I C.S.£ Pre-l999
I ol Ill
1
MATHEMATICS T he number of arbitrary conslant in lhe complete pn1mtive of the diiTenltl\ial
"'b
¢.{:r, y.t~i'l Jx, tl' ~·~ = 0 dr
is
om
equ:\IIOn
d. 1" tun n9 = a The equati1111 whose s.olution famil) 1s self orthogomd 1s I , d1• a. p- -~-- p = -"p dr:
2
b \'PX + yl(x ,. yp)- J..p= 0. p :
c. 3
file dlt.Jerentinl equallon of Ute system of circles 1ouclung lhe ax1~ at origin is , , dv ;!. (1C-y' ) d~ 2Ay = 0
-
, ' dl' b ( >c-y·) -- +2~ = 0
dx , ., d\' o. (x- 1,.-) --- + 2xv =(I - tlx •
b 17y =-lx~
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
w
dy
p~
dt Which of the followtog smtemen(s associated with n lirst order non-linear diJrerential equation llx. y. d) /dx) : (I are
ra
J_ y=()
5,
7
m
.e
c. 27y = -lx' cl .nl = 4x r'he solutions \lf 1he diiTerential equa1ion 2y(y'+2l-~'Y'~ are the i'tmction
w w
4.
xa
:t 27y = 4x
(p-.. <(l'~ is d.~J
d. (px+y-)(X·)'p)-A.[F 0,
l
The smgular solution of the equauon p~4xyp+8p'=c)
dy =-, uX
correct'/
., ~ Jr d (x-+y' ) dx -1 2xy = () 3,
c, (px-y)('"YP)-A.p = o, p
.c
2,
~
ce
cl
dv ck
Its general solution must contain ooly one arbitrruy cooslrutt. 2 Its singular solutioo 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 d:v
particular
integral
dy
or
,
---"c - (1> +b) - + l!b,l' = Q(.t)LS
d~
u. e'~ Jk ...'" ~ (le"~t)}t~
enj ~~~~ •·,. ~~11! "'ttt}jL\: c. e·"" Jk""~ Q""'dx)}w d. e"' '"~ Qc'..dx)jnb.
n··
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Z ot JO
9.
a. lr ·lr -
.
IJ n. ax1by= c b• • ,.: - by = I) cJ aX: • b).! = 1
c. 10
b. c.
X
b
u
)'
- +- ~
l
3.
by~O
om
l
d. a. ,.
t7,
xa
a purabol~
c. u hyperbola ,1. au eIIipse
rr lho nnrnt31 dr.lwn at ~ pomt l "'11 2olt) nf !he parabola y ~~~~ m~et.~ it again in point
(•'i . 21tl) . then
Th" stllndord "'JUntion nf fhc ellipse .lhc:lt ngth whose nt~joo· a:<.i.~ is 8 3)1d thO> dislOoce b"twccm who~" dil'b:lrico~ I!; 16 I!; J,tiven hy
or
1>. ~ . ,• - , 16
.e
1:>.
p
•- ~. • r "'' ;:l
c. .
12
Ill.
po1M equation
represents '~~ a strnighf line
14.
t' l
ra
I
w w The
2h
.::.~
m
nx'-2hxy
""d y -
b. a.- (3
b
d. J,xl. l hX} - ayl=O lf u ax! -o-21LX)• bi 12g.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
w
13.
The dn:les ,=2 a cos (ll - a)
.:os (t)- 131 intersect at au angle
c. bxl- 2h!<) +-0=0 12
:C" + ,~• - h: + 2) - ·I = 0 .~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 II).
~
The joint equation of tht t>air of the lines through l11c Hrigln wbieb ~nl purpendicul~r to the line~ l'epresented lly n x 1-+ 2hxy+by'Z,o i• •· bx:, 2t~yray2 ~o b.
..
wei~
d. neither intc.rsed nor ·touch c;ach other
d. .!..w!. =2
II.
'I'&...,e lwo and
'
~, .! =2
"
IS.
h- J -
. a
.,
.c
-'
.L
11 I
d. l olr • I
d. ~X I t;j - 0 A vari:oble lines p3S$t.."S through the lixcd 110inJ '"' b). The locUJ! ••f Ute utiddl~ t•
''
.!L :
ce
I0.
t..... -
l
19.
