E R K HSS ERUMIYAMPATTI Page 1 +2 STUDY MATERIALS

Transcription

1 E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

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3 E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS COME BOOK TEN MARKS [6 Numer Of Te Mrk Questios To Be Asked For Full Test] UNIT:. MATRICES AND DETERMINANTS (ONE QUESTION FOR FULL TEST) EXERCISE.. () Fid the djoit of the mtri A d verif the result. A A ) A dj ( ) A dj ( A (6) Fid the iverse of the mtri A d verif tht A A. (7) Show tht the djoit of A T A. (9) If A, prove tht T A A. EXERCISE.. () Solve mtri iversio method ech of the followig sstem of lier equtios:, 7, 9. () Solve mtri iversio method ech of the followig sstem of lier equtios:, 7,. () Solve mtri iversio method ech of the followig sstem of lier equtios: 6,, 8. EXERCISE. () Solve the followig o-homogeeous sstem of lier equtios determit method: ; ; () Solve the followig o-homogeeous sstem of lier equtios determit method: ; ; (6) Solve the followig o-homogeeous sstem of lier equtios determit method: 6 ; ; (7) Solve the followig o-homogeeous sstem of lier equtios determit method: ; 6 ; (8) Solve the followig o-homogeeous sstem of lier equtios determit method: 6 ; 6 ; (9) Solve the followig o-homogeeous sstem of lier equtios determit method: ; ; () A smll semir hll c hold chirs. Three differet colours ( red, lue d gree) of chirs re ville. The cost of red chir Rs., cost of lue chir Rs. 6 d the cost of gree chir Rs.. The totl cost of chir Rs.,. Fid tlest differet solutio of the umer of chirs i ech colour to e purchsed. EXERCISE.. () Emie the costec of the followig sstem of equtios. If it costet the solve the sme: (i) solve : 9 ; 7 ; 6 (ii) solve : 6 ; ; 8

4 E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS (v) solve : ; ; () Dcuss the solutios of the sstem of equtios,, for ll vlues of () For wht vlues of k, the sstem of equtios,, k k k hve (i) uique solutio (ii) more th oe solutio (iii) o solutio Emple. : If A,verif I A A A dj A dj A Emple.8 : Solve mtri iversio method, 6, 9 Emple.8: Solve the followig o-homogeeous equtios of three ukows. () ; 7 ; () 8 ; ; 6 () 6 ; 8 ; Emple.9: A g cot tpes of co mel Re., Rs. d Rs.. There re co moutig to Rs. i totl. Fid the umer of co i ech ctegor. Emple.: Solve: ; ; Emple.: Verif whether the give sstem of equtios costet. If it costet, solve them. 9,, 7 Emple. : Emie the costec of the equtios. 7 9,, 7 Emple.: Show tht the equtios 7,, 6 re costet d solve them. Emple.: Verif whether the give sstem of equtios costet. If it costet, solve them:,, Emple.6: Ivestigte for wht vlues of, the simulteous equtios,, 6 hve (i) o solutio (ii) uique solutio d (iii) ifiite umer of solutios. Emple.7: Solve the followig homogeeous lier equtios., 6, Emple.8: For wht vlue of the equtios,, hve (i) trivil solutio, (ii) o-trivil solutio. UNIT:. VECTOR ALGEBRA (TWO QUESTIONS FOR FULL TEST) EXERCISE. () Prove tht B A B A B A si si cos cos cos EXERCISE. (7) Prove tht. si cos cos si si B A B A B A EXERCISE. () If k j c k i k j i,, Verif tht c c c..

5 () Verif c d d c c d ; d i j k where i j k ; i k ; c i j k () Show tht the lies d EXERCISE.7 itersect d fid their poit of itersectio. EXERCISE.8. (7) Fid the vector d Crtesi equtio of the ple cotiig the lie d prllel to the lie. (8) Fid the vector d Crtesi equtio of the ple through the poit (,,) d prllel to the lies d (9) Fid the vector d Crtesi equtio to the ple through the poit (-,,) d perpediculr to the ples d 8. () Fid the vector d Crtesi equtio of the ple pssig through the poits A (,-,) d B (-,,-) d prllel to the lie () Fid the vector d Crtesi equtio of the ple through the poits (,,) d (,,) perpediculr to the ple. () Fid the vector d Crtesi equtio of the ple cotiig the lie d pssig through the poit (-,,-). () Fid the vector d Crtesi equtio of the ple pssig through poits with positio vectors i j k, i j k d 7 i k. () Derive the equtio of the ple i the itercept form.( oth i vector d crtesio form ) Emple.6: Altitudes of trigle re cocurret prove vector method. Emple.7: Prove tht cos A B cos A cos B si A si B Emple.9: Prove tht si A B si A cos B cos A si B Emple.: Show tht the lies itersectio. Emple.: d itersect d hece fid the poit of Fid the vector d Crtesi equtios of the ple through the poit (,-,-) d prllel to the lies. d. Emple.: Fid the vector d Crtesi equtios of the ple pssig through the poits (-,,) d (,-,) d perpediculr to the ple Emple.: Fid the vector d Crtesi equtios of the ple pssig through the poits (,,-), (,,) d (7,, E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

6 UNIT:.COMPLEX NUMBERS (ONE QUESTION FOR FULL TEST) EXERCISE. i (8) (i) If P represets the vrile comple umer. Fid the locus of P, if Im (iii) If P represets the vrile comple umer. Fid the locus of P, if (v) If P represets the vrile comple umer. Fid the locus of P, if EXERCISE. () If d re the roots of the equtio p p q q si ; N si d Re i rg q t Show tht p (6) If d re the roots of Prove tht i si ; N d deduct 9 9 (8) If cos d show tht (i) cos m (ii) i si m () If cos i si, cos i si d c cos i si Prove tht (i) c cos (ii) cos c () Fid ll the vlues of the followig: (iii) i () Solve: c c m m EXERCISE.. () Fid ll the vlues of i d hece prove tht the product of the vlues. Emple.: (i) If P represets the vrile comple umer, fid the locus of P Re (ii) If P represets the vrile comple umer, fid the locus of P Emple.: If d re the roots of d cot, Show tht si si ; N Emple.: Solve the equtio 9 Emple.: Solve the equtio 7 rg i m m E R K HSS ERUMIYAMPATTI Pge 6 + STUDY MATERIALS

