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I PART FM B.Sc.

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I PART FM B.Sc.
B.Sc. PART
I
Annual Examination 2010
Subject; Math€malics
Paper: BMG : 101
Abstract Algebra
Ma. Meks:50
NoteAtteDpt rtry FM quesiioDs. ADser .ry two prrt of
each of lhe fiBt live q!.stioDs rld ttrree pots of
qustion six. All questiors crrry equil m3rk.
Q] (a) Define
R
=
{(\
d
equivalence relarion
od
show
liat
y) lx =y modulo m, x € I, y e I} is
o
the
relation
equivalen e
(b)Define equivalence class and show thar
[email protected] relation in a set A rhen [a] = Ibl
(c)
ed
State dilision aleorithn
i)
d
ira
R is an
Rb
whqe
find the remainder shen
(i) -7 is divided by 5 (ii) 5'r is divided
(d) Give
il
example ofrclation which is
b,
12.
:
reflexive and s)ametic bur not lresitive.
ii) symetric and [email protected]
but not reflexive.
iii) reflexive ad trositive bul not
symelriq
Q2(a)showrhatrhcsetG={......-lm,-2m,,m.0,m,2m,,3 ."
. ... ) ot mulrrples ol.rresss b. a t-red inle8er n i5 a
Croup with respect to addition.
i)
Identit
element in a soup is unique.
ii) (ab)r=lrrar.Vab
(c)
o
Prove thar ofthe lA pemurations on n symboh
even
(d)
€
pemutatioB mdE2
Define order
ofd
elcment
l!2
are
arc odd.
i.
a
sroup dnd show rhat ifrhe
elemont a ofa group G h oforde. n thm a" = c
iffn
is a
Q3 (a) Srale and prcve cayteys theorm.
/b\ Defi nc
q.'i.
Aroun and
-'o\c it ar c\e^ gloup o. p. ne
(c) IfH, &d
4
arc two
is also a subgroup
subercuF ofa srcup G, then H,
ofc.
^
4
(d) state and prove Lmgmge's theoren.
Q4 (a) Define nomat subsoup
H is
a
ad
shos'that
ifc
is
a
group
md
subgroup ofindex 2 in C, H is a nomal subgoup
of
G,
(b) ?rove that
rc,
cdre ofa erotrp
Provelhar ahomomorphi\T
with Kemol K is
d
is a
nomal sub group.
lola goupC inloagouPG
isonoQhism
ofc
itrto
G'iffK = {e}-
(O If f n a hooomorphism of grouP G inio a group G', then
show that (i) (e) = e', whde e, e' are the idenlilv ofc md
c'espectively (ii) (a.r)
-l(a)llV
a
€ G.
O5(a) Prove that every field is an iftearal domain bui convc6e
(t) If
R is a ring such that a: = a V a € R, prove
(i) a+
a- 0 v a € R(ii)R
is
mmutarive.
(c) Prcve thal inle6ection oftwo subrinss in
a
rdl Show rhdr rhe .et ollumors oithr 'om
and b as .alional numbe6 h a Reld.
Q6 (a) De6ne
lhat
Intgral don.in, Idcal,
Coset and
subrins.
a bOwit,
Nomdidr.
O) snow that the serR = {0,2,4,6. 8) is a rine wirh uniry
with rcpect 10 addnio. and multiplicadon modulo I O.
(c) Defire codjusate elements and show thar the relation
conjugacy is
e
of
equival€nce Elation on C.
(d) showthatinarinsRa(-b)=
(ab) = (-a)b.
(e) Define cosl decomposilion ed index.
(D ll
:
H md
subsoup
K
e srbsroups
ofc iffHK
=
ota sroup C,
KH.
Lhen
qK
is d
*,zrv,.*........B.Sc. PART
I
Annual Examination 2010
Subiect : Mathematics
Paper: BMG I 102 (II)
Calcnlus
Md.Mr*s:50
Time:3H6.
