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I C.S.£ Pre-l999 I ol Ill

MATHEMATICS 1

2,

3,

4.

5,

T he number of arbitrary conslant in lhe complete pn1mtive of the diiTenltl\ial

equ:\IIOn ¢.{ :r, y.t~i'l Jx, tl' ~·~ = 0 is dr

"' b 2

c. 3 cl ~

file dlt.Jerentinl equallon of Ute system of circles 1ouclung lhe ax1~ at origin is

, , dv 0 ;!. (1C-y' ) d~ - 2Ay =

( , ' dl' b >c-y·) --+2~ = 0

dx , ., d\'

o. (x- 1 ,.-) --- + 2xv =(I - tlx •

d ., ~ Jr

(x-+y' ) dx -1 2xy = ()

The smgular solution of the equauon p~-

4xyp+8p '=c) (p-.. <(l'~ is d.~J

:t 27y = 4x b 17y =-lx~

c. 27y = -lx' cl .nl = 4x r'he solutions \lf 1he diiTerential equa1ion 2y(y'+2l-~'Y'~ are the i'tmction J_ y=()

2. y =--lx

Select the correct ans11'er usillg the codes gh•er1 belo". a None or I nnd l1s as1ngulnrsolution

b. both I .nnd ?. arc smgular solullons

~ I is a singular solution but2 is not

cl 2 is a singular so!Uiion bu! l is not

The onhogc)llnl trajectories of ibe system ol curves 1" sm ffil=K" are

a y"cos oa = a

b. y cosO = a

c f cosnO=a

7

d. 1" tun n9 = a

The equati1111 whose s.olution famil) 1s self orthogomd 1s

I , d1• a. p- -~-- p = -"-

p dr:

dv b \'PX + yl(x ,. yp)- J..p= 0. p : ck

dy c, (px-y)('"YP)-A.p = o, p =-,

uX

dy d. (px+y-)(X·)'p)-A.[F 0, p~

dt

Which of the followtog smtemen(s associated with n lirst order non-linear diJrerential equation llx. y. d) /dx) : (I are correct'/ l Its general solution must contain ooly

one arbitrruy cooslrutt.

2 Its singular sol utioo can be obtained by subsututing partJculnr ' 'alue of U1e arbitrary constam in JJS general solullon

3 Its stngular solution ls an envelope of hs general solution which also satisfies tile equal ion

Select lhe correct answer using the codes given 11elow: a. L land .3

b I and 2

c. I and 3

d. 1 and 3

A particular integral d:v dy , ---"c - (1> +b)-+ l!b,l' = Q(.t)LS <Lr· d~

u. e'~ Jk ... '" ~ (le"~t)}t~ enj ~~~~ •·,. ~~11! "'ttt}jL\: b.

c. e·"" Jk""~ Q""'dx)}w d. e"' n··· '"~ Qc' .. dx)jn-

or

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9.

n. ax1by= c

b • • ,.: - by = I)

cJ aX: • b).! = 1

d. ~X I t;j - 0

I 0. A vari:oble lines p3S$t.."S through the lixcd 110inJ '"' b). The locUJ! ••f Ute utiddl~ t•<litol of the ~egtnent inten:epte<l hc!ween th~ a~cs is given ~y

II.

12

13.

14.

' h il . - --J a '

b. ~, .! =2 . ~ X b

c. -+-~ l u )'

d. .!..w !. =2 " b

The joint equation of tht t>air of the lines through l11c Hrigln wbieb ~nl purpendicul~r to the line~ l'epresented lly n x1-+ 2hxy+by'Z,o i•

•· bx:, 2t~yray2 ~o b. nx' -2hxy I by~O

c. bxl- 2h!<) +-0=0

d. J,xl. l hX} - ayl=O

lf u ax! -o-21LX)• bi 1 2g.x • 2fy + e 0 rcprcscniJI lw() ~trnigbl Hnos .then the Utird pair of strnight lines through the four poinlll iu which lh.e lines u ~ 0 nteot tl1e axis is

a. u ~ 41ft!' ch)xy =0 \>. e<o"'4jfg- ch~)"'ll

c. cu •(fg-ch)!<)= O

d. u II fg-cb)xyoO

The po1M equation

represents

'~ ~ a strnighf line 1:>. a purabol~ c. u hyperbola ,1. au e II ipse

rr lho nnrnt31 dr.lwn at ~ pomt l "'11 2olt) nf

!he parabola y ~~~~ m~et.~ it again in point ( •'i . 21tl) . then

IS.