11
,,.t
-·
- j
t"
d.
L.L - t
3.
X 1 X_l~X,!j
h.
~I X':X3 X,, - c
il.
:i l '
d.
1C1 x~ x1
lb ·~ lf n drclo cuts lhc rcctangulnr hyp.,·boln XY - ._~ in points Cx,.. y,) I r = 1, 2.3.4 ) .tl1en
TII.e powr
cirdc
- ·(!1
.
j
X:t == c·
of Ute center of Ute y - 6 cos (II- u) ""'
~oortlinate$
•. (0. 0)
h. (0. a.)
c. (6, a.) d. (3. t;t) www.examrace.com
1 ul IU
If~ slr.light line lllllk.;, 3ngle. 6fi 45" ron
d.
26.
to tj ll
3]- Si
h. 31)0 ~.
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
27,
?., (2r +j + 4kl •nd the t•laoe ; (-2 7+0 u. 1
b. ,fi
3-M"$q. unil<
2-M"sq. llnlts
c.
3
The: diameter af U1c clrde x~+J'+l :9. xtyTr-" 3 is.
fl.
2./G
(l'la """"Lion
10 the oonc wbich passa ~tree co-ordinotes •~<"" •~ well
xa
I
l
3
3
I
I
29.
.e
c. yz +7-x + J>.-y ~o 11 :lyz + I(izx + 15:
w w
I
"'
b, a(llx
h )"
f•
b(n.)'·mz}·. n·'
31!.
w
.
c. a(m,:: - lzf·bttty·m:>.)"'n: il. ;l(nx- ~)'·b(ny-mz)"'n~ A unit vector ·perpandiculnr to th" two vccto"' i J j -Inod 2 i •3j l ,i i•
•• si .. 3j -.t h.
i-< 51-3i·i ) s
c..
; .,. i - 2~
; = Hii • 3j I 5i)• 1.12i-3 j -5i)
b. ;=('2i- 3/ -3.i)+ i..(6i-3j-Sk j
nnd bavintt tl1e ellipse z:ll . a.x>1 by'-= 1 • as the guiding CIUYclS
. .' =u = ~)--b(ny+!J1Z)
111e vector eq~•ation of the line pa"-
Thd equation bf tho oylinJcr. generated b.)' llnos pa~'IIUd to the li'l'ed line .!.,.L= .!.
a. a(nx -
"' ;=kh + ii d. r k;;~ !
n
•• y-.t: - 2n + 3:o.-y 0 b. 3y2 - L'< ~ xy - 0
25.
,
r
through the ns the Unes !. ~ .l_ !. and !.. • L =:: .!.is
24.
The snlut i11n of the vecll!l' cquntinn t = ~ ~ i$ (where II' 1Hny rea l11umbe:r) a. xz-; b. ; = ~;;; ~ h
ra
Jf,
m
13,
CL
d. 7~
28,
•. 2.5 .... .5
7/ .f'
ce
22.
t
The tli~t1nQ
b. 35 .\i
""d.
are
a. fin iso~cele> lri
45"
d. 6o• 21.
The point$ A, B.C whoso> po~ition vectors are a 3; · 4/ - 4 i, 6 •2i-j-:t.~ ; .
om
Q.
~ {i ' j . 2 k J
.c
20
3 t.
c.
7 = (6i- 3 / ·5 hl /.~2i-3 j-5i )
d.
i
(·2 i + 3j
l
Sk) • i..(6i -3 j · Si )
If tllt'ee fnrcctcd UflOU by Q<>p!Jlnor forces b~ in www.examrace.com
.1of 10 d. (rug)t (mJ)
::.7.
,Ji!
•. nJ2
b.
"
d. 11111
~t-45 J·
*
'
m
\ ) C<
xa
~ 2v,e1
d. 2v1e"
d.
39.
l
b,
.J().
.e
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.
w w
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
w 36.
If" body of mnss m kg is carried l'Y n lift m
n, mf -mg
'1liC04il7
If
The l!iCap<: vcloeit)' (tf ~ pr'ljt:ctil~ !\'om lite earth i.i. •pproximuJcl)' a. 101 knwcc b 1 km $01;
e. 11.2 kmloL'C d.. 112 k.tnlsee
I he le:~st for
'35.