Notei Atlempt
,!y trI\.E
Min
Pass % 40
q{estious. Eicb qrestiotr
Ql. tuswdeytwoparts:
l{ rr\ rd, r rLll -- lrn'ho$
trE.y"=
ir_lf
"rry
'{n l)l
Sin"o sin(no) whele e=collx and v, denot's lhe n
dtrivadve ofY w...t.x.
(b)
Ify
= e"inr them prove thal (1-x1)y",? - (2"+1)xy,,, -
(n, + a)
yi:0
(c) Expmd los sin (x
(d) Exped
Q2.
a"
ed
ofh
+ h) in power
e* by
tvlaclaurin
s
by Taylor's [email protected]
Lheor
Answe. sny two quesliom :
(a) Find the extrm. values olthe tunction
(b) Ewruale
:,Ero(T)r.
rc)llu=sn II i
, rhen pro\ e rhd
u =
xi6 x
y).
au
\ ;;\' y;y
Si.2u(d)
Ifx-
c Cos
4-9=
Q3.
uCoshv,y=c
!e
Sin u Sin hV then P.ove that
rcos uu - cosh
zvr
A$Banytwopans:
(a) Show that tho ladius of
a
cwatue
at a Point (a Cosro.
sinre) o, lie cuNe x:,r + ya = a,/r is 3a singcos0.
O) Find the asymptots ofthe cune
xr+x:y xy,-yr_ 3x-y_ I :0
(c) Find rhe cnvelope offanily
x: Sin d +
y'7
Cos o
(d) Trace tho cufle r =
-
a')
4l
:
ofcN*:
where d is pdamelq.
+ cos e)
04-
Arsw{ dy
two parts :
I su"eae =
Sin" 'ocos€ +
(b) Find the limit who
$lsi,-eae
-+
tr
NLL
"-
{o Eurmte
los
[-.F
rar Pmve ttrat
Q5.
Aswer
I
*:-.+ _--_
+
,fr"5,
l+rl (r.
-
l'=jjl
=r
any two parts
:
4a
d
(c) Fidd ihe surface of solid generated by the evolution
ol
(a) Find rhe
ma
included betwccn the cunes y: =
(b) Pind the length orrhe asEoid :
the leminiscat
f
=ar Cos(e aboul initia! line.
(d) ftove tha! Ini ln+ I/2
Q6
A.sse.
G) F,n,l
rhp
- i{T
I
2m where m
any thrcc patls :
n'h.liffer.nri,l.6efn.ienr 6f
(b) Ifu = (r) wherc
-
f - x'zr'yr then show that
a'u . ?q .,,. ..
|
",.
,
-
0.
(c)Evaruat€{rl}ro*+
(d) Find rhe6)anptoGs ofthe ouwe y1(x1 ar) = xr Gl - 4a,)
which ep allel toeirhei otrhedes.
re
)
frove that
(0 lind
o
ld+t- nr
-Ei;-i
=
B
m
-
the volume of solid generated by the r€volution
arc ofthe catenary y = c Cosh (x/c) about x axis.
of
B.Sc. PART
I
Annual Examination 2010
Subject I Mathematics
Paper : BHM : 103
3-D Geometry and Trigonometry
Mu. Marks:50
Time : I t]Is% 40Min. Pds
tso P!rts ot
Nol.:Atlemol anv five quesrioN selectitrgtry
''""
aDd alv rhre p'n! otQ6
o,l*o"i
'oQs
"""'r'
if yon select it to mtseL
r-'bt
foDr
a), C (3, -a' 7)' D (0' 2' 5) d'
Ql(a) A (6. 3, 2), B (5, l,
AB on the liDe
points. Iind the projections ofthe segment
CDandftesegmentcDonthelineAB
!
s
(bi
ilnd lhe area of
venices are
(c)
the
tdtuslq
C2.4,l). C4. l,
rhe cordinares of whose
-2), (3.2,
t).
5
Show that the distances belween rhe parallet plees
2x-2y
z+3 =0
+
(d) A plane meeh
ild
4x- 4y +
22 + 5 = O is
l/6.
5
rhe co-o.dinates axes at rhe poinls
A, B,
c
such thar the cenrroid ofthe triangte is the point (a,
b. c),
uEn show thal the equation offtic plhe is
{vz^
:
5
;._E-_a=r
Q2(a) Show thar the li.es
II
,rr.4'=0
-
iiy
2x +
ordinatcs
:
6).