II).

t7,

Ill.

19.

l a. l r ·lr -

'' IJ t .. - .!L . ... :

c. 10 11 I .L ., d. l olr • I

Z ot JO

'&.. .. " • 0 'I .,e lwo wei~ :C + ,~ - h: + 2) - ·I = and .~t-t y' - IOx - lly ~ I M ·0 u. touch each oUter uuern•Jly b. wuch c:aoh oth.:re:<1emally c. inlet'S'"' eoch 1)\her

d. neither intc.rsed nor ·touch c;ach other

The dn:les ,=2 a cos (ll - a) ""d y - 2h .:os (t)- 131 intersect at au angle

tl 3. '

b. a.- (3

.::.~ l

d. a. ,. p Th" stllndord "'JUntion nf fhc ellipse .lhc:­lt ngth or whose nt~joo· a:<.i.~ is 8 3)1d thO> dislOoce b"twccm who~" dil'b:lrico~ I!; 16 I!; J,tiven hy

•- ~. r "'' • 1>. ~. ,• - ,

16 11

;:l ,,.t c . . -· - j

12 t"

d. L.L - t lb ·~

lf n drclo cuts lhc rcctangulnr hyp.,·boln XY - ._~ in points Cx,.. y,) I r = 1, 2.3.4 ) .tl1en

3. X 1 X_l~X,!j - ·(!1

h. ~I X':X3 X,, - c j

il. :i l '<l -x.x.., = ·e . d. 1C1 x~ x1 X:t == c·

TII.e powr ~oortlinate$ of Ute center of Ute cirdc y - 6 cos (II- u) ""' •. (0. 0)

h. (0. a.)

c. (6, a.)

d. (3. t;t)

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20

21.

If~ slr.light line lllllk.;, 3ngle. 6fi 45" ron<l fl~ rt;:\ipectivei.Y with the J :<eot crf co­ordinntes ()X, OY ami OZ , lheo (lis equal to Q. t j ll

h. 31)0

~. 45" d. 6o•

If U1c plnno ! r.!. • !.- I <:Ul~ tha axe. of 2 l f

c•J-ordin:.t~ nl p<~i111 ;\,R.C: ,then ~~~ ore3 oft11e trlnnglc ABC l9

•• 18 sq. on its

b. 35 .\i<l, unito

"" 3-M" $q. unil<

d. 2-M" sq. llnlts

22. The: diameter af U1c clrde x~+J'+l3:9. xtyTr-"3 is.

•. 2.5 .... .5 CL Jf,

fl. 2./G 13, (l'la """"Lion 10 the oonc wbich passa

through the ~tree co-ordinotes •~<"" •~ well ns the Unes !. ~ .l_ !. and !.. • L =:: .!.is

I l 3 3 I I

•• y-.t: - 2n + 3:o.-y 0 b. 3y2 - L'< ~ xy - 0

c. yz +7-x+ J>.-y ~o

11 :lyz + I (izx + 15:<y= t)

24. Thd equation bf tho oylinJcr. generated b.)'

llnos pa~'IIUd to the li'l'ed line .!.,.L= .!. I " ' f•

nnd bavintt tl1e ellipse z:ll . a.x> 1 by'-= 1 • as the guiding CIUYclS

. . = a. a(nx - ~)--b(ny+!J1Z)' =u . . ' b, a(llx h )" b(n.)'·mz}· n·

c. a(m,:: - lzf·bttty·m:>.)"'n:

il. ;l(nx- ~)'·b(ny-mz)"'n~ 25. A unit vector ·perpandiculnr to th" two

vccto"' i J j -Inod 2 i •3j l ,i i•

•• si .. 3j -.t

h. i-<51-3i ·i ) s

c.. ; .,. i - 2~

1 ul IU

d. ~ { i ' j . 2 k J

26. The point$ A, B.C whoso> po~ition vectors

are a 3; · 4/ - 4 i, 6 •2i-j-:t.~ ; . 3]- Si are

a. fin iso~cele> lri<tngt.:

b. • acuta-aoglad tJi angle

c:. on nbtllse -angl~d u'iangle

d. a rlglu - ~ogled lriangle

27, The tli~t1nQ<!' between the line ;:{ i + t ~ ?., (2r +j + 4kl •nd the t•laoe ; (-27+0 : SIS