UtlUJJ
ra
A .sleam bo.-.l ts movmg whh veloc:uly VJ when gJe:mt is shulot't: ff the retatlhl ion :ol ml) subs<:
the par.tkle is PI'C\i.ecled from a horizonllll plane wbiclo volooity u nt "" angle <1, thea the time or fiignt of the JlBt1 icle will be a.
~. +, .l(lt
(l,
u·
ce
tl.
11/ 8
" · 2111
.3)1.
c. 3, - 2;-
If Otc equMlon ilf motion of a particle excc:uting tt slmplo h1umonic motion _U
-~·· • r~r ••, then it., frequency will be
7 ~, F3=2; -2J -lt :md F., then "• i~ egu:ol to
b
~mf
om
.
c. ms
.c
equilibrium n. that the sum oflhe momenu of all the fprce< :lbout {1, a t)()int i.s 7.ero b. any 11vo pootts is zoru c. any three non-oollineru: pointS IS zeoo ,1. ;In ttxi• i& 7,A!t"i) lf a p.miclu in equilibrium i• •ubjected to four fm·ccs viz F 1 2t - 5 J1 6t. F~ ' r3!•
'11oe h~adedmal number (A W. D)of Wh<:n coo~crtcd into the dcdmnl system is cqunllo
•. 2608.6125
b. 2007.8125
c. 2507.8125 .Jt.
d. 26'07.0125 ln ·• flow chart • reclllngh: fs a u, blart/ .!!lllp hnx t.. deciSion box c. t!omputatiou box
d. inpuU output bo.x
An J)gorilhn•
~~
a. • Uthle t'lflogarilhm• b. • t9llcotion of rcsulll c. • "hnrl offormulao
h. ma- mf www.examrace.com
s or Ill d~
such thnt "fc tn. • M !
R~>ru;o>t
CR): Fvr ony •
1
b, le•s 1h••ttl,3.13333 by - --, 3.10 c. g•'<.Utl.,. tlton 0.3333333 b' -
1 -.
om
:uo
1 • utan o------d. gre3'-er .~~..J~~"'.=t~ hy - .
-18.
3,10 Lot zt and z, be 1wo non•zcro complex nwnber'l, then iz 1 ·z~'-tz1 ] 1+ z~ 2, it'
a. z1·z! is purely imagiru1y
.c
44.
47.
c. x< z < _y d. x~ z- )' Th.e ti':tclion I '3 is • · equa l to 0.3333333
b. Z;-Z: is !'Col e. ~~-2~ ;, J1Urdy imugm:ory
ce
-13.
a
lirut of A c. A IS true but R •~ f~l
d, 1:1+z.1 is rca I
....
l, /• '.r.=lOf.~
1) ill J point in tltc Argond clingrnm repre"enting tl1e CQmple.." number 4 4 4( cQt " 1 1sin " ) and UP 1. rot.1ted 3 3 2Jf through on angle - - in Ute •nti3 olud:w~e direction , then 1' in th~ n~w Oll$illon represenfM [f
m
ra
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'
49.
45.
xa
the eorrcct cxplafintiou "fA .:. A is true bill R is false d, A is r~lse butR is true
·t ' ,.
dJ·
nf
dilfl:J1)Tltin l
c:quMinn
.e
' .i - 4 - + 4y = 0 'rhe genera l solut1on
w w
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
si11 n/2)
50.
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
diffcrenlial cquolion _ !hen au I bv i5 the
c. nolo eube
genern l ~olution of the equatinm where a
d. divi•ihle by I0 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
nnd b ore arbitrary conslnnL
w
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.
1
~' ~ <:, 4(~0• 7113- • ~in ;t/3)
All.lutinmi
• · -!(cos nil
tfx = ' ..fii. y = •.fij, z = "· :t .,.... Y z b. X " y~ 'L.
•.Ji7. tben
Sl,
3.
b. c.
LI - b
(;x - b)/(a l (.~ 11 )/(b) tJ - b (,t -a)/(h) - (.~- b)[(riJ
" r
d. Nune ofthe nbove www.examrace.com
tlul Ill
If x l ~ ll! - ~ lilctor . x:~ -a-x·~.:~x-ll, ··· Ihlh ,.,, ~ value ora i$ u. 1)
1\>;;, B
d. A.'"IB " = A = Ill
B
C
b. ]
L<:t X 1Je. ;t no n-cm))l)' finl\e •d .For a
c. 2 d. 3
sub~c.t
quadro1i~:
u.
x'-411~ J~
equation wilh rntion.1l coeflicients and with one rooa M l- ..fi i•
•'-
1(~ • .~.,
b. 2
t ~o
c, ~
II' <1.. fl •re ilte rOOts Qf the <>qulllion otx' l bll h! 0 . tbllD llu>valu~: of(u.- p) !.!
h.
