of the (4marks) taagem to the curre at P_ Let x siny = 2002. (5 mrucks) 18. Let f(x) be a polynomial. whei-e -2 5c x S:10 _ Figure 5 (a) sho"W'S a sketch dJ' ofthecurve y=f'(x). where f'(x) denotesthefirstderivativeof f(x). Find (4marks) dx (a) (i) Write down the :range of values of x fo:r which f(x) is 14. increasing. (ii) find the x--c.oordinates of the maximum and minimum points of the cunte y=f(x). i p (ii,) In F,gure 5 (b). draw a possible sketch of the curve y=f(x). X (6marks) B C (b) InFigure5(c).sketclithecurve y=f"(x). I FigureS (2marks) Figure 5 shows an isosceles triangle A.BC with AB = AC and BC= 4. (c) Let g(x)=f(x)+x.whe<e-2sxsl0. D is the foot of -peipendicu1ar from A to BC and P is a point on AD. Let PD=xand r=PA+PB+PC, where OsxsAD. (i) In Figure 5 (a), sketclt the curve y = g' (x). (a) Suppose fbat AD= 3 . (11) A student makes the following note Since the functions f(x) and g(x) are diffuent, dr 2x (i) Showfbat I. the graphsof y=f"(x) and y=g"(x) should be dx ,Jr,--: x2 +4 different (it) Find the rnnge of values of x fur which Explain whether the student is correct or not. (4marks) (1) r is increasing , y (2) r is decreasing . Hence, or otherwise, find the least value of r. --,;- --,''---_,,,,---.+--+c-+r {iii) Find the greatest ,,,due of r. -� 4 8 10 (9marks) Suppose fbat AD = 1. Find the least valoe of r. ! y = f'(x) (b) Figure 5 (a) (3 marks) P, 22 M2 Past Paper Differentiation (2003) t6. 4. 13, Let f(x}=lsinx-r for O sx ,;;,r . Find ibe greo""1t md least "lllue, of f(x). (I mad:s) (2004) In Fia;urec 9', AB-CD is a qwidri!ateral inscribed in a circt,e ecntrcd at O tnd 9. wi1h radiu, r, 1och lhat ,!8 II DC ed O 1;.., i,..i<!e ibe quadrilatml. Lct.CCOD =Wand reflex. LAOB·,::::.2p. where 0< 8 <f < /J<K. Point E de..,,., !be; foot of' P"!'<Ddlculor from O to DC Lei S be di¢ are, of ABCD. - l (ol Sltowmat S:' /rin20-sin2/1+2,in(P-0)J. l (lmub) {b} &,ppo,c /I is i!Xed. Let s, be the J!l12-"" ,.iu, of S ., O �3 In FlpM 3, P{.r,li) it a p:,1n1 • llio - C:y a z:'. 1'k ...... IC C al P ,-,. llroup ffi!IJll"III (0. l}. (Hint; t'OUmi!.yu:selhcidcmit)< sin3a-=3.sina-45ffl1a.J (a) (6 -) (b) Fil!d 1hc Ylilllcl of d S /; , {<) A"'°"£ •II J'O"iol< ..iu.s of /J. t1x, quodr,1.,.,.i ABCD bccomts :a :square ""hen s,, in. (b) ana,lll$ ml _..i... (2005) 9. (a) (b) Let xy+ y2 =2005. Find dy. dx (6marks) P. 23 M2 Past Paper Differentiation 18. (b) (continued) (ii) ·work platform work platform '.X:,.. l· work platform xm Q rail Figure 10 Figure 11 xm I< Figure 10 shows an elevating platform for !lt'ting w-0rkcrs to work at different heights. The horizontal work p]atfonn is supported by two identical pairs -Of .steef shafts. Figure 1 l sho•\'S 1i cross-section of the elevating Figure 12 platform in a vertical plane c:ontain,ing one pair of shafts PQ at1d RS. The two shafts, each of length 4 m. are hinged at their mid-points. Th-c ends P and R of the shafts can mo·ve afong a straight horizontal rail witt Figure 12 shm.vs a vertical cross-section of a scissors-type elevating platform which can bring ide:nlical uniform speed and in opposite directions. Suppos;c that the elevating pfatfom1 is operated under the workers to a greater height. Two more identical pairs of shafts are instaUed on each side of following conditions the elevating platfom1 as shown. Suppose that this elevating platform is operated under the same conditions {*) as described above. Do you thlnk the operation of this elevating platform will comply witft the safety regulation ? Initially, PR= 3.6 m , The work platfonn is then lifted upward by moving the ends P and R. of the shafts towards eacl1 other such that both PR and ff 'Yes', state your reasoning. (*) SQ decrease at a Ltniform rate of ..!:, m s-1 Let PR ""x m at time r s. It is 2 <lx lf "No'. find the range of possible values of in order for the operation of this elevating given that 0.8 5 x 5 3.-6 , dJ platform. to comply with the safety regulation. (J mark,) In this question� numerical answers should be eorr�t to three significant figures. (a) Let- h m be the height of the-work platfonn above the rail at time ts. (2006) (i) Find the range of possible values of h. " d 1'in (2�+ l)l l. Fi!nu-· dx x dh .Y (ii) Show that - = � . GlmamJ dt 2Vl6 �x 1 (5 marks) 12. (a) Let (b) Suppose that the operation of elevating platfo:rms has to comply with the following .safety regulation : Find 11,e equirtloaofthenon••ho the curve x' -,;:v+y' ; 1 attbe pomr (l, l). , (t,) (5 marks) At any instant, lhe ekvating spe-ed of work. platforms should not exceed 2 m s.