TEST CODE O2234OIO FORM TP UNIT2-Paper0l. t hour 30 minutes. 0l JUNE 2016 (a.m.)

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1 -L TEST CODE OOO FORM TP 68 MAY/JUNE 6 CARBBEAN EXAMNATONS COUNCL CARBBEAN ADVANCED PROFCENCY EXAMNATON@ PURE MATHEMATCS ANALYSS, MATRCES AND COMPLEX NUMBERS UNT-Paperl hour mnues l JUNE 6 (a.m.) READ THE FOLLOWNG NSTRUCTONS CAREF'ULLY. Ths es consss of ems. You wll have hour and mnues o answer hem. n addon o hs es bookle, you should have an answer shee. Do no be concerned ha he answer shee provdes spaces for more answers han here are ems n hs es. Each em n hs es has four suggesed answers lefered,, (C),. Read each em you are abou o answer and decde whch choce s bes. On your answer shee, fnd he number whch conesponds o your em and shade he space havng he same leer as he answer you have chosen. Look a he sample em below. Sample ern The expresson ( +.6 )' s equvalen o l l+"6 +Jl Sample The bes answer o hs enr s * +.6,,,,o has been shaded. 6' f you wan o change your answer, erase conrpleely before you fll n your new csoce. ff 7 ' When you are old o begn, urn he page and work as quckly and as carefully as you can f you canllo answer an ern, go on o he nex one. You nay reul.'l o ha enr aer. 8. You nray do any rough work n hs bookle. 9' The use of slen, non-prograrrnrable scenfc calculaors s allowed. E E -E E E - - Exanr naon Maerals: A ls of,raher,acal fornrulae and ables. (Revsed olz) DO NOT TURN THS PAGE UNTrL YOU ARE TOLD TO DO SO. /CAPE 6 Copyr glr A Carbbean Exanr naons Counc All rghs reserved.

2 -- The complex number z = can be Whch of he followng s r a skech - of he locus of he pon represened represened on an Argand by dagram he as complex number z, gven ha lz + :? m z - z7, ) Re -.r m - z (, -) Re - fm z Gl' l) Re fm - T, & - 7'7 Re - /CAPE 6 GO ON TO THE NEXT PACE

3 ,, )_.l - l l l,l!l r l : r rl l l l l l J l, j, J,, Theexpressson [(l + )r-(l - ),] sequal 7. The dervave of ln xl s o -. lf xzy - x' =, hen lz s equal o + '^-" x'-zry Y'.-zxY xy. The value of - -l -J- l x-y 8. lft cos-+rsl'l- S 9 -. X -x x x ff *:Zxy,henhe value of * pon (, ) s f*= "'" (;)-' Zan- (x) + c an- (x) + c ahe { '' : n l l., sec'x 6 Zan x -6s lnlr".'*l+" _ lnlan xf+c - lnf sec.xf +c +l lnf an.rf+c -l -an (, +c A curve s gven paramercally by he equaons x = -, y = + Z. The expresson f", * s gven by +l r -l -l +l +l -l GO ON TO THE NEXT PAGE

4 -- l J(cosSx cosx)dx=. lf f(x, y) s such ha l J{cosAx + coslx)dx J(cos8, - cosx)dx (cos8x + cosx)dx (cos8x - cosx)dx ' J One square roo of - s J -z J +z - +!,d*= J (x-l)(x+) dx x-l (x+) r(x-l) (x+) -+_ (,+- ) (x-) (x+) _+-- l dx dx ( dx (x-) (x+) 6. * =,'[-sn (r+y) + cos (x + y)] and ox af,. b= -e'- sn (x+y), hen whch of he followng s TRUE? f(x,y) =e'sn (x+y) f (x, y): d cos (x + y) f (x, y) : e'sn (x + y) +cos (x +y) f (x, y): e" cos (x + y). Gven secr.r= * anx, L *rr,, f n r l- f he erns of he sequence ur, u, u j..,, un... sasfy he recurrence relaon u,n, = u,*, n Zl, hen he r,l, erm may be expressed as, * $77 u, * u+(n-l) u,+6(n-l) GO ON TO THE NEXT PAOE

