2022
Mathematics
Time: 3 hours Maximum Marks: 200
IIqq: 3 Eit srEonq oio: zoo
Question paper specific instructions
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i. This paper is divided into two Sections, Section-A and Section-B.
i crtqe d {dd n ffl{rfud t, t s-A sitt Eis-B
ii. Each Section contains four (4) questions.
c-&6 {rs t qn crq tt
iii. Candidate has to attempt five questions in all.
cfieilpf 6.r ciq csit 6I rfl fuqcr tt
iv. Question Nos. 1 and 5 are compulsory and out ofthe remaining, TIIREE questiors are to be
attempted choosing at least OIYE question from each Secti'on.
crr €€qr r silr s 3rM t 3ik nrs crif n t Rtd d-r iD'I stn fusnr t rS6 ds t go c{r 6} Ef,
orcr t r
v. Word limit in questions, where specified, should be adhered to.
crS i yr< frqr, sd f4frfre i, in sgstq fuqr qnr qGc t
rri. The number ofmarks carried by a question/part is indicated against it.
rd6 crHlqr.r d ffrc trTd oi6 ssd qrqi ftc .rc t t
vii. Questions are printed in English & Hindi languages. In case of any ambiguity in
trarslation of any question, English version shall be treated as final.
c-d6 q:rc f.q 3ilr 3iffi +it.TNI3il i oqr tt c{.if d $gql< d ftffi erssar
d fufr n.3iffi ss-rq d A ortdq qnr qrgrn I
1
SECTIONA(qsE-A)
Q 1. Write answers of the following questions.
ffiR+aqeilbufifrfus
l(a) State differential equation oforbit in a central force two body problem. Find the (8)
forcelaw for the orbit givenby r: e0 .
\1o. &-fi
q ild d trfi rq' q{qr i aeff 6-t er+o-e qfro-rq sdTt( I . : e gm d rr€ offt ]-
"
6qwftqq{rdaltrst
l(b) Is the set [(1,1,1 ,7) ,(1,2,7,2) ] linearly independent in lR{? Justifu' Extend the given (8)
set into a basis for IRa.
arr m4 t vgan {( I,l, l, I ), (t J,r l) } tfttr6-d, wi7 er .:ifurq qod r EV sgau+l na
be{ftrroqrffid6tl
1(c) (8)
Find complete solution ofx2fi +2x9= logx-
,'oj+zro]= lrer61qf 6e tfrdt
1(d) Find the area above the z -axis bounded by y2 = 4.x and the line : +y : 3' (8)
yz = 4r,t€t, * y : 23fu ,- qUfr qfr{dEot &*d$t
1(e) Identiry the following surfaces and find their intersection. (8)
xz+yz+22-2x+22:7 and xz + yz + 22 *2x -22 --7
ffi Ed e-rd 61 qrqri ofc{tr E{o.r ctr#fi {rd t
xz + yz + z2 -zx +22:7 8Rx2+y2+22+2x-22:7
Q 2. Write anslyers of the following questions.
ffiRaqfrbEflfrtus
2(a) A bullet is fired at an angle of60o with an initial velocity of200 meters/sec. For how (10)
long it remains in the air? What is the maximum height it reaches?
\rm ffi zoo fdv+irs b qr{ff 6 trr t 60" t olut q{ Tdr{ slfi e I 116 fu -ilfi tt il6
6qr fr T{fr A qE s{fuo-ilq frtfrfi iffi do. qf{fi il
2
(15)
2(b) Findgeneralsolutionofrs - @x-t)fl+('-
1)v: 0providedv = e* isone
solution of given equation'
t) ! t" - 1)v : o 6'I qlq6' sqlqFl dfr ffisot q(r- qclqH
, ! - {2, - +
y: e' dl
(1s)
I: R3 be linear map defined by
2(c) Let R{ -+
: (1,-1'1)'?(0'0'1'0) : (1'o'0) and
r(1,0,0,0) = (1,1,1), r(0,1,0,0)
?(0,0,0,1) : (1,0,1)' Find rank and Nullity of
"'
qH dfrg r, R4 -+ R3 qftqTfrdlRff crqffiGtero t ffieh Erir
: 3ln
?(1,0,0,0) (1,1,1), ?(0,1,0,0) = (1' -1'1)',r(0o'r'o) = (1',0',0)
r(0,0,0, rF(1,0, r o1 olE cllt {qar qn at t
1 ).