28,

u. 1 b. ,fi

c. 7/ .f'

d. 7~

The snlut i11n of the vecll!l' cquntinn t ,

= ~ ~ i$ (where II' 1Hny rea l11umbe:r)

a. r xz-; b. ;= ~;;; ~ h

"' ;= kh + ii d. r k;;~ !

29. 111e vector eq~•ation of the line pa"-<ing Lhroush the point with position vector 2 i · n -5 i and perpendiculnr to the pl;tne

j. .( i -3/·Si) - 2 = (\ ls (A. i• • scalar in each case)

a. ; = Hii • 3j I 5i)• 1.12i-3 j -5i)

b. ;=('2i- 3/ -3.i)+ i..(6i-3j-Sk j

c. 7= (6i- 3 / ·5hl /.~2i-3j-5i )

d. i (·2 i + 3j l Sk) • i..(6i -3 j · Si )

31!. If tllt'ee fnrc<!s. acting upon a rigid body, be rcprcs~ottld in m«gnilud6. dircctioll nnd line or aclian by ~10 ,.;d .. of n trln"llle

3 t.

1<1 kim in ord et. th "" • · they will be in equilibrium b. Utuy will ttdueb l(t n •in&Je force

c. they will 1-educe to a couple d. they will reduce to a s ingle for.:e and a

couple

A ne=.~ary nn<l su()i~ient cl\ndition Jhot • body >ctcd UflOU by Q<>p!Jlnor forces b~ in

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'35.

36.

equilibrium n. that the sum oflhe momenu of all the fprce< :lbout {1, a t)()int i.s 7.ero

b. any 11vo pootts is zoru c. any three non-oollineru: pointS IS zeoo

,1. ;In ttxi• i& 7,A!t"i)

lf a p.miclu in equilibrium i• •ubjected to

four fm·ccs viz F1 2t - 5 J1 6t. F~ ' r3!• . "

7 ~, F3=2; -2J -lt :md F., then "• i~ egu:ol to

b ~t-45 J·<U

c. 3, - 2;- * ' tl. ~. +, .l(lt

A .sleam bo.-.l ts movmg whh veloc:uly VJ when gJe:mt is shulot't: ff the retatlhl ion :ol

ml) subs<:<tutmt lime Is equnl t~ the mngnltudc of tho vclodty al ~tal lime. then thc.~·etodty In timet a tier the •team is shut oft ,.

l (l, v1e

b, \ ) C<

~ 2v,e1

d. 2v1e" I he le:~st for<e Utot wnt move a weoghl W ,olcrng u m uglJ hori?.ontal r lan;:, when: }., i• the angle of friction Is a. W tan ).

b. \V en• .i.

c. W sm ?.. tL Wcot ;..

Tho magnitude tlf • lot·~e wb idt ill ucliug on n body of mn~~ I k'g for 5 ~ec<m(l~ 10 r mduce vtlr!cit) nf I ml~cu in it. i~ o. 1000 dyne•

b. 2000 dynes

c. 10.000 dyooes

d. 211.(100 dy~

If" body of mnss m kg is carried l'Y n lift m<Win!l 11 ill• up11 An:l ueceli:TaliQn f .then the prc:$suro uu the pI nne of Lbc lij\ is

n, mf -mg

h. ma- mf

::.7.

.3)1.

39.

.J().

.Jt.

.1 of 10 c. ms ~mf d. (rug)t (mJ)

If Otc equMlon ilf motion of a particle excc:uting tt slmplo h1umonic motion _U

~·· - • r~r ••, then it., frequency will be ,Ji!