7 Emple.: Fid ll the vlues of i UNIT:. ANALYTICAL GEOMETRY (THREE QUESTIONS FOR FULL TEST) EXERCISE. () Fid the, verte, focus, equtio of directri, ltus rectum, legth of the ltus rectum for the followig prols d hece sketch their grphs. (iv) 8 6 (v) 6 () A cle of suspesio ridge i the form of prol whose sp mts. The rod w mts elow the lowest poit of the cle. If etr support provided cross the cle mts ove the groud level fid the legth of the support if the height of the pillrs re mts. EXERCISE.. (6) Fid the eccetricit, cetre, foci, vertices of the followig ellipses d drw the digrm: (ii) (iv) (7) A kho-kho pler i prctice sessio while ruig relies tht the sum of the dtces from the two kho-kho pples from him lws 8m. Fid the equtio of the pth trced him if the dtce etwee the poles 6m. (8) A stellite trvelig roud the erth i ellipticl orit hvig the erth t focus d of eccetricit /. The shortest dtce tht the stellite gets to the erth kms. Fid the logest dtce tht the stellite gets from the erth. (9) The orit of the plet mercur roud the su i ellipticl shpe with su t focus. The semi-mjor of legth 6 millio miles d the eccetricit of the orit.6. Fid (i) how close the mercur gets to su? (ii) the gretest possile dtce etwee mercur d su. () The rch of ridge i the shpe of semi ellipse hvig horiotl sp of ft d 6ft high t the cetre. How high the rch, 9ft from the right or left of the cetre. EXERCISE. () Fid the eccetricit cetre, foci d vertices of the followig hperols d drw their digrms. (iii) 6 6 (iv) EXERCISE. () Prove tht the lie 9 touches the hperol 9 9 d fid its poit of cotct. (6) Show tht the lie tget to the ellipse. Fid the co-ordites of the poit of cotct. EXERCISE.. () Fid the equtio of the hperol if (ii) its smptotes re prllel to d 8, (,) the cetre of the hperol d it psses through (,). EXERCISE.6. () Fid the equtio of the rectgulr hperol which hs for oe of its smptotes the lie d psses through the poits (6,) d (-,). Emple.7: Fid the, verte, focus, directri, equtio of the ltus rectum, legth of the ltus rectumfor the followig prols d hece drw their grphs. (iv) (v) E R K HSS ERUMIYAMPATTI Pge 7 + STUDY MATERIALS

8 Emple.8: The girder of rilw ridge i the prolic form with sp ft. d the highest poit o the rch ft, ove the ridge. Fid the height of the ridge t ft, to the left or right from the midpoit of the ridge. Emple.: O lightig rocket crcker it gets projected i prolic pth d reches mimum height of mts whe it 6 mts w from the poit of projectio. Fill it reches the groud mts w from the strtig poit. Fid the gle of projectio. Emple.: Assume tht wter suig from the ed of horiotl pipe, 7.m ove the groud, descries prolic pth. The verte of the prolic pth t the ed of the pipe. At positio.m elow the lie of the pipe, the flow of wter hs curved outwrd m eod the verticl lie through the ed of the pipe. How fr eod th verticl lie will the wter strike the groud? Emple.: A comet movig i prolic orit roud the su which t the focus of prol. Whe the comet 8 millio kms from the su, the lie segmet from the su to the comet mkes gle of rd with the of the orit. Fid (i) the equtio of the comet s orit (ii) how close does the comet erer to the su?( Tke the orit s ope rightwrd ). Emple.: A cle of suspesio ridge hgs i the form of prol whe the lod uiforml dtriuted horiotll. The dtce etwee two towers if ft, the poits of support of the cle o the towers re ft ove the rod w d the lowest poit o the circle 7ft ove the rodw. Fid the verticl dtce to the cle from pole whose height ft. Emple.: Fid the eccetricit, cetre, foci, vertices of the followig ellipses: (iv) 6 7 Emple.: A rch i the form of semi-ellipse whose sp 8 feet wide. The height of the rch feet. How wide the rch t height of feet ove the se? Emple.: The ceilig i hllw ft wide i the shpe of semi ellipse d 8ft high t the cetre. Fid the height of the ceilig feet from either wll if the height of the side wlls ft. Emple.: A ldder of legth m moves with its eds lws touchig the verticl wll d the horiotl floor. Determie the equtio of the locus of poit P o the ldder, which 6m from the ed of the ldder i cotct with the floor. Emple.6: Fid the eccetricit, cetre, foci d vertices of the hperol d lso trce the curve. Emple.7: Fid the eccetricit, cetre, foci, d vertices of the followig hperol d drw the digrm : UNIT:.DIFFERENTIAL CALCULUS-APPLICATIONS-I (TWO QUESTIONS FOR FULL TEST) Emple.6 : A o, who stdig o pole of height.7m throws stoe verticll upwrds. It moves i verticl lie slightl w from the pole d flls o the groud. Its equtio of motio i meters d secods 9.8 t. 9 t (i) Fid the time tke for upwrd d dowwrd motios. (ii) Also fid the mimum height reched the stoe from the groud. E R K HSS ERUMIYAMPATTI Pge 8 + STUDY MATERIALS

9 Emple.7 : A ldder m log rests gt verticl wll. If the ottom of the ldder slides w from the wll t rte of m/sec how fst the top of the ldder slidig dow the wll whe the ottom of the ldder 6m from the wll? Emple.8 : A cr A trvellig from west t km/hr. d cr B trvelig towrds orth t 6 km/hr. Both re heded for the itersectio of the two rods. At wht rte re the crs pprochig ech other whe cr A. kilometers d cr B. kilometers from the itersectio? Emple.9 : A wter tk hs the shpe of iverted circulr coe with se rdius metres d height metres. If wter eig pumped ito the tk t rte of m / mi, fid the rte t which the wter level rig whe the wter m deep. EXERCISE.. () A msile fired from groud level res metres verticll upwrds i t secods d t t. Fid (i) the iitil velocit of the msile, (ii) the time whe the height of the msile mimum (iii) the mimum height reched d (iv) the velocit with which the msile strikes the groud. () The dtce metres trveled vehicle i time t secods fter the rkes re pplied give : t / t.determie (i) the speed of the vehicle (i km/hr) t the tt the rkes re pplied d (ii) the dtce the cr trveled efore it stops. () The ltitude of trigle icresig t rte of cm / mi while the re of the trigle icresig t rte of cm / mi. At wht rte the se of the trigle chgig whe the ltitude cm d the re cm. (6) At oo, ship A km west of ship B. Ship A silig est t km / hr d ship B silig orth t km / hr. How fst the dtce etwee the ships chgig t. p.m. (8) Two sides of trigle hve legth m d m. The gle etwee them icresig t rte of / mi. How fst the legth of third side icresig whe the gle etwee the sides of fied legth 6? (9) Grvel eig dumped from coveor elt t rte of ft / mi d its corseed such tht it forms pile i the shpe of coe whose se dimeter d height re lws equl. How fst the height of the pile icresig whe the pile ft high? Emple. : Fid the equtios of the tget d orml t Emple. : to the curve si, cos. Fid the equtios of tget d orml to the curve 6 9 t, ) where d. Emple. : Fid the equtios of the tget d orml to the ellipse ( cos, si t the poit Emple.7 : Fid the gle etwee the curves d t the poit of itersectio. Emple.8 : Fid the coditio for the curves, to itersect orthogoll. Emple. : Prove tht the sum of the itercepts o the co-ordite es of tget to the curve cos, si, equl to.. EXERCISE.. () Fid the equtios of those tgets to the circle, which re prllel to the stright lie 6. E R K HSS ERUMIYAMPATTI Pge 9 + STUDY MATERIALS

