IAS. (MAIN) MATHEMATICS - PAPER- 2008
MATHEMATICS
Cuni^idcr Ibc fo]lairing sUlcmiml^:
I* For M> nqiura] nmnbi^r n (i c-. n t N}, Id In
= {re R:^nirsn),
Tlxn Ul„ =R-
2, For snv posiltvr re^l number r. Id Ir = {x^Ri ^ r < s < r). Then cotilams
e^41cll^ OK elemem Which of the smcmenLs givci] above isVore
iL 1 cmly
2 only c. Buih I a]Kl 2
Nctlbcr I iicr 2 Ans. li
If \" + ^ I =0, then ^^hal if the ^aluc
Four An^H h
Consider llic poKiiomial + 2x + il over integers Ifn. |l Y arc fools of = 0,
then uhai iiihc talucofir + + iL 9 b- -I c. -6
ir ihc set nf rfl1cgcr^"^^Hi ihc opcTa[inn dcHncd bv 311 * n = 1^-*- n^l-forms 9 group, ftliat is the
O
is * V-
Consider (lie rollc^^iiig Totir iii^iiibcrs inhere is
n cube roo( of unil^
I. FViiicipat bninch of 1'
3, w + if I 4.
Hoi\ man> of the flboie arc real numbers? a. One
r. Ilirce
Fn (Z. +t. nZ dcndcs the subgroup of all inlegral multiples or n. If pZ = jZ nkZ^ ihen v^h^L is p equal LD? *L jk b. j-^k
LCMoFjandk GCDofjnndk Ans. I!
In s tjroupiG. if (a + b) ' = a ^ * b ' fpr all a, b ^ G. ihcn^^halls G? JL An Abclian
b. F]nitc
c. Cyclic
it. None oflllc above Ans, a
Consider ibe follaiving smiemenls in respecl ofn li]iitc p"oiLp G
1. 0(3) =0(p 'jfor aUa e G
2. 0
\Mueh of the ^atcmenlE^ gi^crn ^bo^c i^are corrcci'^
a. I onh
b. 2 onl^
c* Bolblpnd2
d. Neither I nor 2
l.eL H l>e a yifliip ol" order ?3 Wbal is the tiumber of subgroups of H?
n.
13.
14.
15.
2
d. I
Am. i:
WTitch Dnc of ihe ]bllow']nB la cf»rrecl7
if ct p t Sn, then a""p""ap is m c\xn pcfiThUEalion
iL lor all tx- p E
onty ifCL P^odd
t cnty d"a- p are even
d. onlv ifonc of a. p Oildand dlher iievoi
L*t M dcnol*; the ael cif uU 2 ■■ 2 mfllnces over the reais Addition and mulnplrattf^ on M Qii ^cllo^;^u.
A-viitjJ and B - (h^X Ibcfl A + B - (CL|), ^WC CJ|
= ttq - and A4J = (d,j}. where =
ThcD which cne at Ebt foliowmg VAIUI for (M
ji. KT n n field
bn M IS Em ititegr^l dimmin which is nota fkld M IS B cotntnuLdlave nng wbch ia n^fPl nilgai di:4UaLi]
d. M IS Fi non-commulalrvc nng ^^-^
<:omhEdef Z\\l the sei of a^J^i^^Mi^l^ with
integer cciefficienls and (J| ^ ptiie set L>f all
teal numbers of Ihc f^^rm - hVT ivilh a^ b raEjojiul numb^ni ^MULM V'E~ [her folluWii^ is
correciabouTZlxl .tfii:
A- &ithjtfcj-iD^"s ^iJU>flK onehas^□itv
b. B^th ar^j;L:iii:in^ut.LU\c nogs, buC o£il) one an inU^^d^^m
c. Bolh arc lntc^rdl domains, but only one \A a fidifa
iider the field R of real numbers and the followu^ spAcra-
1. Set or rcal-vaEucd Kjncliotta on [0. 1] havmg (ii^ontiniiity atx= P"2
Whidiof ihe above i^:jrevectJ>r spacefs)^
0. I onlv
b. 2 only
c Both 1 and 2
d. N'erther I rior! Aiu. d
("onsid^ the reai i^wlor space V = and fDllowir^ of JLs subsets
Which one of the following statoimls is comsl?
d. T md W Are wbspaera Octfy S, W are subspMc* t t>nly T, W are subapaces d Only S- T fire subsprtces Anfi. d
Lcl V be s vector sp^PiJ**^'^ '■^^^ ^ ^ dtTTKaiaLonn Ct^naidertfl^Mawtfig sHtemmts
L. Kvery yuhtel coaucung n demenu ua
basisof^. i/W
2- No ImterF^inaiKndcntsuh^d ofVeoolains
more ih^n n elmient^
Whicli.^^Tb^_^A^e i^Entm^ts ]Sv"are correct'^
^Eoih I and 2
>Ji.']lhcr 1 nor 2 Am. h
If T -rR' is i linear iraibiformatiaii Kuch Lhil T[l.'» = a 3. Dana TCI. I)-(3,0.2). ihoi «ihi(di Oncof thelnllnwuig slnCcmcnti in cnrra;!''
b. T(:ty) = (2>L-Fy.3x-3y.ji + y)V(jty>e t y) = i2i^- y. 3x * 3v, jt -y) V K y) &
Ain.b
Lei T --> R'' be the Imear tmnrifomuilinr liiven by
y.i) = (:t. y)V(K.y,i)eR^
With rcspctc in siandird h.i^is of iind ihe bftsic 0). (L, IJ) of R^ ^^hal la the matns reprfflffitaiinn nf T"
I 0"
d.