59.
b 1 ~ 4ac
~
Wb.iclt ort]J.: loUowin£! proporti~~ bold for a function f: X • ~· and subsets U.V c X.M. N ~Y
ce
~-
d.
om
A
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. x~+4:<+ 1~1 c. x'-4, - 1=0 54.
A v B = Ifl ~
.c
53.
c.
1l - 4ac
I. [ 1(1\i.lv N) = [ 1(1\f)v ( 1tN)
ra
2. f (l l \' ~(( Ll) / (I)
·Ia•
b'
3. r'
2
;t.
0
b.
-1
c... I d. 2 56.
llto values ofi11•
ll!d
cquution
hlr
nrr . 12
~
SU\- T I .:08
l2
II
611.
C05 -/ llf "!- I• SU\
12
J.
w
6
it C
6
57
b. I and 2
For an\' a. b.;;N. Ute~et of namcol ·numben;
" I'.l. ~~
Symmelric ond
transitive
bul
no
reOesive c. Rel1e..xtve b11l nol tr.msotive and $ytrunetric:. d. Refle.xwe ond trnn$i11vc: hut nnt
17.
21
• -+ mr .' co~ c. sut
a. I and 4
b.
I. s. 9. 13. !7.
mr II - 1 -. -. • 12
giv~tl be-l ow:
a. An equivale:nee relation
21
b,
'(Nl
-1. f(U V)~f(U}I"o/(V) .Sele
c. land 3 d. 1,2 •ud3
Arc
w w
'~
of
xa
A r~cated cool xJ· ~x~-+3-.""<
.e
55.
~rTC
b'
d.
m
2
symmetric
111r
-,
6
,, .," n = 0 , 1• 2.". ,, ....
!!!:!.. n ~ 6
61.
The
l, 2, 3.4.S, 6
.For •ny llll'e~ Set> A.B aod C wblch one of the followinl' stot=ents i.> U(!l'n:(;t? ~.. .'\!'Ill - ~ A = :::; A~ B www.examrace.com
n
In Ia
In
Ia
II
m
In Iq
Ia It>
ld In
m
n
11
m
IP
Ia
-
.OJ
II
Ill
IU
Ul
Ul
Ill
n
m m
n
m
n
In lq
m
ID Ia
Then wh1ch one
or
Ill
m
~ubset I)( S
In Ia
66.
(ll.~
b.
a. (R, ~ ) i< • group , Inn (R, -t,.J ,. not• l'ing
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>
m
ra
d. (R • ..) is a non-conuliUiliUV
• dy. l•x + cy)
ce
cortc(':t•/
•· r~ :a b. (:
~)
a. (Z, ...) is a rin!l b. (l, +, X) i.< not•
c.
(~
,'j
tl.
c :J
xa
lllltll.tplicalinn 'X' ofrcalnumhen; b<:caus~ ll~ld
(R. x) is not • group d. O rdinary multiplianlioo of roa1 number,; does not dll'line a sca1ar multipHcotion of/' liy R. Whicb of the followin!l set' ol' vecto1~ In Rl arc linearly ind"Jlc-ndenl I. lC1. 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
w
w w
.e
g.
Select the correct answer nsmg Lhe codes
given below: :t. I and 2 b. 2 ond3 c. 3-ond ~-
65.
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)
the folio\\ fng l•
h. (R, +, .1 i• ~ ring with unity (R. ~ •. ti ~ comm~llltl\·c ring
64.
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 $<:! S .then V ean l>o !!OOOt'.t
1!..
6"3,
I
om
"'n
.c
m
7 oi JO a. 11tere i.• no vector
d. I and4 Which one of lhe following statemenl' is
corre.:t'l
63.
If A =~ ~J fhcn ,,J, A-u whl;J11:H:r a.
ctiJ ~ II
b.