10 (7) Let P e poit o the curve d suppose tht the tget lie t P itersects the curve gi t Q. Prove tht the slope t Q four times the slope t P. () Show tht the equtio of the orml to the curve cos ; si t cos si cos. () If the curve d k re orthogol the prove tht 8 k Emple. : Evlute : cot si Emple. : Evlute : () t cos lim lim lim si EXERCISE.6. Emple.8 () : Fid the solute mimum d solute miimum vlues of si, Emple. : Fid the locl miimum d mimum vlues of f EXERCISE.9 () Fid the locl mimum d miimum vlues of the followig fuctios:. f. (iii) 6 (iv) (v) si, (vi) t cos t Emple. : A frmer hs feet of fecig d wt to fece of rectgulr field tht orders stright river. He eeds o fece log the river. Wht r the dimesios of the field tht hs the lrgest re? Emple. : Fid poit o the prol tht closest to the poit (,) Emple. : Fid the re of the lrgest rectgle tht c e cried i semi circle of rdius r. Emple. : The top d ottom mrg of poster re ech 6 cms d the side mrg re ech cms. If the re of the prited mteril o the poster fied t 8 cms, fid the dimesio of the poster with the smllest re. Emple.6 : Show tht the volume of the lrgest right circulr coe tht c e cried i sphere of rdius 8 7 ( volume of the sphere ). Emple.7 : A closed (cuoid) o with squre se to hve volume of c.c. The mteril for the top d ottom of the o to cost Rs. per squre cm. d the mteril for the sides to cost Rs.. per squre cm. If the cost of the mterils to e the lest, fid the dimesios of the o. Emple.8 : A m t poit P o k of stright river, km wide, d wts to rech poit Q, 8 km dowstrem o the opposite k, s quickl s possile. He could row h ot directl cross the river to poit R d the ru to Q, or he could row directl to Q, or he could row to some poit to etwee Q d R d the ru to Q. If he c row t 6 km/h d ru t 8 km/h where should he ld to rech Q s soo s possile? EXERCISE.. () Show tht of ll the rectgles with give re the oe with smllest perimeter squre. () Show tht of ll the rectgle with give perimeter the oe with the gretest re squre. () Fid the dimesios of the rectgle of lrgest re tht c e cried i circle of rdius r. Emple.6 : Dcuss the curve with respect to cocvit d poits of iflectio. Emple.6 : Fid the poits of iflectio d determie the itervls of coveit d cocvit of the Gussio curve e E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

11 EXERCISE.. Fid the itervls of cocvit d the poits of iflectio of the followig fuctios : () f 6 () f si i, (6) UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II (ONE QUESTION FOR FULL TEST) EXERCISE 6.. () Use differetils to fid the pproimte vlue for the give umer (iii).. Emple 6.9 : Trce the curve Emple 6. : Trce the followig curve : () Emple 6.8 : If w Trce the curve v u e where EXERCISE 6. u d v log, fid Emple 6. : Verif Euler s theorem for f, Emple 6. : Usig Euler s theorem, prove tht u u t u w if d u si w () Verif u u for the followig fuctios: EXERCISE 6.. (ii) u (iii) u si cos (iv) u t () Usig Euler s theorem prove the followig : (i) If u u u t Prove tht si u. UNIT: 7. INTEGRAL CALCULUS (TWO QUESTIONS FOR FULL TEST) Emple 7. : Fid the re etwee the curves, - d the lies d Emple 7.6 : Fid the re etwee the lie d the curve. Emple 7.7 : Fid the re ouded the curve d the lie. Emple 7.8 : Fid the re of the regio eclosed Emple 7.9 : Fid the re of the regio commo to the circle 6 d the prol 6. Emple 7. : Compute the re etwee the curve si d cos d the lies d E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

12 Emple 7. : Fid the re of the regio ouded the ellipse Emple 7. : Fid the re of the curve 6 (i) etwee = d = 6 (ii) etwee = 6 d = 7 Emple 7. : Fid the re of the loop of the curve Emple 7. : Fid the re ouded - d rch of the ccloid t si t, cos t. EXERCISE 7.. () Fid the re of the regio ouded the curve d the etwee = - d =. (7) Fid the re of the regio ouded the ellipse etwee the two ltus rectums. (8) Fid the re of the regio ouded the prol d the lie. (9) Fid the commo re eclosed the prols 9 d 6 () Derive the formul for the volume of right circulr coe with rdius r d height h. Emple 7. 7 : Fid the legth of the curve etwee d Emple 7.8 : Fid the legth of the curve Emple 7.9 : Show tht the surfce re of the solid otied revolvig the rc of the curve out - log 9 si from to Emple 7. : Fid the surfce re of the solid geerted revolvig the ccloid t si t, cos t out its se ( - ). EXERCISE 7.. () Fid the perimeter of the circle with rdius. () Fid the legth of the curve t si t, cos t etwee t d. () Fid the surfce re of the solid geerted revolvig the rc of the prol, ouded its ltus rectum out -. () Prove tht the curved surfce re of sphere of rdius r itercepted etwee two prllel ples t dtce d from the cetre of the sphere r d hece deduct the surfce re of the sphere.. UNIT: 8. DIFFERENTIAL EQUATIONS (TWO QUESTIONS FOR FULL TEST) Emple 8.7 : d Solve : d Emple 8. : Fid the cuic polomil i which tt its mimum vlue d miimum vlue t = - d respectivel. EXERCISE 8.. d (7) Solve the followig : d E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

13 Emple 8. : Solve : d d Emple 8. : Solve : d d Emple 8. : Solve : e d e d Solve the followig : () d d Emple 8.8 : Solve : Emple 8.9 : Solve : d t d Solve the followig : d t () d d () si d d give tht, where EXERCISE 8.. EXERCISE 8.. d (7) d d e sec d (9) Show tht the equtio of the curve whose slope t poit equl to through the origi e Solve the followig differetil equtios : (6) d d d EXERCISE 8.. e whe log, d, d () D 6D 9 e () Solve the differetil equtio D cos si d which psses Emple 8. : I certi chemicl rectio the rte of coversio of sustce t time t proportiol to the qutit of the sustce still utrsformed t tht tt. At the ed of oe hour. 6 grms remi d t the ed of hours grms. How m grms of the first sustce ws there iitill? Emple 8. : A k ps iterest cotiuous compoudig, tht tretig the iterest rte s the tteous rte of chge of pricipl. Suppose i ccout iterest ccrues t 8% per er compouded cotiuousl. Clculte the percetge icrese i such ccout over oe er. [ Tke e 8. 8 ]. Emple 8.7 : For postmortem report, doctor requires to kow pproimtel the time of deth of the decesed. He records the first temperture t..m. to e 9. F. After hours he fids the temperture to e 9. F. If the room temperture ( which costt) 7 F, estimte the time of deth. (Assume orml temperture of hum od to e 98.6 F) log e.6. d log e.9... Emple 8.8 : A drug ecreted i ptiets urie. The urie moitored cotiuousl usig ctheter. A ptiet dmitered mg of drug t time t =, which ecreted t Rte of t mg/h. (i) Wht the geerl equtio for the mout of drug i the ptiet t time t >? (ii) Whe will the ptiet e drug free? Emple 8.9 : The umer of cteri i est culture grows t rte which proportiol to the umer preset. If the popultio of colo of est cteri triples i hour. Show tht the umer of cteri t the ed of five hours will e times of the popultio t iitil time. E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