5 J l lj For-l<Zn<,lZn),= rn. The express o" hcan be smplfed l-n -n n +n and wren as! n- n(n-l) n l-n. *r( *)= 8 Gven hauorepresens he nh erm of a sequence, whch of he followng converges? u,= n - u,,= Z(-l),,-, n+ n+l n n+l n+ (C) u,=* n n+l {,,=r( fl-l u j j 9 f::: :::'':"*:l l l.' j,:? f;,: - -. Gven na, + l)=g hen, for m ( r?, _ *-n+l k (k + l)-- ^ -s m +l S.*,-{, S,, - S, S,, - S,, lr. ryl ro nfny of he geornerc seres s The equaon e* -.xa = has a roo beween and and (C) and and :. GO ON TO THE NEXT PAGE

6 -6-. By usng he Newon-Raphson mehod wh a frs approxmaon x,, he second approxmaon 'x,*, for a roo of he equaon x = xj + may be expressed as x,,-xl,-l,+ sx { x,, - x: - xl, -zs sx -xj 8. Arelay eam of eachers s o be chosen from a group of l eachers. n how many ways could hs relay eam be chosen? l! l!! l! r! 6 C, equals +xl, - lx) -zs rl, -Zxj + sxl -xj Cr*'C, 9 l!! l! The values ofx for whch he expanson of L Jloo-sox; svaldare 'Cr*C, -l <x<l 8 X,C, g x 7c J -Z <x < ll -- <x <- )) x < - and x> 7 f he coeffcen of.f n he expresson of (6 - ax)e s -8, hen he value of a s Le f be a connuous funcon wh f () = and f (.8) = The frs approxmaon o he roo n [,.8], usng lnear nerpolaon, o decmal places s....s GO ON TO THE NEXT PAGE

7 . n how many ways can he leers P, Q,, R,,S and be arranged so ha P and Q, are always ogeher, and.r and.s are always ogeher?! s! s.z l x x -7 - (-z. fp= and Q:,O= lr r [-, [ -7-7 l , hen A s a x marx wh deermnan f he marx of cofacors of A s - - -'7-7 hen A-r : The marx P-r equals (PQT' (QPT' P l jl a Gven ha y : a.r =, he general soluon of he dfferenal equaon y"+6y'+9y=q,- sr * Bx!= Bxe-' (C) Y=e*(A+Bx) y: e-'+.br l,: -7-7 l { l GO ON TO THE NEXT PAGE

8 -8- A sample space X consss solely of muually exclusve evens, p, R and S. f P(8) =. and P(R) =.6, hen P(S) =.r..6.e em 7 refers o he followng able whch shows he number of males and females and her preferences for Drnk A and Drnk.B. Drnkl Male Female Toal 8 6 A school debang eam comprsng eachers, boys and grls s o be chosen from eachers, boys and 6 grls. The number of ways n whch hs eam can be chosen s Drnk.B Toal r 8 6 One person s randomly seleced. Wha s he probably ha hs person s female and prefers Drnk B? 6 l 8 The FRST ROW ofhe produc PQ ofhe wox marces P- -l 6 J,,, =[ - - -l rs ()) (6 (6 ( ) ) 7) -r) GO ON TO THE NEXT PAGE

9 -9 - " r"l., ',., :,,., rf,",', :,,: 9 A, B, C and D are four x marces. Gven ha AB = J, BC = K, CD = L, ABC = p and BCD = Q, where J, K, L, p and e are marces, he produc of ABCD s JQ JKL LJ PD A suable negrang facor for he soluon of he dfferenal equaon dy v l =- ls dxxx b lnx f ek Two cons and a de wh faces numbered o 6 are hrown ogeher once. Assumng ha hedeand cons are far, he probab ly of obanng heads and a number ess han s em refers o he Venn dagram below whch shows he probables assocaed wh evens.( and L n a sample space S.," 'l : - '., 8 sgl. ',: J J.s The probably ha occurs, gven hal( occurs, s. The general soluon of he dfferenal equaon * *=: y=d y= b y=x'lk )=lnx+k....9,r:r: ::::. GO ON TN T]r Nr].T D^ T:r

10 _ l _ The number of possble values of x. whch sasfy he sysem of smuaneous equaons, rs x+y*=- x + 6y * z= - 6x+9y+6=- The marx A represens a sysem of lnear equaons afer some elemenary row operaons have been performed. r ^=l; Whch ofhe fo low ng saemens s TRUE? The soluon s unque. There s'no soluon. (C) There are nfnely many soluons The soluons are dependen. END OF TEST F YOU FNSH BEF'ORE TME S CALLED, CHECK YOUR WORK ON THS TEST. m?/cape 6

Chapter Lagrangian Interpolation

Chapter Lagrangian Interpolation Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and