-l ,/.ljandl\-2y_/-<
ly + 42 - 4 and
ofrheirpoifl
@pla.qr. Find also rhe coof inters€crio. and rhe equalion
ofrhe plane in which tney
rie.
(b) Findue line which itrteBccls
rhe
5
lires,
5
x+y+z=1,2x-y-z=2.
\-y-z=3,2x+4y-z-4
andpassesihroughthepoinr(1, l, l).findahorhepoinrs
(c)
Prove that rhe equarions lo rhe Iine ahroueh (c,
P, y) at
: -,L Z.-\ -LZ
xo
zr
ti -0
frlrL m,n, - nr: n.l, =Gnri
,.aht
aatci.o
rnr
line.
(d)
Find the equ.rions ofline ttrough
to the
and
pasllel
4
$e cirle x,
Show lhar the pltue
xr+f+zl2x
(c)
is
5
the [email protected] to the sphere
(o,0,7)
(b)
t.2,3) which
line:
r:]=t!21
Q3(a) Find
(
+
2x,2y
I
+
z
phich pNes rhiough poid
= a:, z
+
12
-
0.
= 0 touches the sphere
4y+22=3 andfindthepointof
Show that the two
5
contacr.
5
chcts:
5
-t + 2z=0. x_y+z-2=ol
x1+ r2 + 21+r 3y+z-5-0.2x-y+42-t=0.
yt+2,
x'/ +
lie on the sme sphere and find the eqution.
(d) Find the equalion of sphere which has ils cenre
at
rhe origin ad which louches lhe Iine 2(x +
1) = 2 . y _
z+3.
Q4(a) Show
5
rhat ihe eene.at equarion ofcone which touches
the
cNrdimtes plms
three
is :
i6.1Jsv+lhz=0.
wherc
i gad
h
beins
(b) Find U€ equation or
3tr+
2yr =6,
y+ z=
parmee6.
the .one with v€nex (5. 4,
0 as basc.
5
rc) rro,crhdr\r-J.
.i^Jhr
neh,
[email protected]
): 4^t 1, b/.l2 -nr,prcnnrisms r_e^
D rne porr,
r.r ,. -,),
{hose"x., i( pM,tel,.Oyao
_ho:e,onr \eni.a,dgre
n 1rn
(d) P.ove
rhat lhe cones axz
r by,+cz.=0an{l!+
= 0 are recjprocal.
5
o5,,, prcLe$d..osa.
r.[, $.$rem. u.hg t.rgonome.ri,ar
i;;,iii,
(b) Ifttu
(0 + iO)
=x+
....Le.,, r",,"
,o,
*.i,
iy, prove rhar.
5
.i'r'{*ffi=t*a
0
yn
(c) Suo
-
cos-o
|
5
si.o+nsi!(d+p)+1?sii(e+rp)+....,o
/n
rhe
-l,
(d) rird
fotlowing series
rerms. n beinea ponriue
in.eetr.
Lte
sstralvaiue otlog Gi
+
i).
5
Q6 (a) Find &e direcrio.osires ofa
straiglr line which is equany
inclined wiLlr ihe rrwe
co-ordinare
a\s.
(b) Find the equarions
ofthe line
poinr G. -1, I t) ard
*hich pa$es through tbe
isperyendicultuiotheiine
i
L=tl=!!
234
* I I z:-r makes
. .,,
an<: ^r i Z
a, b, c on the co_ordlnttes
rcr,lfielange
rntercepts
sphne
Plane to lhe
:
respectively' $en Prov€ that
rllt
?* b'-7-7
ofnultiples
(d) Ex"and Sin'o in a se'i's ofsines
relshoq
'
thal the eqJatron
onei4 and bak
i.
-
olfe
dde T' K
ofe'
on€ *ho5e vmexis rhe
--xK vK.r=0'
t{ )r-oisf(e ?
+ %)
+
cos {(2m + %) aa) i sin {(2m
Ia)
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