14 EXERCISE 8.6. () Rdium dppers t rte proportiol to the mout preset. If % of the origil mout dppers i ers, how much will remi t the ed of ers. [ Tke A s the iitil mout ]. () The sum of Rs. compouded cotiuousl, the omil rte of iterest eig four percet per um. I how m ers will the mout e twice the origil pricipl? ( log e. 69 ) () A cup of coffee t temperture C plced i room whose temperture C d it cools to 6 C i miutes. Fid its temperture fter further itervl of miutes. () The rte t which the popultio of cit icreses t time proportiol to the popultio t tht time. If there were,, people i the cit i 96 d,6, i 99 wht popultio m e 6. ticipted i? [ log.7, e. e ] () A rdioctive sustce ditegrtes t rte proportiol to its mss. Whe its mss mgm, the rte of ditegrtio. mgm per d. How log will it tke for the mss to e reduced from mgm to mgm. ( log e. 69 ) UNIT: 9. DISCRETE MATHEMATICS (ONE QUESTION FOR FULL TEST) Emple 9.8 : Show tht Z, * ifiite eli group where * defied s *. Emple 9. : Show the set G of ll mtrices of the form where R, group uder mtri multiplictio. Emple 9. : Show tht the set G /, Q ifiite eli group with respect to dditio. Emple 9. : Let G e the set of ll rtiol umers ecept d * e defied o G * for ll, G. Show tht ( G, * ) ifiite eli group. Emple 9. : Prove tht the set of four fuctios f, f, f, f o the set of o-ero comple umers C defied f f, f d f C forms eli group with, respect to the compositio of fuctios. Emple 9. : Show tht Z, forms group. Emple 9.6 :Show tht, 7 7 Z forms group. Emple 9.7 : Show tht the th roots of uit form eli group of fiite order with usul multiplictio. EXERCISE 9.. () Show tht the set G of ll positive rtiol forms group uder the compositio * defied * for ll, G. (6) Show tht,,, group with respect to mtri multiplictio.,, Where, form (7) Show tht the set M of comple umers with the coditio forms group with respect to the opertio of multiplictio of comple umers. (8) Show tht the set G of ll rtiol umers ecept - forms eli group with respect to the E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

15 opertio * give * for ll, G. (9) Show tht the set,,,, 9 forms eli group uder multiplictio modulo. () Show tht the set of ll mtrices of the form, R forms eli group uder mtri multiplictio. () Show tht the set G / Z eli group uder multiplictio. UNIT:. PROBABILITY DISTRIBUTIONS (ONE QUESTION FOR FULL TEST) Emple. : A rdom vrile X hs the followig proilit mss fuctio X 6 P ( X = k k k 7k 9k k k ) (i) Fid k. (ii) Evlute P X, P X d P X 6 (iii) Wht the smllest vlue of for which P X? Emple. : A ur cot white d red lls. Fid the proilit dtriutio of umer of red lls i three drws oe oe from the ur. (i)with replcemet (ii) without replcemet Emple. : The totl life time (i er) of er old dog of certi reed Rdom Vrile whose dtriutio fuctio give, for F Fid the proilit tht such five er old dog will live, for (i) eod ers (ii) less th 8 ers (iii) where etwee to ers. EXERCISE. (7) The proilit desit fuctio of rdom vrile X f k e,,,, elsewhere Fid (i) k (ii) P ( X > ) Emple.6 : If the umer of icomig uses per miute t us termius rdom vrile hvig Poso dtriutio with.9, fid the proilit tht there will e (i) Ectl 9 icomig uses durig period of miutes. (ii) Fewer th icomig uses durig period of 8 miutes. (iii) At lest icomig uses durig period of miutes. EXERCISE.. () The umer of ccidets i er ivolvig ti drivers i cit follows Poso dtriutio with me equl to. Out of ti drivers fid pproimtel the umer of drivers with (i) o ccidet i er (ii) more th ccidets i er e.98. Emple.9 : If X ormll dtriuted with me 6 d stdrd devitio fid. P X 8 (ii) P X 6 (i) E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

16 [Are tle: P(<<.) =.89, P(<<.) =., P(<<) =.77 ]. Emple. : The me score of studets for emitio d S.D. 6. (i) How m cdidtes c e epected to oti mrks etwee d 6 ssumig the ormll of the dtriutio d (ii) determie the limit of the mrks of the cetrl 7% of the cdidtes. [ Are tle : P(<<.) =.987, P(<<.6) =.8, P(<<.) =.] Emple. : Oti k, d f of the orml dtriutio whose proilit dtriutio fuctio give ke X. Emple. : The ir pressure i rdoml selected tre put o certi model ew cr ormll dtriuted with me vlue psi d stdrd devitio. psi. (i) Wht the proilit tht the pressure for rdoml selected tre () etwee. d. psi () etwee d psi (ii) Wht the proilit tht the pressure for rdoml selected tre eceeds. psi? [ Are tle: P(<<.) =.98, P(<<) =. ]. EXERCISE. () The me weight of mle studets i certi college i pouds d the stdrd devitio pouds. Assumig the weights re ormll dtriuted, fid how m studets weigh (i) etwee d pouds (ii) more th 8 pouds. [ Are tle: P(<<.67) =.8, P(<<.667) =.6, P(<<.667) =.88]. (8) Fid c, d of the orml dtriutio whose proilit fuctio give f ce X. ************ E R K HSS ERUMIYAMPATTI Pge 6 + STUDY MATERIALS