-I 0
0 0
1
0 0
I I
1
0
]
0
-1
0 0
0 0
Am. c
19. Let T. ^ cfcflned by Tfs, ?> = fx,
|>1 and S ^ be Jefmtd by 5(Jt. y) ^ £23L iy), be linc^ iraitsftttinslioos ini Ihc rcnl vccl« spaces and re^peclively Itien which wie irf ihc Ibllo^ ij^ IS correct'^ A. T and S ur^ bolh smgular b. T and S ore both non-singular C- T u s]ngLikr arid !^ i^t^on-aingLjlAr (L S i£ singular and T is nnn-Aingulnr
20- Consider ibe linear transfomalifln TR"' -given by
Tl[Jt y, i u) - (K. y, 0.0) CJ^ u) E Thcn^'hich one aflhe following ia direct? fl, knnkcLfT-^Kui]il>'ofT b. NLilh[vofT>RaokofT t RflnkDrr = NuElityofT = J d. RBnLorr = NuLli^orT=2 AiH-d
24.
If A and B are tivo odd order skew-synunclric in
1 Q
lis kh out of
21» If the valuer ^ the d^lfflnmstil
I b
positive.
a.
e. d.
abc^-a abc >-3 abc<:-S abc - -R
Ihcrn which one of tbc foUox^^^
36.
0
Alu. h
0 -I
If A =
0
dt 0
Am, h
\i A m J B arc I hen wlml c;(A+
of «iiK i>Fdcr,
B)
AaA + B)
"B-^^a'A A'B f B^A 4 AB* + BA^ H ABA
17.
then when isA"-^ A = 0? a. lfap = U
b- If op--J
c. If op =-1
d. If 0^1=-1 Am. a
A sliaighL line diD^vn
\ y
cf
0
-3 2
S -5
equal 10 which one of fliefoilowiiiB'' I I
then rnriii'j^ A la
L
I 0
I I
0
I
Ain. ii
d.
IS.
Lnlcrscciifm of ibc aitaiuhl hnci —-^—=1 and
—-H^-^l mfcls iBc c4ioii!]DjiicT a>[« La A anilB. b A
Whai in ihe lociu of ihe midpaim ctf AB? a. 2(ji + b):ty = ab(x*y) b- d(ft + yj = b(j£-yj c. abOi-y) =x-*-y
Am. n
Whnl vt ths HiiM nf ihe rhombus eclovd by thff llDcu ax n: by n c = U?
a.
2SI.
30.
31,
If (4^ 4) iA dfic md Df n focal chord of the pambolra = ihcn whal ia The ather end?
(-::, I)
b. t-l. 1/4) t f-l--l'4) d. C-4.4) An\- b
poini nn eliipue ^--^ = 1 Mid iLs cnreipimtiing poini on ihe aii>dhar>' cucLi?
b.
u. —
b. -
3&
7
A™, b
Mjtich Ltift-I with Lisi-]l and ^Feci ih? Diiawer ming iht code gjvm bflow ihc
lUtA (Stnighl Linr)
A. 11+1=0
C. x-y-3 = 0
D, 4x-3v-4 = 0 LJbl-II >jriiri: Pn-uibung lu du- pjrjibulj
I. 'lung^l 1 Focal
3, No
4. D Code
A-2. [VI, r-^, DA
c. A-4, [J-l. CJ,
d. A-2.B-3,C-].D-4
Am. *:
Whal ]» Iha urea of Ihc ciroJc + ^ -fct
2v -2z-5^ = [' = 2>i-^y + 6z-627 u- 15 71 sq uiul b. hjn^ unii
d f>4 n ^. unit
4of n
Whal IS Bie shnctKl distance I'rnm ihc poinl (1-2, -L) lo the aiufdcf ot ihc ?^hac T
c. ^ cL 2 Ani. r
Whal K the equaiion nf the ih?S:>'|inder whoi ihc cqualiiiD of the gdidlnc ^clc of ihc riBlit girculnr cylinda ajji^Tiyt - 3 ■■ 0 R -
l-"ktn& Llimn^ X and Y axes inciiide a cunslanL ^knglL! IS Ihe locus Ihcit line of
^ffpfr^Mrtn' lane
t COM A Cylmder Ani.