~~~ = I
u
c. o.lh
d. aJJ = • I (19.
lf
01e
~ "' ··~ ": h,
,,
rl,
V~ltUt:
OJ
hnYe a,,
6,
&,
<.t
c,,
the
of
detenn irmnl
i(i "<(Ual to K, then the
,,,
d,
of ~;\a,
..
q ~ ~ IJ-
'I,:!,, ,, "· +.l
the
.., b,
",,
delc:nrrinsnl.
is -cquol lo
~- 3K
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8 o l Iii
h, - K
74.
"· K d. - 3K If A
'
H~~ -;]·
U1cn
iowrs~ of m~lrf:<
.. [-~ l ~l ~ ~~ - 1
d.
.r' -
•
- ;,)
0
0
(I
by
(0 < -S'• 0/
.c
[~ ~ ~)
I
a. ;, not cominunu. ((), ' )
b. is nol differentiable oo(O. m}
u-! -~J
c. iK dillerentinhle 1111 (0, "')
d. i• differcmti•blo on (b."') ""copl nl x •
Co1uidcr the equation A.'\~ B. where A=
(-~ -~) a~ (n. llt~u lhe equ•tlon h•~ no solution
h.
[~l is- a solution nf theequorion
m
~.
•
76.
lfy = sin x . then for any positive lnteg..r n. .!!:..!. ~< gi1•cn bv tN'' Y
ra
71.
75.
will be a, - 2. -1 b. 2, I c. - 2. I d. 2. ·l llt~ funetinn
ce
c.
b . where b is fLniie
.then U1
A will b.,
h. [
Lf ht11 Aln.!r osmx.. .... !. xl
om
70.
1
. l' "' Tn~ )
b.
sm
xa
c. l.hor<: e.s.ist< u non-~ero lllliquo ~olut.ioo d. lbe equation solutions
72.
bas infinil.,ly
many
il.
Ote liotit of a convergent sequolnce or
lx
77~
.e
ralioonl numbun
- .1n :t for all even n
uo· 1 1_ _,.z
IIy
u. need n()l exist ot all
equnllo
b. exisiJ; and is otway• ratiOnal
ilit
w w
c.. exisb; and i~ n.hvays ([mtinnnl
d. crins bul it ulir)' irrational
J.c
(!- •>"
b. 1 e. 2
d. 1/2 78.
'Otc derivative ol'tnn' 1(sec s + ~'Ul xt wiTh
respect to xis
w
7.3.
(:., rational or
I t~
rutd z • IAu·'r:; llu:n ~is
" · bas U1c v•loc 1/2
I
a.
b. hM U1o value> I
h. 2
c. has the value 2 d. does not
c.
..,.; .1
tn
d. (sec' 1(1an x - SIX x))z
7-J.
lfj'(:<.) = -~m. - 1-s x -s 0 3nd/ (x) = x 1'J , 0 < x s l. then www.examrace.com
RoUe'$ theorem docs not apply to fin
c. - 1. ·I
1·1. I)
d.
b. Rolle"s theorem applieo, to/ in (· I, 1]
86.
fis not eonlinuou;, ~~ x 0 d. f '(0) = 0 The expansion of tan ~ in pow~r~ 11r1' by Maclaurin '5 theor<:rn is v~lid m tb" ll.
NQ
a (0 + 8in 1:1) y = 3 fl· cos !!)
h.
c. sin2 9 12
87.
.c
~
ce b. z
...!.,> .!..,J "l
1
c..
t 'I t--I .s • • r-r. l I
"'
·' ,.J_l,..J :: 6
88.
The minimllm v~lue of ~x • \)Is ' ~) ls
m
.-- tx,.-- ~>
xa
b. - 1m
c. - 119 d. - l 12
w w
89.
cl None of the above T he nomllli to ~.t c parobola ).l..~a at th.: pnint (am1 - 2llm ) is
w
b. L 1
2~
ifu
- 1~). Utctl
3.
x - - v- =0
b.
tl11 i"JJ x-)1-= f.!x
ily
\)
c.
Cu x-• t1x
y- ~
_,,
d.
~--
OIL
1
1!11
ex . DJ·
011
81. l')y
iltt
y- - 0
crr · IY
If z= l'(x+ayj + (l<- •Y) .lhen fJ' : a;r &I t;-
J {rl:
·;;r at•
il'· il't. -·- ~ o'--
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 n, 1,. · 1
d.
b. (!v' -
b. ~· = 2 mx - 2llm - 2llm
rc;:spcclivc;Jy
l.'l
3.