17 E R K HSS ERUMIYAMPATTI Pge 7 + STUDY MATERIALS COME BOOK SIX MARKS [6 Numer Of Si Mrk Questios To Be Asked For Full Test] UNIT:. MATRICES AND DETERMINANTS (TWO QUESTIONS FOR FULL TEST) EXERCISE. () Fid the djoit of the mtri A d verif the result. A A ) A dj ( ) A dj ( A () Fid the iverse of ech of the followig mtrices : (i) (ii) 7 (iii) (iv) 8 (v) () If 7 A d B verif tht (i) A B AB (ii) T T T A B AB (8) Show tht the djoit of A A itself. () For A show tht A A. PROPERTIES: () Stte d prove reversl lw for iverses of mtrices. [OR] Prove tht (AB) - = B - A -, where A d B re two o-sigulr mtrices. EXERCISE.. Solve mtri iversio method ech of the followig sstem of lier equtios: () 7, (), 7 EXERCISE.. Fid the rk of the followig mtrices: () () () () () 7 6 (6) 6 7 EXERCISE.. () Solve the followig o-homogeeous sstem of lier equtios determit method: (iii) 8 8 ; 9

18 E R K HSS ERUMIYAMPATTI Pge 8 + STUDY MATERIALS EXERCISE.. () Emie the costec of the followig sstem of equtios. If it costet the solve the sme. (iii) 6 ; 8 ; 7 (iv) ; 8 ; 7 Emple.: Fid the djoit of the mtri A Emple.: If, A verif the result I A A A dj A dj A Emple.: (iv) Fid the iverse of the followig mtri: A Emple.6: If A d B verif tht A B AB. Emple.7: Solve mtri iversio method 8, Emple.: Fid the rk of the mtri 7 Emple.: Fid the rk of the mtri Emple.: Fid the rk of the mtri Emple.: Fid the rk of the mtri 7 6 Emple.6: Fid the rk of the mtri 7 Emple.7: Solve the followig sstem of lier equtios determit method. () 6 6 ; 8 Emple.8: Solve the followig o-homogeeous equtios of three ukows. () ; ; () 6 ; 8 ;

19 UNIT:. VECTOR ALGEBRA (TWO QUESTIONS FOR FULL TEST) EXERCISE. Prove vector method: () If the digols of prllelogrm re equl the it rectgle. () The mid poit of the hpoteuse of right gled trigle equidtt from its vertices. () The sum of the squres of the digols of prllelogrm equl to the sum of the squres of the sides. (8) Forces of mgitudes d uits ctig i the directios 6 i j k d i j 6 k respectivel ct o prticle which dplced from the poit (,-) to (,,). Fid the work doe the forces. EXERCISE. (9) Let,, c e uit vectors such tht.. c d the gle etwee d c tht c 6. Prove EXERCISE.. () Prove vector method tht the prllelogrm o the sme se d etwee the sme prllels re equl i re. (6) Prove tht twice the re of prllelogrm equl to the re of other prllelogrm formed tkig s its djcet sides the digols of the former prllelogrm. (8) Forces i 7 j, i j 6 k, i j k ct t poit P whose positio vector i j k Fid the momet of the resultt of three forces ctig t P out the poit Q whose positio vector 6 i j k. () Fid the mgitude d directio cosies of the momet out the poit (,-,) of force i j 6 k whose lie of ctio psses through the origi.. EXERCISE.. () Show tht the poits (,,), (,,-), (-,,) (,,-) re lig o the sme ple. ( Hit : It eough to prove three vectors formed these four poits re coplr). (7) If i j k, i j k d c i j k (8) Prove tht c c, show tht c c iff d c re collier.(vector triple products o-ero ) () Prove tht. c d c. d c. d () Fid c d i j k i k, c i j k. if,, d i j k EXERCISE.6. () A vector r hs legth d directio rtios (,,), fid the directio cosies d compoets of r. (6) Fid the vector d Crtesi equtio of the lie through the poit (, -, -) d prllel to the vector 9 i 6 j k. (7) Fid the vector d Crtesi equtio of the lie joiig the poits (,-,) d (,-,) EXERCISE.7. () Fid the shortest dtce etwee the prllel lies (i) r i j k t i j k d r i j k s i j k (ii) d () Show tht the followig two lies re skew lies: r i j 7 k t i j k d r i j k s 7 i 6 j 7 k () Fid the shortest dtce etwee the skew lies () Show tht (,-,), (,-,) d (,-,6) re collier. 6 7 d 9 E R K HSS ERUMIYAMPATTI Pge 9 + STUDY MATERIALS

20 EXERCISE.8. () Fid the vector d Crtesi equtios of ple which t dtce of 8 uits from the origi d which orml to the vector i 7 j 8 k () The foot of the perpediculr drw from the origi to ple (8,-,). Fid the equtio of the ple. () Fid the equtio of the ple through the poit whose p.v. i j k d perpediculr to the vector i j k. (6) Fid the vector d Crtesi equtios of the ple through the poit (,-,) d prllel to the ple r i j k 7. () Fid the Crtesi form of the followig ples: (i) r s t i t j s t k (ii) r s t i s t j s t k EXERCISE.9. () Fid the equtio of the ple which cot the two lies () C ou drw ple through the give two lies? Justif our swer. i j k t i j k d r i j k s i j 8 k r 6 () Fid the poit of itersectio of the lie r j k s i j k d ple. () Fid the meetig poit of the lie r i j k t i j k d 8 d the ple 7 EXERCISE. () Fid the vector equtio of sphere with cetre hvig positio vector i j k d rdius uits. Also fid the equtio i Crtesi form. () Fid the vector d Crtesi equtio of the sphere o the joi of the poits A d B hvig positio vectors i 6 j 7 k d i j k respectivel s dimeter. Fid lso the cetre d rdius of the sphere. () Oti the vector d Crtesi equtio of the sphere whose cetre (, -, ) d rdius the sme s tht of the sphere r i j k. (6) Show tht dimeter of sphere suteds right gle t poit o the surfce vector method. Emple.: With usul ottios i trigle ABC prove tht cos A c c Emple.: With usul ottios prove (i) cos C c cos B Emple.: Agle i semi-circle right gle. Prove vector method. Emple.: Digols of rhomus re t right gles. Prove vector methods. Emple.: If d q 6 i j k the fid p q. Verif tht p d p q re perpediculr to p i j 7 k ech other d lso verif tht q d Emple.: p q re perpediculr to ech other. If the positio vectors of three poits A, B d C re respectivel Fid AB AC. Iterpret the result geometricll. Emple.6: Prove tht the re of qudrilterl AC BD i j k, i j k where AC d BD re its digols. d i k 7. E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