Q 3. Write answers of the
following questions'
ffituaqfrils{rRfrs (10)
3(a) For what values ofzl the equations
x*Y*z=L
x *3Y l9z : A
x*9Y*1.8z:72
in each case'
have a solution and solve them completely
rtat{t{dfifrqqffi
x*y*z:l
x *3y *92: I
r*9y1l8z:12
6.r qqFlr{ A 3{tr e-ed crcd fr s€ qfr a-re t us of t
3
30) Using Laplace transform, find the solution ofthe initial value problem (lS)
dry
Trztl: stttx, y(o) = o:y'(o)
ilqrs Fqi6q 6'r gqfr'r 6-{b, sRftIfi qs qII*II 6r qqrEH E+q
dzv
On*Y= sinx, y(0) = 0:y'(0)
3(c) Find the characteristic roots and characteristic vectors ofthe matrix (1s)
12023 1 1 r
,q=l o zozz o l.
Io o 2oz7]
on-qodftr)A =['T'
lo
rin I la.,u"un*woirqnffiq{Fmqfrqr
o zozi
tfrSr
Q 4. Write answers of the following questions.
frsfrBdcfrbts-f,rfrEs
4(a) Find the directional derivative of f(x,y,z) = xzyzz2 at the point (1,1,
-1) in the (15)
direction of tangent to the curve r : et,f :2sint * l,z:t- costatt:0.
tlfi x: et,y=Zsint *I,z: t-costq6f t: o]fVfftOf61RCfffrfr-g
( l, l, - I ) Tr f (r, \ z) = x2 I z2 Or E{ -er+oe-q W olfuqt
4(b) 11p: (2xz - * zxyj - ar2fi, then evaluate I I Iua*
4z)i Fdv, whercv is (15)
bounded by the planes r: O,y: O,z: 0,r ly + z= L.
u6 p :
- 4z)i + zxyj - a:r[, d qsio-r o\ [ [ [, aio Fdy, q6i v qffird
(zr,
r= 0,y: O,z=O,x*y*z: rtqffefltt
4(c) Find the maximum and minimum of f(r,y,z) = 4y - 2z srtbject to the constraints (10)
2x- y -z:2andx2 I y2: !.
4
SECTION B (qus- B)
Q 5. Write answers of the following questions.
ffifuocfrt ts-f,rftTEs
5(a) If 6 is a group in which for all @,0 E 6with(ai)i: aiE', where
i: 2O21,2O22,2023. Show that G is abelian group.
qE c rto qW t ftrsfr qrfi a, E e 6 t. fr( (aD)i : a' ai t, q6i r : 2o2t,2o22loB A | (s)
Mtuconifrs{d tr
5(b) LetU,V are ideal ofR. l-etU +V *{u*vzuEll,ueV},W:{uv:ueU,veV} (8)
and U UV : { r e U tll x € Vl. Are U + V,W and U U I/ ideal of p? Justift.
rnin&'enqqf u,v trcIil&'
U * V : {u * t z u e U,e eY},W : {uv z u E U, v e Iz} Oilt
u uv : {rt x E U or .r € y}. iFrr u+vuv .rilq uuv n} strdsf A .lilfuecartr
5(c) Let/: *, -r R be tunction satisfuing f(r * y) : f(x)f(y),vr,y € lR and (8)
h*-o.f (r) : 1. Then prove that f(r) : f (1)E ,Vr € R.
qHdfuqrfurcitRr{)f:R -+ 4 f@+y): f(x)f(1r),vr,y E R.o{tt
lim.-o /(r) = 1 a1 €qg 6{t t r fu-r erkd 6t fu f (') : f (I )., vx € R..
