sfo ffi FORM TP CARIBBEAN EXAMINATIONS COUNCIL

Transcription

1 t- FORM TP sfo TEST CODE O234O2O CARBBEAN EXAMNATONS COUNC CARBBEAN ADVANCED PROFCENCY PURE MATHEMATCS UNT -Pape02 AGEBRA, GEOMETRY AND CACUUS 2 hous 30 minutes MAY/JUNE 2O7 READ THE FOOWNG NSTRUCTONS CAREFUY. l. This examination pape consists of THREE sections. 2. Each section consists of TWO questions. 3. AnsweA questions fom the THREE sections. 4. Wite you answes in the spaces povided in this booklet. 5. Do NOT wite in the magins. 6. Unless othewise stated in the question, any numeical answe that is not exact MUST be witten coect to thee significant figues. 7. f you need to ewite any answe and thee is not enough space to do so on the oiginal page, you must use the exta lined page(s) povided at the back of this booklet. Remembe to daw a line though you oiginal answe. 8 f you use the exta page(s) you MUST wite the question numbe clealy in the box povided at the top of the exta page(s) and, whee elevant, include the question pat beside the answe. Examination Mateials Pemitted Mathematical fomulae and tables (povided) - Revised 2012 Mathematical instuments Silent, non-pogammable, electonic calculato DO NOT TURN THS PAGE UNT YOU ARE TOD TO DO SO. Copyight O 2015 Caibbean Examinations Council ffi All ights eseved /cApE 20 7 lffilllllllllll llffilllllililllllilllffilllllll J

2 t (a) et p and q be two popositions p : t is aining. q : John is sick. SECTON A Module Answe BOTH questions. Wite EACH of the statements below in tems of p and q (i) t is not aining o John is sick. (ii) f it is aining then John is not sick makl [1 mak] (b) An opeation * is defined on the set {, 2,3, 4} as shown in the following table. * ) J (i) Pove that * is commutative [1 makl (ii) Show that the identity element of * is 3 [2 maks] CO ON TO THE NEXT PAGE 02t34020lcAPE 2017 fiil t]ll ilt] flil tilfl ilil ffi ll]t!illt ffi fl til

3 t- 5 (c) The polynomial f(x) : ox3 + 9x2 - llx + 6 has a facto of (x - 2) anda emainde of 12 when divided by (x + 2). (i) Show that a:2 and b: -30 ffi /cAPE 20t7 021 iltil ilililll!ilil ilil ilil ffi ll]t ilil ilil il tilt [4 maksl J

4 -6- (ii) Hence, solveax3 +9x2 - llx+b:0. [9 maks] CAPE 2017 ilil til ilil ilil ill lill llll lill llll ffi llll llll

5 -7 (d) - Use mathematical induction to pove that n:4n(n + l) foall n e N. [7 maksl ffi 02t34020tcAPE 't iltil illl ill] tffi ilil ilil ffi ll]t ililt lilt ll] ilt Total25 maks CO ON TO THE NEXT PAGE _t

6 t (a) (i) Given lhat a2 * b2 : l4ab, pove that ln ( a+b\ t, t;l=,-{'n a+tnb) [5 maksl 02t34020lcAPE 2017 ilil til ilil tfl ililt tilt ilfl til l]l lill il lil CO ON TO THE NEXT PAGE J

7 9- -1 (i i) Solve the equation 2* + 312'1: 4. [You esponse may be expessed in tems of logaithms.] ffi [6 maksl 02t34020lcAPE 20t7 J ilililil ilililililil ilil tffi ilil ilil ffi ililil

8 -10- (b) The following diagam shows the gaph of the function f(x): x+4 On the diagam, (i) inset the asymptotes fo the function f 12 maksl (ii) sketch the gaph of f-, the invese of f showing the asymptotes fo f-. [4 maksl b x CAPB 20t7 tilil ffi ilil til! ililt illll llfl lllll lllll lllll l]l lil

9 - ll - -1 (c) Civen that o,, p and y ae the oots of the equation + 3x + 2:0, fom an equation whose oots ae 0T, oy and ob. [8 maksl ffi 02t34020lcAPE ililil tililfl!iltil lil[ilt ilil lil ffi lllll lil llll Total25 maks

10 t SECTON B Module 2 Answe BOTH questions. ---:.'...1;(li 3. (a) (i) Pove the identity td.ttt,.7t t)t: tan(a) + tan(.b) -tan(a) tan(.b) - [4 maks] /CAPE 2017 il!ilil ilil1 tilililililil ]il ffi tilt ffiilffi ill ilt GO ON TO THE NEXT PACE J

11 t (ii) Given that sin,a: andcos B: *nee angle A is acute and angle.b is obtuse, expess tan (A + B) in the fom a + b J1, whee a and b ae eal numbes. [6 maksl ffi 02t34020lcAPE 2017 J ilil ull ilil ililt ililt ilil ffi il]t il] tilt til til

12 -14- (b) Solvetheequation sin2 0-2cos2 0*3cos 0+5:0fo0 <0<4n. [6 maks] ffi /CAPE