21 Emple.7: If,, c re the positio vectors of the vertices A, B, C of trigle ABC, the prove tht the re of trigle ABC Emple.8: c c Prove tht i trigle ABC with usul ottios, Emple.: For three vectors Emple.6: If i j k, i j show tht the re ot equl. Emple.7: Let Deduce the coditio for the poits c si A si B si C,, c prove tht, c, c c 6 k, c i j k,, c to e collier. vector method.,fid (i) c (ii) c,, c d d e four vectors the (i) c d d c c d (ii) c d c d c d Emple.8: Prove tht, c, c,, c d Emple.9: Fid the vector d Crtesi equtios of the stright lie pssig through the poit A with positio vector i j k d prllel to the vector i 7 j k Emple.: Fid the vector d Crtesi equtios of the stright lie pssig through the poits (-,,) d (,-,6) Emple.: Fid the shortest dtce etwee the prllel lies r Emple.: i j t i j k d r i j k s i j k Show tht the two lies r i j t i k d r i j s i j k dtce etwee them. Emple.: Fid the shortest dtce etwee the skew lies r i j i j k d r i j k i j k re skew lies d fid the Emple.6: Show tht the poits (,-,-), (,,-) d (,-,-) re collier. Emple.8: Fid the vector d Crtesi equtio of ple which t dtce of 8 uits from the origi d which orml to the vector i j k Emple.9: The foot of perpediculr drw from the origi to the ple (,-,-), fid the equtio of the ple. Emple.: Fid the equtio of the ple pssig through the lie of itersectio of the ple d d pssig through the poit (,,) Emple.: Fid the equtio of the ple pssig through the itersectio of the ples 8 d d perpediculr to the ple Emple.: Fid the dtce from the poit (,-,) to the ple r i j k s i j t j k E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

22 Emple.6: Fid the dtce etwee the prllel ples r i j k d r i j k Emple.7: Fid the equtio of the ple which cot the two lies E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS d Emple.8: Fid the poit of itersectio of the lie pssig through the two poits (,,-) ; (-,,) d the -ple. Emple.9: Fid the co-ordites of the poit where the lie r i j k t i j k r i j k Emple.6: meets the ple Fid the vector d Crtesi equtios of the sphere whose cetre i j k d rdius. Emple.6: Fid the vector d Crtesi equtio of the sphere whose cetre (,,) d which psses through the poit (,,) Emple.6: Fid the equtio of the sphere o the joi of the poits A d B hvig positio vectors d i j k respectivel s dimeter. Emple.6: Fid the coordites of the cetre d the rdius of the sphere whose vector equtio r r 8 i 6 j k i 6 j 7 UNIT:.COMPLEX NUMBERS (TWO QUESTIONS FOR FULL TEST) EXERCISE.. () Epress the followig i the stdrd form i (ii) i i i (iv) i i 8 9 i 6 i i i () Fid the rel vlues of d for which the followig equtios re stfied. i i (i) i i i (ii) i i i i i (iii) 8 i i () For wht vlues of d, the umers i d i re comple cojugte of ech other? EXERCISE.. () Fid the squre root of 8 6i () Prove tht the trigle formed the poits represetig the comple umers 8i, i d i o the Argd ple right gled. () Prove tht the poits represetig the comple umers 7 i, i, 7i d i form prllelogrm.( Plot the poits d use midpoit formul). (7) If rg d rg the prove tht 6 (8) P represets the vrile comple umer. Fid the locus of P, if (ii) i i (iv). EXERCISE.. () Solve the equtio 8 if i root. () Solve the equtio if oe root i () Solve : 6 give tht oe of the root i EXERCISE.. k

23 () Simplif: cos i si si i cos () If cos cos cos si si si,prove tht (i) cos cos cos cos (ii) si si si si (iii) cos cos cos (iv) si si si (v) cos () Prove tht cos (i) i i cos (ii) i i si cos cos si si (iii) cos i si cos i si cos (iv) i d i re rel d purel imgir respectivel cos (7) If cos prove tht (i) cos (ii) i si m (9) If cos i si ; cos i si prove tht cos m EXERCISE.. () Fid ll the vlues of the followig: (ii) 8i () If,, show tht (i) (ii) where the comple cue root of uit., the () Prove tht if (i) c c c c c (iii) () Solve: (i) Emple.9: Fid the modulus d rgumet of the followig comple umers: (i) i (ii) i (iii) i Emple.: If i i... i A, prove tht ib (i) A (ii)... B t t... t Emple.: Prove tht the comple umers comple ple. Emple.: k t m B, A i, i, i Prove tht the poits represetig the comple umers vertices of rectgle. k Z re the vertices of equilterl trigle i the d i o the Argd ple re the i, i, i E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

24 Emple.: Show tht the poits represetig the comple umers the Argd digrm. Emple.6: Fid the squre root of 7 i 7 9i, 7i, i Emple.7: Solve the equtio 8,if oe of its roots i Emple.9: Simplif: cos i si si i cos Emple.: If positive iteger, prove tht Emple.: si i cos si i cos If positive iteger, prove tht i i form right gled trigle o cos i si PROPERTIES: () Stte d prove the trigle iequlit of comple umers. () For two comple umers Z, Z, show tht (i) Z Z Z Z (ii) rg Z Z rg Z rg Z () For two comple umers Z, show tht (i) Z Z Z, Z Z (ii) rg rg Z rg Z Z Z () Show tht for polomil equtio P with rel coefficiets imgir roots occur i cojugte pirs. UNIT:. ANALYTICAL GEOMETRY (ONE QUESTION FOR FULL TEST) EXERCISE.. () Fid the, verte,focus equtio of directri, ltus rectum, legth of the ltus rectum fro the followig prols d hece sketch their grphs. (iii) () If prolic reflector cm i dimeter d cm deep, fid the dtce of the focus from the cetre of the reflector. () The focus of prolic mirror t dtce of 8cm from its cetre (verte). If the mirror cm deep, fid the dimeter of the mirror. EXERCISE.. () Fid the equtio of the ellipse if (ii) the foci re (,-), (,-) d e = /. (iii) the foci re, d the vertices re, (iv) the cetre (,-), oe of the foci, d (v) the cetre t the origi, the mjor log, e = / d psses through the poit, (vi) the legth of the semi mjor, d the ltus rectum re 7 d 8 / 7 respectivel, the cetre (,) d the mjor prllel to -. (vii) the cetre (,-), oe of the foci (6,-) d pssig through the poit (8,-). (viii) the foci re,, d the legth of the ltus rectum /. e cos 6 E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