• . nJ2

b. 11/8

" · 2111

d. 11111 u· the par.tkle is PI'C\i.ecled from a horizonllll plane wbiclo volooity u nt "" angle <1, thea the time or fiignt of the JlBt1 icle will be

a. UtlUJJ

'1liC04il7 d. If

The l!iCap<: vcloeit)' (tf ~ pr'ljt:ctil~ !\'om lite earth i.i. •pproximuJcl)' a. 101 knwcc b 1 km $01;

e. 11.2 kmloL'C

d.. 112 k.tnlsee

'11oe h~adedmal number (A W. D)of Wh<:n coo~crtcd into the dcdmnl system is cqunllo

•. 2608.6125

b. 2007.8125 c. 2507.8125

d. 26'07.0125

ln ·• flow chart • reclllngh: fs a

u, blart/ .!!lllp hnx t.. deciSion box

c. t!omputatiou box

d. inpuU output bo.x

An J)gorilhn• ~~ a. • Uthle t'lflogarilhm• b. • t9llcotion of rcsulll

c. • "hnrlofformulao

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d~ a <tep by $lep proce<lure for finding the Sll lution of n problem

-13. As.<~ertion (A) : If~ a.nd yare l)dd po•itivo integers. then thet·e e.'\osts no 1111turnl number n such lh~l ,?,y'· n•.

Reason (R): lf n is nahn•l numher, ~ten ,,.ooer ~~~ olm for $Ome Integ« m ~ 1 or n::_ .!/ + l for son\e inr.:ger I ~ 0. :t. Anth A and R arc true anti R ;, the

eon'!:cl cl(planolilon ,.fA h. Bo.lh A :ond R. are true but R i< NflT

Ute ~on·~et c:otpllln>lirut of A

c. A IS true but R •~ f~l<e d. A is false but R i~ true

44. Ass<'I'Lion (A): Giv<n any arbitrorUy Iorge 0nmber.• L. nnd M.. lhere e.'<Jsts • n.umber ~

" such thnt f c tn. • M !

.... R~>ru;o> t CR): Fvr ony • l, /• '.r.=lOf.~

rutd foro • uibblc A log A cau.cxol:ell M .•

n, Both A •nd R are tnoe nnd R Ill the coll't:cl cxplnnnJlon nf A

b. BoUt A and R arc (ru.: hut R is N01' the eorrcct cxplafintiou "fA

.:. A is true bill R is false

d, A is r~lse butR is true

45. All.<ertion (A): y - c'lx ami y - :<e"' aro two ~·>lutinmi nf dilfl:J1)Tltin l c:quMinn ·t' ,. dJ· ' .i - 4-+ 4y = 0 'rhe genera l solut1on tk tlx

of the .:quntlrm i y = tn +bx)c=-, ll'ht:r.: u nnd b "'" orbitrol') con•lol\l,. R.ea~nn (R): II' u and v Me twn solut.lnn.< 11f

the scC<lnd onlcr homogo:ncoul llno:or diffcrenlial cquolion _ !hen au I bv i5 the genern l ~olution of the equatinm where a nnd b ore arbitrary conslnnL

a. llolh A "od R ore true ~nd R cy the ~UJn-ect explanation of A.

b Ruth A 3nd R ,,rc [rue hvt R. Ill NrYr Uit: ~o;r.:cl cxrlannrjon of A

c. A .strue bu1 R u f~~e d, A is fal•c but R is true

46. tfx= ' ..fii. y = •.fij, z= •.Ji7 . tben

"· :t .,.... Y z b. X " y~ 'L.

47.

-18.

49 .

50.

Sl,

c. x< z < _y d. x~ z- )'

Th.e ti':tclion I '3 is

• · equa l to 0.3333333

b, le•s 1 h••ttl,3.13333 by --1 -,

3.10

s or Ill

c. g•'<.Utl.,. tlton 0.3333333 b' -1-.

:uo

d • u o------- h 1 . gre3'-er tan .~~..J~~"'.=t~ y --.

