pioneer junior college 2008 pron,iotionai, examin,4.tions

8/14/2019 Pioneer Junior College 2008 Pron,Iotionai, Examin,4.Tions

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PIONEER JUNIOR COLLEGE2008 PRON,IOTIONAI, EXAMIN,4.TIONS

Can r1o the nhole question. ,r/Can do part of questiononly. 1,2

Diffcrcntiate each ofthe following wilh respect to x:(a) ire"O) s,r'2r . t2lI2tIt is givcn Ihat l().) =r.(r+lxr-6).(i) Sketch tbe graphs of, = f(r) and y = f'(-t) on the sanre diagram, indicating

r Ic.rr I' thr c,r'rrln rre, of rhe d\r.,1 inrFr, ctrs trf anyr. I2l(ii) Stalc drc st of vaLues of .t for which the graph oa t = f (r) is concavett I

Witcr is drippirg fiom the vertex of a conical fitter \\,irh a semi veftical angle ol_30'.

Show that thc curved surface area,l cmz, ofthe conical tiltcr thai is in contact rvith/.llrc waret rs Br!cn by,4 ,rl'wherc/rcmrsrh.depthot thcw ei. t'jWhen the water level io the conical filter is 8 cm, the rate ofdecrase of the watcrlevel is 0.16 .- s l- Find the ratc of decrease of the curved sudace area of theconical filter lhat is in cootact with tho water at this inslrnt- tll[The curved surface arca ofa cone with base mdius r and sla[t height / is given by2vrl .l


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/

fhe lunction fl,) ar'-br+-l has an asymptot" r l.'Thelincy=Zr .l rsr-da tangent to the curve at the poilt (2, l). Civen that the curve also passes tbroughthe point (-1, l),findthexactvaluesofa,b,c^idd. t5lSolve the inequality ir+2


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Fxnrcss ) 'n nr.'nl fracLrons and shou rnat I I' r,\rt2 a' l' 'Hcncc findrhcvnrre of i-- L2 r'+3r +2

t5l2(.n + 2)

10 A culve is defioed paramotrically by (he equations.2.i-2x' Jnd v--The point P on the curve has paramctcr u = 1 .(i) Find the equation ofthe tangent at P. t3l(ii) Delenniue whelher this tangent al P intersects lhe curve agair. Ifl(iii) Find the value oLll at the point where the normal to the curvc is parallel to

Ithe linr i:l+ 1.'4I I fl,r curvc a ha. c"ludnon v 5 I' ' r-l rrl(i) S{ate the coordinales ofany poirls oIintcrscction with thc a\cs.(ii) State the equations ofthe asymptotes.(iii) Sketch a. making clear the rclcvant lcaturcs ofthe cufl'c.

t2l

t2t

t2l

trlt3lt4t

12 {a) F,nd [**" Vt_lr

.i/ = (r. + 1) ln -r

13v

(b) By using thc substitution r =secd , or otherwisc, cvalua{c| -- a^ , cr. rnJ- \our ar\tt et in rhe e\acr lorm t6l

(a) The diagram shows the cun,e withequation y=(]r+l)lnr, -r > 0. Thellnc , 2r 2 r. thc Lrntcnt ro lhccurvc al thc poinl ( 1,0). Thcshadedregion R is bounded by the curve, theline y =2\ 2 and lhe line-! = 2.t5lFind the volumc ofihc solid lbrmcd$,hetr the regior R is rotatedcomplctely about the r axis, givingyour arlslvcr corrcct to I significant

F;nd the exact area ofl?.(b)

figures. ttl


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14 (a) The second, fifth and tenth terms of an aritbmefrc progression areconsecutive terms ofa gcometric pmgression. The eighth term ofthefiithmetic progression is 6. Find the sum ofthe fi.st 12 terms of thearittmetic progression. t51

t5

l.4l

The diagmm shows the graph ofl, - f(Jr). The curve has a minimum point at C(1, 3)and a horizonlal as,,rnptote y:2. The curve passes through the pointsI (-2,0) and A (4,0).

(b) The sum ro irfinity ofa geometric series is 8 and thesum to infioity of asecond series formed by taking the first, third, fifth, seventh, ... terms (thatts, 'l;+ l; + l;+'t; +....) of the tirst series isf . Find the common ratio of the3

Sketch, on sepamte clcarly labcllcd diagrams, the graphs of(i) ,=f(2')r2,{ii) t=-L.r(r)(iii) r,: = f(.x).

t3lt3I

A (-2,0\ n (4,0)c(r, -r)

t4l


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Answer Kev1. Objectives : Differentiate expoDeDtial aDd trigonomelry functions and use ofproduct rule(a) el(21+D G) 6sirr? 2:ccos2x2. Objcctives : Use GC to sketch graphs off (x) ard f'(r); To find range ofr forwhich curve is concrve downw:rrds, \For (,) lo be con.a!ed downward,, \ ' .- or Lb/{3. Objective : Use of b|sic trigonometry to forrn equatiou with one variable , use ofchain rulc

7 68r cn's '4. Objective: Ability to forrn equatiotrs based on given instructions. Us ofAPPS inGC

t15a b -. t-ttl5. Objective : 'fest basic idea of inequality i.c. not allorved to crosr-multiply an

unknown- Idea of{actorizatior nnd simple application.r


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9. Obiective : Express a fraction in its partial fractions, use of method ofdifference, find the limit to a sum.

) 2 2 \r I - lI .i | _l,,,rn ,n-,rr'kr,.t,t2 2(nt 2)'llr, tJrt2 210. Objctive: Test parametric differeotiatioo and higher order thinkiDg itrintcrprting tangents aod normals-

IEqn ottangent.l = - x+l& : I or u = -2 . . the tangent will intersect the curve again-

ll. Otiectives: Us a graphic calculator to graph a given fu[ction.(i) (-l.s.o) and (0, {)(ii) vrtical xslrmptotc: ;u = I, ;r = -l

horizontal as),rnptote: ), = 0

(iii)

( l 5,0)y=0(-2.62, 0.7641 'l( q 182. 5.2 (0, 6)

(D(it

12. Objectives ; lntegration by usiog formula list and by substitutiona.:=.r[+tn(z+.6)r"r lfs;"-'{.12,.)+c (b) I' r+1J'' t

t0


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(r)

13. Objectives : Find area bounded by 2 curves and liDding volume of solid ofrevol(tiol by using the difference between the volume of 2 solids ofrevolutiol(a) A-red of/l - qhz llrbt Volume - 0.22)414. Objectivc : Form eq[ations by using properfy ofa GP, Fitrd sum to n tcrms ofAP, ideDtify the common ratio of a newly formed CP, find sum to i[ftnity ofCP(a)/-o(NA, ,t !.a-t: s,-1t) rtr. Ne) ,-tt'7')15- Objectives: Apply AMIIL{ rule, and sketch the Sraphs of y = ^ ,l , , f' = t ( t)t (x)based on thc graph of y = (r.).

x-2 Y(iD

=4

A, ( t,z)

A.) ( 2,0) 1 i B. t4.o)c.t1---i^. ti

lt


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pioneer junior college 2008 pron,iotionai, examin,4.tions

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