25 (i) the vertices re, d e () Fid the locus of poit which moves so tht the sum of its dtces from (,) d (-,) 9. () Fid the equtios d legth of mjor d mior es of (ii) 9 6 (iv) () Fid the equtios of directrices, ltus rectum d legth of ltus rectums of the followig ellipses: (iii) (iv) EXERCISE.. () Fid the equti of the hperol if (i) focus : (,) ; correspodig directri :, e (ii) Cetre : (,) ; legth of the semi-trsverse ; e = 7 / the cojugte log -. (iii) cetre : (,) legth of the semi trsverse 6 ; e =, trsverse prllel to -. (iv) cetre : (,-) ; legth of the trsverse 8 ; e = / trsverse prllel to -. (v) cetre ; (,) ; the dtce etwee the directrices, the dtce etwee the foci d the trsverse prllel to. (vi) foci :, 8 ; legth of trsverse (vii) foci :, ; e = (viii) cetre : (,) ; oe of the foci (6,) d the correspodig directri = 9 /. (i) foci : (6,-) d (-,-) d pssig through the poit (,-) () Fid the equtio d legth of trsverse d cojugte es of the followig hperols: (iii) () Fid the equtios of directrices, ltus rectums d legth of ltus rectum of the followig hperols. (ii) () Show tht the locus of poit which moves so tht the differece of its dtces from the poits (,) d (-,) EXERCISE.. () Fid the equtios of the tget d orml to the prols. (i) t, 6 (ii) 9 t, (iii) t, (iv) to the ellipse 6 (v) to the hperol 9 t, () Fid the equtios of the tget d orml (i) to the prol 8 t (ii) to the ellipse t t (iii) to the ellipse 6 t (iv) to the hperol t 9 t 6 () Fid the equtios of the tgets (i) to the prol 6, prllel to (ii) to the prol 6, perpediculr to the lie 8 (iii) to the ellipse, which re perpediculr to t, (iv) to the hperol 6, which re prllel to 9 () Fid the equtios of the two tgets tht c e drw (i) from the poit (,-) to the prol (ii) from the poit (,) to the ellipse 9 6 (iii) from the poit (,) to the hperol 6. E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

26 EXERCISE.. () Fid the equtio of the smptotes to the hperol (ii) 8 () Fid the equtio of the hperol if (i) the smptotes re 8 d d (,) poit o the hperol. () Fid the gle etwee the smptotes of the hperol (iii) 6 EXERCISE.6. () Fid the equtio of the rectgulr hperol whose cetre, poit,. d which psses through the () Fid the equtio of the tget d orml (i) t (,) to the rectgulr hperols (ii) t, to the rectgulr hperol 8 () A stdrd rectgulr hperol hs its vertices t (,7) d (-,-). Fid its equtio d smptotes. () Fid the equtio of the rectgulr hperol which hs its cetre t (,) oe of its smptotes d which psses through the poit (,-). (6) Fid the equtios of the smptotes of the followig rectgulr hperols. (ii) (iii) 6 6 (7) Prove tht the tget t poit to the rectgulr hperol forms with the smptotes trigle of costt re. Emple.7 : Fid the, verte, focus, directri, equtio of the ltus rectum, legth of the ltus rectum for the followig prols d hece drw their grphs. (iii) 8 Emple.9: The hedlight of motor vehicle prolic reflector of dimeter cm d depth cm. Fid the positio of ul o the of the reflector for effective fuctioig of the hedlight. Emple.: A reflectig telescope hs prolic mirror for which the dtce from the verte to the focus 9mts. If the dtce cross (dimeter) the top of the mirror 6cm, how deep the mirror t the middle? Emple.: Fid the equtio of the ellipse whose foci re (,) d (-,) d eccetricit /. Emple.6: Fid the equtio of the ellipse whose oe of the foci (,) d the correspodig directri =8 d eccetricit /. Emple.7: Fid the equtio of the ellipse with focus (-,-), directri d eccetricit /. Emple.8: Fid the equtio of the ellipse with foci, d vertices, Emple.: Fid the equtio of the ellipse whose cetre (,), oe of the foci (,) d eccetricit /. Emple.: Fid the equtio of the ellipse whose mjor log -, cetre t the origi,psses through the poit (,) d eccetricit /. Emple.: Fid the equtio of the ellipse if the mjor prllel to -, semi mjor, legth of the ltus rectum 6 d the cetre (,). Emple.: Fid the equtio of the ellipse give tht the cetre (,-), focus (,-) d pssig through (8,) E R K HSS ERUMIYAMPATTI Pge 6 + STUDY MATERIALS

27 Emple.: Fid the equtio of the ellipse whose foci re (,) (-,) d legth of the ltus rectum 6. Emple.: Fid the equtio of the ellipse whose vertices re (-,) d (-7,) d eccetricit /. Emple.6: Fid the equtio of the ellipse whose foci re (,) d (,9) d eccetricit /. Emple.7: Fid the equtio of poit which moves so tht the sum of its dtces from ( -, ) d (,). Emple.8: Fid the equtios d legths of mjor d mior es of (iii) 9 6 Emple.9: Fid the equtios of es d legth of es of the ellipse Emple.: Fid the equtios of directrices, ltus rectum d legth of ltus rectum of the followig ellipses. (iii) 8 Emple.: Fid the eccetricit, cetre, foci, vertices of the followig ellipse : (iii) 6 Emple.: The orit of the erth roud the su ellipticl i shpe with su t focus. The semi mjor of legth 9.9 millio miles d eccetricit.7. Fid how close the erth gets to su d the gretest possile dtce etwee the erth d the su. Emple.6: Fid the equtio of hperol whose directri, focus (,) d eccetricit. Emple.7: Fid the equtio of the hperol whose trsverse log -. The cetre (,) legth of semitrsverse 6 d eccetricit. Emple.8: Fid the equtio of the hperol whose trsverse prllel to -, cetre (,), legth of the cojugte d eccetricit e =. Emple.9: Fid the equtio of the hperol whose cetre (,). The dtce etwee the directrices /, the dtce etwee the foci d the trsverse prllel to. Emple.: Fid the equtio of the hperol whose foci re 6, d legth of the trsverse 8. Emple.: Fid the equtio of the hperol whose foci re, d eccetricit /. Emple.: Fid the equtio of the hperol whose cetre (,), oe of the foci re (8,) d the correspodig directri =. Emple.: Fid the equtio of the hperol whose foci re, d the legth of the trsverse 6. Emple.: Fid the equtio of the hperol whose foci re, d pssig through (,). Emple.8: Fid the equtios d legth of trsverse d cojugte es of the hperol Emple.: Fid the equtios of directrices, ltus rectum d legth of ltus rectum of the hperol E R K HSS ERUMIYAMPATTI Pge 7 + STUDY MATERIALS

28 Emple.: The foci of hperol coicide with the foci of the ellipse. Determie the equtio of the hperol if its eccetricit. Emple.: Fid the equtio of the locus of ll poits such tht the differeces of their dtces from (,) d (-,) lws equl to. Emple.: Fid the eccetricit, cetre, foci d vertices of the hperol d lso trce the curve. Emple.: Fid the eccetricit, cetre, foci d vertices of the hperol d lso trce the curve Emple.8: Poits A d B re km prt d it determied from the soud of eplosio herd t those poits t differet times tht the loctio of the eplosio 6km closer to A th B. Show tht the loctio of the eplosio restricted to prticulr curve d fid equtio of it. Emple.9: Fid the equtios of the tgets to the prol from the poit (,). Also fid the poits of cotct. Emple.6: Fid the equtio of the tget d orml to the prol t (,). Emple.6: Fid the equtios of the two tgets tht c e drw from the poit (,) to the ellipse 7 Emple.6: Fid the seprte equtios of the smptotes of the hperol 7 Emple.6: Fid the equtio of the hperol which psses through the poit (,) d hs the smptotes 7 d. Emple.66: Fid the gle etwee the smptotes of the hperol 6 9 Emple.67: Fid the gle etwee the smptotes to the hperol 7 Emple.68: Prove tht the product of perpediculrs from poit o the hperol costt d the vlue Emple.69: Fid the equtio of the stdrd rectgulr hperol whose cetre through the poit,, to its smptotes d which psses Emple.7: The tget t poit of the rectgulr hperol c mkes itercepts, d the orml t the poit mkes itercepts p,q o the es. Prove tht p q Emple.7: Show tht the tget to rectgulr hperol termited its smptotes ected t the poit of cotct. E R K HSS ERUMIYAMPATTI Pge 8 + STUDY MATERIALS