3,10

Lot zt and z, be 1wo non•zcro complex nwnber'l, then iz1 ·z~'-tz1]1+ z~2, it' a. z1·z! is purely imagiru1y

b. Z;-Z: is !'Col

e. ~~-2~ ;, J1Urdy imugm:ory

d, 1:1+z.1 is rca I [f 1) ill J point in tltc Argond clingrnm repre"enting tl1e CQmple.." number

4( cQt 4" 1 1sin

4" ) and UP 1. rot.1ted

3 3 2Jf

through on angle -- in Ute •nti-3

olud:w~e direction , then 1' in th~ n~w Oll$illon represenfM

• · -!(cos nil 1 si11 n/2)

~' ~ <:, 4(~0• 7113- • ~in ;t/3)

220 tannolllc the sum of tlle flflil n cube• fi>r" suital>le n , bec;ruse 2211 ls :1. not on Qdd number

b. nut n square

c. nolo eube

d. divi•ihle by I 0 lf l'(xl is • pqlyollJI!iol in Jt • nd n, 1• -are unequ~l .then tl1e remainder m lhe div1sion ofll'l(.l by(;.:- AI (~-b) is

1.~-1•)/(11 ) - (.t-b)/ (bl 3 .

LI - b

(;x - b)/(a l (.~ 11 )/(b) b. tJ - b

(,t -a)/(h) - (.~- b)[(riJ c.

" r d. Nune of the nbove

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- . ~ ··· hlh If x l ~ ll! ~ lilctor x: -a-x·~.:~x-ll, I ,.,, ~

value ora i$ u. 1)

b. ]

c. 2

d. 3

53. A quadro1i~: equation wilh rntion.1l coeflicients and with one roo a M l- ..fi i• u. x'-411~ J~

b. x~+4:<+ 1~1 c. x'-4, - 1=0 •'- 1(~ • .~., t ~o

54. II' <1.. fl •re ilte rOOts Qf the <>qulllion otx' lbllh! 0 . tbllD llu>valu~: of(u.- p) !.!

55.

56.

57

b 1 ~ 4ac ~-

1l - 4ac h.

d.

b' ·Ia• 2

b' ~rTC

2 A r~cated cool of ll!d cquution xJ· ~x~-+ 3-.""< <t-l==O is ;t. 0 b. - 1

c... I

d. 2 llto values ofi11

• Arc

hlr ~ nrr '~ SU\- T I .:08 - . II I. s. 9. 13. !7.

l2 12 2 1

/ llf • mr 1 • " I' 17 b, C05 - "!- I SU\ -. II - -. J. ~~ .l. . 12 12

2 1

• mr . 111r 0 1 2." ,, " c. sut -+ ' co~ -, n = , • ..,, .... ., 6 6

it C<l5 ~i~in !!!:!.. n ~ l, 2, 3.4.S, 6 6 6

.For •ny llll'e~ Set> A.B aod C wblch one of the followinl' stot=ents i.> U(!l'n:(;t?

~.. .'\!'Ill - <I> ~A= <ll orB =<ll

b. A- B = <I> :::; A~ B

59.

611.

tlul Ill c. Av B = Ifl ~ 1\>;;, B

d. A.'"IB " <b. A r C "' d> = A B C = Ill

L<:t X 1Je. ;t non-cm))l)' finl\e •d .For a sub~c.t Y of X let n(Y) denote lite numbcT of elemeol' in Y. Let n!"X1~ 15 nnd let A.B.C be subsets of X such d13t n(A<.JB) = .5. n(C)=7 >nd n( A'nB' ~,t~) = 4 . (where for n subset Y of X Y denotes tlu; compltmlml of Y m X). Then nl( A '"IC) ..,(Br;Cll equnls

• . l

b. 2 c, ~

d. ~

Wb.iclt ort]J.: loUowin£! proporti~~ bold for a function f: X • ~· and subsets U.V c X.M. N~Y

I. [ 1(1\i.lv N) = [ 1(1\f)v ( 1tN)

2. f (l l \' ~(( Ll) / (I)

3. r'<M 'J=F'<Mr r '(Nl -1. f(U V)~f(U}I"o/(V)