5(d) Prove that there does not exist an entire function g, such that g(O) : 1 and (8)
ls(z)l < .-zt't,lzl 22.
sTfud 6't fu tfi €qlf q-{E?GtNn s +{d =rfr t, q6 .e (0) = I Gtt{
lg(')l < e-2t"1 ,bl > 2
5(e) Find partial differential equation by eliminating arbitrary function 6 from equation (8)
e (x + y,x -,l-r):0, where z = z(x,y).
qfiq;tul c(r + y , * - 'l-r): 0, q6i z : z(x,y) t rDcFI c a1ffiqT 6-tb qiRr6
er*osqftorqdsr
5
Q 6. Write answers of the following questions.
ffifuaqfrbEf,rftRs
6(a) Obtain Hamilton's equations of motion for the Lagrangian (15)
L :'; (e' + sinz e i2) + (ri, + O co s0)2 - msl cosg, where rr, Ir, m, s and I
|
are constants.
AilFqqr b frq efu€q o1 rrfr b ed-s-rur qrq 6t
L :'; (e' + sir,z e $2) +
I (ri, + i cos0)2 - mg t cosl,fldl I r, Iz, m,g olf{ t or+t
tt
6(b) Find singularities e1::t(tn 66 trence find residues at these singularities. (1s)
'"5d 61 fudqlrdnt dS orti qsfds q+ fuoerurdroil qr orq*q dS r
6(c) Find the general solution of :G2 + yz)p + xyq - f,@ + y)z : o. (10)
){,'+ f)r+ ,aa- }{,+r), : sE'Iqrqrqqm-6.dfrt
Q 7. Write answers of the following questions.
ffiffiicfrilrtrrfrEs
7 (a) For a steady flow under pressure ofviscous fluid through a cylinder ofradius 'a' (15)
obtain the velocity in the form rr, : L 1az - 12) .
E-qr,
"'
& qs Rrfi -sr b crqc t EqErE a{e } rsrd fr RR rsrd t, fr s fu r
o,= fi{"' -r'z) t'wfrurratr
7(b) Find the complete integral of ff * ,#,* '# : ,"'* . (15)
afi :
# *, #, t xe' +r orEf 66 sq +lfrv
5
7(c) Using the bisection method, find an approximate root of the equation .trrr:1, that (r0)
lies between r=l and x: I.S (measured in radians). Carry out computations up to
the 7th stage.
Ertro-r Efu or sqftr ort gq. sfi-6-rq 16, = 1 q;y 3qqfr6 qo cfrd. d *: I ettt *=
r.s ttBqr fr cT qr qqn & d-s tr zd qtur ilo,.runrtotI
Q 8. Write answers of the following questions.
ffifuas?if &udrfrEs
8(a) Classi! the partial differential equation, r - 2t :0 and convert it into the canonical (15)
form.
o{iRr', i.*6E qfi -6-rur, r - 2t : o sn ErfE-d 6t .r}i gS Etra sq fr qffia of r
S(b) Let f,.: R -+ R be a sequence of functions with
/,, + f uniformly on R- prove or (15)
disprove the following:
(i) f is bounded, if each f" is bonded on R
(ii) h"...-.(./" (x)dx = fif(r)dr,orovided each f. is integrable over R-
un fr o1 egmu t/") t, qO f", n * n t efu f" r / \,rr.wrn R q{ t ffifuo +i
f
ft-gqrqfuaot'
(D f qfws e qtrffi5 4 qfrq6 s qq I I
(iD lrm [/"(r) dr: [rf(r)dr,wAq-&6f"qqrfi-mfiqnqrt r
8(c) kt p be a prime. Prove that every group of order p? is abelian. (f 0)
qFr dfug p eFrq t r fusolfuq fr. ot p2 or u-&o qg6 endft t I