29 UNIT:.DIFFERENTIAL CALCULUS-APPLICATIONS-I (TWO QUESTIONS FOR FULL TEST) EXERCISE.. () A Prticle of uit mss moves so tht dplcemet fter t secs give cos t. Fid the ccelertio d kietic eerg t the ed of secs. K. E. m v, m mss kt () Newto s lw of coolig give e, where the ecess of temperture t ero time C d t time t secods C. Determie the rte of chge of temperture fter s, give tht 6 C d... k e. (7) Two sides of trigle re m d m i legth d the gle etwee them icresig t rte of.6 rd / sec. Fid the rte t which the re of the trigle icresig whe the gle etwee the sides of fied legth /. Emple. : Fid the equtios of the tgets d orml to the curve t the poit (, ). Emple. : Fid the equtios of the tget d orml to the curve t the poit (, - ). Emple. : Fid the equtio of the tget t the poit (,) to the curve c. Emple.6 : Fid the equtio of the tget to the prol, with the. Emple.9 : Show tht d c cut orthogoll. EXERCISE.. () Fid the equtio of the tget d orml to the curves (i) t (ii) si cos, t (iii) si t (iv) 6 si cos t () Fid the poits o curve t which the slope of the tget. () Fid t wht poits o the circle, the tget prllel to the lie 7 () At wht poits o the curve the tget prllel to (i) (ii). (6) Fid the equtio of orml to tht prllel to 8 9. (8) Prove tht the curve d 6 cut ech other t right gles. (9) At wht gle do the curves d itersect? Emple. : Verif Rolle s theorem for the followig : (v) f e si, Emple. : Appl Rolle s theorem to fid poits o curve cos, where the tget prllel to i,. EXERCISE.. () Usig Rolle s theorem fid the poits o the curve, where the tget prllel to. Emple. : Verif Lgrge s lw of the me for f o, Emple. : A clidricl hole mm i dimeter d mm deep i metl lock reored to icrese the dimeter to. mm. Estimte the mout of metl removed. E R K HSS ERUMIYAMPATTI Pge 9 + STUDY MATERIALS

30 Emple.6 : Suppose tht f d ' f for ll vlues of, how lrge c f possil e? Emple.7 : It took sec for thermometer to re from -9 C to C whe it ws tke from freeer d plced i oilig wter. Show tht somewhere log the w the mercur ws rig t ectl 8. C / sec. EXERCISE. () Verif Lgrge s lw of me for the followig fuctios: (i) f,, (ii) f,, (iii) f,, (iv) f,, (v) f,, ' () If f d f for how smll c f possil e? () At. p.m. cr s speedometer reds miles / hr., t. pm it reds miles / hr. Show tht sometime etwee. d. the ccelertio ectl miles / hr. Emple.8 : Oti the Mcluri s Series for () log e () rc t or t EXERCISE.. () Oti the Mcluri s Series epsio for : (ii) cos (iii) (iv) t, Emple. : Fid Emple. : lim si t lim log if ets. si Emple. : Evlute : lim cos ec Emple.6 : The curret t time t i coil with restce R, iductce L d sujected to costt electromotive force E give E e R RT i L Oti suitle formul to e used whe R ver smll. Evlute the limit for the followig if ets. () (6) t lim si lim t EXERCISE.6. E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

31 (8) lim cot cot (9) log. lim () lim () lim () cos lim Emple.7: e Prove tht the fuctio f si cos ot mootoic o the itervl, Emple.8: Fid the itervls i which f icresig d decresig. Emple.9 : Prove tht the fuctio f either icresig or decresig i, Emple. : Dcuss mootoicit of the fuctio f si,, Emple.: Determie for which vlues of, the fuctio, strictl icresig or strictl decresig. Emple. : Determie for which vlues of, the fuctio f 6 icresig d for which it decresig. Also determie the poits where the tgets to the grph of the fuctio re prllel to the. Emple.:Show tht f t si cos, strictl icresig fuctio i the itervl, EXERCISE.7 () Which of the followig fuctios icresig or decresig o the itervl give? (iv) o, (v) si o, () Prove tht the followig fuctios re ot mootoic i the itervls give. (i) o, (ii) o, (iii) si o, (iv) t cot o, () Fid the itervls o which f icresig or decresig. (i) f (ii) f (iii) f (iv) f si,, (v) f cos i, (vi) f si cos i, Emple. : Prove tht the iequlit true wheever d. Emple.6: Prove tht si t, () Prove the followig iequlities :, EXERCISE.8. (i) cos, (ii) si, (iii) t for ll (iv) log for ll 6. E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS

32 Emple.8 : Fid the solute mimum d miimum vlues of the fuctio. f, Emple.9 : Dcuss the curve with respect to locl etrem. Emple. : Locte the etreme poit o the curve 6 d determie its ture emie the sig of the grdiet o either side. EXERCISE.9. () Fid the criticl umers d sttior poits of ech of the followig fuctios. (iii) f (iv) f (vi) si f i, (vii) f si i, () Fid the solute mimum d solute miimum vlues of f o the give itervl: (i) f,, (ii) f,, (iii) f,, (iv) f 9,, (v) f,, (vi) f si cos,, (vii) f cos,, () Fid the locl mimum d miimum vlues of the followig fuctios: (i) (ii) EXERCISE.. () Fid the umers whose sum d whose product mimum. () Fid two positive umers whose product d whose sum miimum. (6) Restce to motio F, of movig vehicle give F. Determie the miimum vlue of restce. Emple.6 : Determie where the curve c cve upwrd d where it cocve dowwrd. Also fid the iflectio poits. Emple.6 : Determie the poits of iflectio if, of the fuctio Emple.66 : Test for poits of iflectio of the curve si,, EXERCISE.. Fid the itervls of cocvit d the poits of iflectio of the followig fuctios: () f () f UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II (ONE QUESTION FOR FULL TEST) Emple 6. : Compute the vlues of d d if f where chges (i) from to. d (ii) from to.. Emple 6. : Use differetils to fid pproimte vlue for 6. E R K HSS ERUMIYAMPATTI Pge + STUDY MATERIALS