.Sele<t lbe .:orrecl answer using the Q()de,~ giv~tl be-low:

a. I and 4

b. I and 2 c. land 3

d. 1,2 •ud3

For an\' a. b.;;N. Ute~et of namcol ·numben; <lefine ·~~ ~ b iJ: ond ou ly if a J b lhcn - is

a. An equivale:nee relation

b. Symmelric ond transitive bul no reOesive

c. Rel1e..xtve b11l nol tr.msotive and $ytrunetric:.

d. Refle.xwe ond trnn$i11vc: hut nnt symmetric

61. The <el ofintege11; modulo 7 is

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m "' II In Ia n n m I q It>

I n I n l d m n

Ia I a I n 11 m

- .OJ II IP Ia Ill IU Ul Ul Ill

n m n m n

I n m I D Ill In lq m Ia m Ia

Then wh1ch one or the folio\\ fng l• cortc(':t•/

a. (R, ~ ) i< • group , Inn (R, -t,.J ,. not• l'ing

h. (R, +, .1 i• ~ ring with unity 1!.. (R. ~ •. t i~ comm~llltl\·c ring

d. (R • .. ) is a non-conuliUiliUV<l ring without unit:y dement and has dh·isau of zero.

6"3, l11e ••t Z of a )I integers 1$ not o Yector spoce eWer the field R of re.•l numbetll und.:r onlinory addition • • • and lllltll.tplicalinn 'X' ofrcalnumhen; b<:caus~

a. (Z, ... ) is a rin!l b. (l, +, X) i.< not• ll~ld

g. (R. x) is not • group d. Ordinary multiplianlioo of roa1

number,; does not dll'line a sca1ar multipHcotion of/' liy R.

64. Whicb of the followin!l set' ol' vecto1~ In Rl arc linearly ind"Jlc-ndenl

I. lC 1. 0, 0), (0, I,OL( I. 1, !I)J 2. I(U , O), (0, L 0). (U. 0.1)1

3. 1(0. l. 0). ( I. 0, 1). ( I. l. O)J -h 1(\).0, 1 ).(0, 1 ,0),((~ l, l)j

Select the correct answer nsmg Lhe codes given below:

:t. I and 2

b. 2 ond3 c. 3-ond ~-

d. I and4

65. Which one of lhe following statemenl' is corre.:t'l

66.

7 oi JO a. 11tere i.• no vector <p~"CC of dimension

I

b. Any three ' ecton of~ vector sp•ce qf di01en>10n 3 are linearly lnd"Jlendent

.:. There IS one nnd only one bas is of • \'ector space of finite dimen~ion

d. If a f)on·zcro ,·ecto.- sr•~ v ~ genernted by a finite $<:! S .then V ean l>o !!OOOt'.t<d h~ a lincorl}' indefl"ndcnt ~ubset I)( S

lf T is ~ linQ:Ir ltons(olJI'l~tion from R1 lo R0 which T (1. 0) : (n. b). TCO. J)- (e, d1 then T(x. y) 1 x. yoz R is

a. {n:'t I by,"" I dy)

b. (ll.~ • dy. l•x + cy)

c. (nx + cy, bx+dyl

d. None of the above

f.1. lf A = (~ I ~1 ond B=[ ~ ~J• ~~en - 1 I

(AB)T lt A13T denotes !Ito tron..~ros<> of i\13}] i>

•· r~ :a b. (: ~ )

c. (~ ,'j tl. c :J

63. If A=~ ~J fhcn ,,J, A-u whl;J11:H:r

a. ctiJ ~ II b. ~~~ = I

c. o.lh u d. aJJ = • I

(19. lf 01e hnYe of the detenn irmnl

~ ": OJ a,, h, 6, &,

i(i "<(Ual to K, then the ,, <.t c,, rl, d, ,,,

V~ltUt: of the delc:nrrinsnl.

··~ "' ~;\a, .. .. , q ~ ~ IJ- b,

is -cquol lo 'I,:!,, ,, " "· +.l<lj "'

,, ~- 3K

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70.

71.

72.