«. y ~ ms - 2llm - am 1
85
c,
l>lto~c
.e
lllc m:1.'rimum :u\:n of o sllC.lot perimeter is 1gi\'en l•y a.. /116 b. Ft l6 c f /4
84.
l tl.:: ,l ' oi,)•
.r' -"t --equn~
•. 1/z
~
.. - 1 3
83.
X
ra
b. •
If ~z ~ :! = ~(1•'- .t:' 1. ihen the expl'eS!Iion i:lz.
x-l .i.' .l_,~s 2
82
d. cos:on
T he lir~l tln.:e lentui 10 ~te power •eric~ for log ( I + rin x) 1U'e a,
1 t.1n fl/2 1 COt (:1.1'2
a.
(· '1',
h. ( -~n/2. 31112) c. ( ·l!.JI)
llte ratitl of the • ubtnngen1 lo the subnormnl for .:my poUtl on tlte c:unrt ~=
inlOJV41 ll.
I, I
om
:~
<"yl
&'
c1=f 1 .. d. - - - 2a 1 - -
a""
90.
~'fl
rr ., · • .. siJl' I \\hen (1(. !AI .,x.y) = ~·sm--y" l ,, y).-(1), 0);
(x, 0)= ~ •in ~ when x..- Q <
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lO clJ Jll
1'(0, yJ = y sin .!. when y = n
••
1{0, 0) ~ 0, then ut (0, 0) o. ·IX is continuous hut not f)'
b.
•
b. I)> i~ continuous hut not fs c. IX and JY;~re hotll coulinuous !17.
c.
tl 1
The ore of the •ine curYe y= ,.[o " from x = () to x ~ 1t revolved •bout Uae /1:-a.'
c.
"l ;ff'J;m; J.ot i~ stnx i
d.
~~x.z.
n./2
98.
2n { Jl +log( .J1 + 1) J
f {Jl • logl ~
i
1H
' J IJl- log(Jl- 1)1 3
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 Ifl.
c.. n•8
ra
d. ~tl6
c. <'onverges lo the sum I
1'hc \'Olue ()( f~(e IS
u. log(e·• - 1)-< c ll. los l!e'"- lJ)·~
'o
xa
.:.. log (e''..:')
(I log ce•-l) ·nce length ut' the arc of tho pornlwl:~ =IJo; mea~un:d from the 'c:rtC); 10 nne <:Xtrcmity ol'thc latus rectum i~
.e
i
~lJr~ lo!O·t-filj
b.
fiE·...,.. Jl))
w w
a,
d. divergl:>!
m
,. - I )
fnlegralion)
?4.
'* ,
J
b. n/4
93.
d.
ce
T he value M
~nab
b. 2lr' { Ji +log( .fi+ Il}
(~. 3)
d. (2. I ) 92.
)
c.
a
1), (3, 4)
~"'"l
.c
91 .
d. neither fx. 3Jld ~v .iJ; eonlinuoll!! I he double point on the cucve (x- 2)'= ,)'()'- 1 )2 is n. tl. 2)
4 nob
om
2
100.
The ~cries 1~ =.1~.._~ f
~.ll
6 1:>. 18
. 1"
u. divergent b. CODVOt'gcnll c. oscillates fl
d. oscillott'i infinitely JVf.,lch list ( \Vilh tisl U lfncl select tlcc Cllrrect answer:
List l
-
A. n j
'1)"·'-'-' (II 1)1
"' klJi' · lot.() • Jill d.
11ce :oroo of the crudiod p a( H cosO I is equal to
w
95.
-i-15-h>£\1--Ji)) 1
iL
4 To:t
b.
Sli3 )lTD!
c. -1d. 2n.;,: 96,
Tha volume of the solid ganerallld by re\'Oiving d1e curve :<= a co~ I .y = b Sill t.
;chooJithe 15 ~~i!! ls
D.
...
n.!
1
_u l
!:t - t)
u ., n I.
z.
Divergenl Coovergen1
Conwrgt:s conditiono.Jly Coover@."'· absolutely .\ c '0 a a. 2 4 l 3 b. 4 3 2 1 c. 4 1 2 3 d. 3 I 4 2 ~.
~.
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