~1. P. 24 (i) Determine whether the operation orthe above elevating platfonn under the condititlns (*} will comply with this regulation. M2 Past Paper Differentiation :l -i' 15. (2007) Q* 9. i Em ''' ' B' Two rods HA and HB • each ofJength 5 m � arc hinged at H. The rods slide such that A • B • H are on the same vertical plane and A , B move in opposite directions on the horizontal floor, as sbov.n in Figure 3. Let AB be x m and the distance of H from the floor be y m . (a) Write dO\VD. an equation connecting x and y. N k-- lm -➔)>!<I(- Im� (b) When His 3m fromtheground,itsfallingspeedis 2ms-1. Fmdtherateofchangeofthe distance benveen A and B with respect to time at that moment (5marks) fn Figure 5, ABCD is a.horizontal� board ofside 2 m for displaying diamomls. Let M, N be the mid-points of BA and Cl> n,spodively. Thm, identical small bull,s are loeal<d at points N, I' and Q 10. y respectively for illumination purpo� where l! and Q m:c at a height Jim wrtfoally above A and B respo:;tivcly, A diam� is p1aeed at a point S along MN and MS =:rm , where 0:S:x�l. 2 2 Let PS+QS+NS =- lm. y = f'(:c) (a) licp!e$S- i in. terms of x. dl 2x Henceshowfbat. -=---!. dz x>+J ✓ (2 marks) {b) Fmd 'fbe 'i'alues of -x at wbkh l :a.ttaios Figure4 (j) the least value, ml I.et f(x) be a fi.mction of x . Figure 4 shows the graph of y = f'(x) which is a straight line with x- and y-interoep,s 4 aod 2 respectively. (6marks) (a) Findtheslopeofthetangenttotbecurve y=f(x) at x=I. (el Supp°"' !hat the int:nsity of liglit m:qy rocoived by tho diamond from =h bulb vatie> invmoly •• tbc "l.""" Qf1h, di-.:,;oftii, bulb from Ibo <WllDOJli!, with/. ( > 0. in.,,.,_blo (b} Find the x-eoordinate(s) ofall theturningpoint(s} ofthe curve y = f(,,) . For each turning point, ll!tit) bciDg tbc variation eoostimL L<t E (in suilllble unit) be the !Otal inl=ily ofligltt-gy rct;eivc:d by tht diamond fi:om the 1hrce bclbs. determine whether it is a minimum point OT a maximum point. (5 marks) (i) Express E in 'terms of k and x • (n) A smdent guesses thm. when l attains its least val:ue, E wm attain its great.e:;t vahw. Explain. wbether the stodant�s guess is correct or not (4marlo!) P. 25 M2 Past Paper Differentiation 16. 13. Let f(x)=x(x-6)2 . (a) Findthemaximnmandminimumpointsofthegraphof y=f(x). A (b) Sketch the graph of y = f(x) , (7marks) 18. • ''• •• '•• '' Flgur.9 !hem'' ''' q is a circle with centre O and radius l . PR is a variable chord which subtends an angle 20 at _..... .. - O,where 0<0<!:.. C2 isacirdewithcentre O andtouches PR. Lettheareaoftheshaded T __ T�-� ..... ... 2 region bounded by C1 , C2 and PR be A (see Figure 9). (a) Show that In a Wmter Carnival. a display item is in the shape of a right circular cone. lt is made of ice and a stabilizer so that the display� lll the shape of a right circular cone with the volwne remaining constant. Within� duration of the Camival, the height of the cone decreases at a constant rate of 2 cm per day. Attime t day, (i) after the beginning of the Carnival, the base radius and height of the cone are r cm and h cm respectively (see Figure 7). (ii) dA =(n--tan0)sin20 d0 dr r (S marli:s) (a) Show that dt = h . (4marks) (b) When A attairudtsgreatestvalue,findthevalueof tan0. (3 marks) (b) Let S cm2 bethecurvedsurfaceareaof1hecone. (c) A student guesses that when A attains its greatest value, the perimeter ofthe shaded region will also attain its greatest value. Explain 'Whether the studenes guess is correct or not "' [Note: the perim- of the shaded region= PQR +PR+ circumference of C:, .] (ii) At the beginning of 1he Carnival, the height of the cone is 1.2 times 1he base radius. Tho ptel<eepor of the Camlval claims that the curved Slll'face area of the display inereases during the {4marks) whole p<riod oftbe Carnival. (2008) Do you agree wi1h the gatekeeper? Explain your answer. (Smarks) 6. Find the equation of the tangent to the curve y = ;"' at the point (2, !) . X +2 (S marks) P. 26