7.3.

h, -K

" · K d. - 3K

If A H ~~ -; ]· U1cn iowrs~ of m~lrf:< A will b.,

.. [-~ l ~l h. [ ~ ~~ - ;,)

- 1 ( I 0

c. [~ ~ ~) d. u -! -~J Co1uidcr the equation A.'\~ B. where A=

(-~ -~) a~ (n . llt~u ~. lhe equ•tlon h•~ no solution

h. [~l is- a solution nf theequorion

c. l.hor<: e.s.ist< u non-~ero lllliquo ~olut.ioo

d. lbe equation bas infinil.,ly many solutions

Ote liotit of a convergent sequolnce or ralioonl numbun

u. need n()l exist ot all

b. exisiJ; and is otway• ratiOnal

c.. exisb; and i~ n.hvays ([mtinnnl

d. crins bul it ulir)' (:., rational or irrational

" · bas U1c v•loc 1/2

b. hM U1o value> I

c. has the value 2

d. does not ..,.; .1

74.

75.

76.

77~

78.

7-J.

8 ol Iii

Lf ht11 Aln.!r 1 osmx- b . where b is fLniie .. .... !. xl

' .then U1<t values Of o .and h respectively will be

a, - 2. -1 b. 2, I

c. - 2. I d. 2. ·l

llt~ I funetinn .r' - 0 (0 < -S'• 0/ •

a. ;, not cominunu. ((), ' )

b. is nol differentiable oo(O. m}

c. iK dillerentinhle 1111 (0, "')

by

d. i• differcmti•blo on (b."') ""copl nl x •

• lfy = sin x . then for any positive lnteg..r

n. .!!:..!. ~< gi1•cn bv tN'' Y

b. . l n~ ) sm ' "' T

il. - .1n :t for all even n lx I t~

IIy uo·11 _ _,.z rutd z • IAu·'r:; llu:n ~is

equnllo

ili t J.c (!- •>"

b. 1 e. 2

d. 1/2 'Otc derivative ol'tnn' 1(sec s + ~'Ul xt wiTh respect to xis

a. I

h. 2 c. tn d. (sec'1(1an x - SIX x))z

lf j'(:<.) = -~m. - 1-s x -s 0 3nd/ (x) = x 1'J , 0 < x s l . then

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:~ RoUe'$ theorem docs not apply to fin 1·1. I)

b. Rolle"s theorem applieo, to/ in (· I, 1]

ll. fis not eonlinuou;, ~~ x 0

d. f '(0) = 0

NQ The expansion of tan ~ in pow~r~ 11r1' by Maclaurin '5 theor<:rn is v~lid m tb" inlOJV41

ll. (· '1', <I'•)

h. ( -~n/2. 31112)

c. ( ·l!.JI)

<l . JtJ2. n/2)

81. The lir~l tln.:e lentui 10 ~te power •eric~

82

83.

for log ( I + rin x) 1U'e

a, x-l .. i.' .l_,~s 2 ~

b. • ...!.,> .!..,J "l 1

t 'I I • c.. • r-r. t-- .s l I

"' ·-' ,.J _l,..J :: 6

The minimllm v~lue of ~x • \)Is ' ~) ls .-- tx,.-- ~>

.. - 13 b. - 1m

c. - 119

d. - l 12

lllc m:1.'rimum :u\:n of o sllC.lot l>lto~c perimeter is 1 gi\'en l•y a.. /116

b. Ft l6 c f /4

cl None of the above

84. T he nomllli to ~.tc parobola ).l..~a at th.: pnint (am1

- 2llm) is

85

«. y ~ ms - 2llm - am1

b. ~· = 2 mx - 2llm - 2llm1

c yt mll' = am1- 2am , 11. ..: ·my =3orn· If the lioo y -=x touches (he pambolo y = ~ ~ a~ + b m the pQ,in! ( I, l) tlten :o ,b ~re rc;:spcclivc;Jy n, 1,. · 1

b. L 1

86.

87.

88.

89.