13 (c) (i) Expess f (A:6 cos d+ 8 sin d in the fom sin (d+ o) whee 0 < a < 90' [3 maks] (ii) Hence, o othewise, find the geneal solution of f 104:2 [6 maksl Total25 maks HH 02t34020tC.APE 2017 tiltil tilil ililt ilil ilil ilfl til il1il ilt ilil til il

14 ,::':,...-) (a) (i) The cicle, C,, with equation *' + f - 4x + 2y -2: 0 and the cicle C, have a common cente. Given that C, passes though the point (-1, -2), expess the equation of C, in the fom (* - h)' + O/ - k)' : k. titi [3 maks] CAPE.2017 ililil ililililil ilil lil il 1 il il] ffi il lill

15 -17 (ii) - The equation of the ine, is x + 3y: 3. Detemine whethe, is a tangent to the cicle, C,, in a (i) on page 16. ffi 02l34020tcAPE 2017 o213/,02017 l lilil til flil tilililil lilffiilil[]!ilililil til [7 maksl GO ON TO THE NEXT PACE J

16 t- - l8 - (b) et P(3,, 2) and Q( l, -2, 4). (i) --) Expess the vecto PQ in the fom xi + yi + zk. [2 maks] (ii) Detemine the Catesian equation of the plane which passes though the point Q and is pependiculu. o f. [6 maksl 02t34020lcAPE 20t7 tffi tlll lill lill flil lill l]ll lill llll llll llil lil

17 t (c) The vecto equations of two lines, Z, and, ae:.,: -i+ j - 2k + o (-2i +.1-3k) :-2i+j-4k+B(i-i+k) (i) Show that, and, intesect. ffi CAPB 2017 fiil ililt ilil ilil lillt till ffi ililt ilil tffi flil ll] [5 maks]

18 -20 (ii) - Hence, detemine the coodinates of the point of intesection of the two lines. [2 maksl Total25 maks ffi 02t34020lcAPE GO ON TO THE NEXT PACE

19 -21 - SECTON C Module 3 Answe BOTH questions. 5. (a) Detemine the value of ft fo which (x): xt- x' -.x*l k, x=1. is continuous fo all values of x. t$:fi t.gjlt tCAPE 2017 ilil ffi ilil ilil ilil il ffi lill lilll llll lil llll [4 maks]

20 t (b) A cuve, C, is descibed paametically by the equations x: 5t + 3 and y: f - f + 2 dy (i) Find; in tems of t. [3 maksl 02l34020lcAPE 2017 llllll ffi illfl ilil tilil ilfl ilfl il]t til ilil ill til

21 -23 (ii) - dy Hence, detemine all points of C such that-:0. ffi 02t34020lcAPE 2017 il]il ilil lilt tffi tilil til illl llill illl llil lil ll] [6 maks] _t

22 -24 - (c) (i) Given that y: 2+2x2, show that a) dy v: -2x: 0 'dx b) d'y _,4 : o dtf [9 maks] /C,APE 2017 lillil ilfl ilil ilil til ilil ffi ilfl ffi ililt!il tilt CO ON TO THE NEXT PAGE

23 -25 - (ii) Hence, find the value of ff *n"n *: o [3 maks] Total25 maks th cAPE 2017 ilil illl lllll lllll lllll lill illl llill llll lllll llll llll

24 (a) Tiangle PQRhas vetices P(0, ) QQ,3) and R(4,2). (i) On the axes below, sketch tiangle PQR. v x makl (ii) Detemine the equations of EACH of the following:.pq.qr.pr.-s.' /CAPE 2017 ililil ilill liltil ililil]t ilil ililt lllll ffi il ilt J

25 -27 - [7 maks] Flf.,'S ffi CAPB 2017 illil illl ililt ilil tilil lllll ffi lill!illl lill lil lil

26 -28 (iii) - Hence, use integation to detemine the aea of tiangle PpR. [7 maks] 02t34020tcAPE 2017 J lilll ilfl ill] ilil ililt ilil ffi ilil lllll ilil il ilt

27 -29 (b) dvv The voftage in a cicuit, 4 satisfes the equatio, d, * -:0. - Given thatv:25 volts when / : 0 seconds, wite an expession fo Z in tems of. [5 maksl,,et ffi 02t34020lCAPE 2017 ll!il ffi ilil ilil tilil fl t]ll il ]il il flil llil

28 f3 (c) Given that J [3f (x) + g (x)] dx: 5 and,_1!.3 * t l,'{') 3 [5f (") dx: l, detemine a -3!, eot a.' /CAPE 2017 tflfl t]ll ilillilil tilil tilt ilil ll]t lilll iltil lilt ll]

29 = $i-:* -::v i-:ls ::f :ll* '.-.8-3l - END OF TEST ' [5 maks] Total25 maks ffi F YOU FNSH BEFORE TME S CAED, CHECK YOUR WOR( ON THS TEST. 02t34020lcAPE2017 ilil illl ilil lil lllll illl illl lill lffi lll] llll llll