90.

c. - 1. ·I d. I, I

llte ratitl of the •ubtnngen1 lo the subnormnl for .:my poUtl on tlte c:unrt

~ = a (0 + 8in 1:1)

y = 3 fl· cos !!)

a. t.1n1fl/2

h. COt1 (:1.1'2

c. sin2 9 12

d. cos:on

If ~z ~ :!=~(1•'- .t:' 1. ihen the expl'eS!Iion X

i:lz. l tl.:: .r'-"t--equn~ ~ ,l' oi,)•

•. 1/z b. z c, l.'l

d. 2~

ifu - 1~). Utctl

OIL 1!11 3. x- - v- =0 ex . DJ·

tl11 i"JJ b. x-- )1-= \) f.!x ily

Cu 81. c. x-• y-~ _,, t1x l')y

011 iltt d. ~-- y- - 0

crr · IY If z= l'(x+ayj + <i>(l<- •Y) .lhen

fJ' : 3 . a;r

&I t;- J {rl:

b. (!v' - ·;;r at• il'· il'-

t. -·-~ o'--<"yl &'

c1=- f 1 .. d. - - - 2a1 - -

a"" ~'fl

rr ., · • .. ' I !AI .,x.y) = ~·sm--y" siJl- \\hen (1(. l , ,

y).-(1), 0); (x, 0)= ~ •in ~when x..- Q <

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1'(0, y J = y2 sin .!. when y = n • 1{0, 0) ~ 0, then ut (0, 0)

o. ·IX is continuous hut not f)'

b. I)> i~ continuous hut not fs

c. IX and JY;~re hotll coulinuous

d. neither fx. 3Jld ~v .iJ; eonlinuoll!! 91 . I he double point on the cucve (x- 2)'=

,)'()'- 1 )2 is

n. tl . 2)

1), (3, 4)

c. (~. 3)

d. (2. I )

92. T he value M "l ;ff'J;m; J.ot i~ i ~~x.z. stnx

93.

?4.

tl 1 n./2

b. n/4

c.. n•8

d. ~tl6

1'hc \'Olue ()( f~(e IS <CIIllit8nl. of ,. - I )

fnlegralion) u. log(e·• - 1) -< c

ll. los l!e'"- lJ)·~

.:.. log (e''..:') ' o (I log ce•-l)

·nce length ut' the arc of tho pornlwl:~ i =IJo; mea~un:d from the 'c:rtC); 10 nne <:Xtrcmity ol'thc latus rectum i~

a, ~lJr~ lo!O·t-filj

b. fiE· ... , .. Jl))

"' klJi' · lot.() • Jill

d. -i-15-h>£\1--Ji))

95. 11ce :oroo of the crudiod p a( H cosO I is equal to

iL 4 To:t1

b. Sli3

c. )lTD! -1-

d. 2n.;,:

96, Tha volume of the solid ganerallld by re\'Oiving d1e curve :<= a co~ I .y = b Sill t. ;chooJithe 15 ~~i!! ls

!17.

98.

100.

lO clJ Jll

•• 4 nob

b. ~"'"l )

c. ~nab

d. '* , The ore of the •ine curYe y= ,.[o " from x = () to x ~ 1t revolved •bout Uae /1:-a.'<is .The lltcu of tltc sucf3cc of Uca solid gcncrntcd i•

a 2n { Jl +log( .J1 + 1) J

b. 2lr' { Ji +log( .fi + I l} J

c. f { Jl• logl ~ i 1 H

d. '3J IJl- log(Jl- 1)1

Tbe sc:rie~~ whose n'' term i~ ~.~..r,;t;l ·n

a. conVc:rj!C$ to the sum 0 b. Cl)nve>'p to the •urn I fl. c. <'onverges lo the sum I

d. divergl:>!

The ~cries 1~ =.1 ~.._~ . 1" f ~.ll 6 1:>. 18

u. divergent

b. CODVOt'gcnll

c. oscillates fl<litely d. oscillott'i infinitely

JVf.,lch list ( \Vilh tisl U lfncl select tlcc Cllrrect answer: List l - ' A. n 1)"·'-'-' -

j (II 1)1

D. ... n.! !:t - t)<t-1

-1 _u l

u., n I . Divergenl z. Coovergen1 ~. Conwrgt:s conditiono.Jly ~. Coover@."'· absolutely

.\ a c '0 a. 2 4 l 3 b. 4 3 2 1 c. 4 1 2 3 d. 3 I 4 2

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