Bangalore University 2008 B.Sc Mathematics III Maths VI - - Question Paper
Ill Year B.A./B.Sc. Examination, November/December 2008 (1999-2000 & Onwards Scheme)
Paper - VI: MATHEMATICS
Max. Marks : '100
Time : 3 Hours
Instructions : 1) Answer all the questions.
2) Answers should be completely in Kannada or in English.
I. Answer any twenty questions of the following :
(20x2=40)
JL)lflr(t) = e x i + log(t2 + l)j - tant k, find the acceleration at t 0.
A f \
/TF A = t2i - tj-(2t+ l)k and B = (2t -1) i + j - tk, find (Ax Bj.
dt
3) Find the spherical polar coordinates of the point whose Cartesian coordinates are (1,1,1). |
Find the direction cosines of the tangent at the point t =j of the curve r = 4 cost
! 2
i + cos 2tj + cos21 k.
5) If <|> = xyz, find V2<j) *
= x3 + cosy + z, find grad <j> at f 0, ~, 1
7)JRrn3 the directional derivative of (j) = xy + yz + zx at (1,2,0) in the direction of i+j+k.
the vector F = (ax + 3y + 4z) j + (x - 2y + 3z) j + (3x + 2y - z)k is solenoidal, find the value ofa.
)valuate : I xdx + ydy + zdz; where C is the curve X = t2 +13; y = t2 + 1,
c
z = et\ where -l<t<l.
P.T.O.
1
AN 1319 -2-
lQyEvafuate : J J (x2 + y2).dydx
o x
ill
jJEvaluate : J J J (x + y + z) dxdydz.
ooo
jifA = ti -12j + (t - l)k and B = 2t2i + 6tk, find jA.B dt.
13) Using Gauss divergence theorem, show that JJr .nds = 3V, where V is the volume
s
of any closed surface S.
JylJsing Greens theorem, show that the area bounded by a simple closed curve
C is given by j xdy - ydx. .
15) State Greens theorem.
sin t i + e 1 cost j, find |
17) Evaluate: Km
dfA
'dty
dt
'V + l
v 1 - zi y
18) Show that f(z) = z is not analytic.
19) Examine C -Requations for f(z) ~z~ z.
(J20)"<Verify whether u = x3 - 3xy2 + 3x2 - 3y2 + 1 is harrhonic.
21) State fundamental theorem of Algebra.
22) State Cauchys integral formula.
23) Evaluate : j 3z2dz> where C is the line segment frJm 0 to 2i.
c ' !
2z-l
-3-
II. Answer any two questions :
AN - 1319 (2x5=10)
r = ti +1 j + t k; find the unit tangent and the principal normal vector at t = 1.
2) Derive the formula for the torsion x of a space curve r = r(u) in the form [rrf]
[r x ?]
)HEfT= 3t2i -(t + 4)j + (t2 - 2t)k and b - sin t i +3e-ti j - 3 cost k , (axb),
find
dt
4) Show that the surface 5x2 - 2yz = 9x is orthogonal to the surface 4x2y + z3 = 4 at the point (1, -1, 2).
III. Answer any two questions :
(2x5=10)
1) Prove that div (curl F) = 0 and curl (grad <|>) 0 *
xi + yj + zk . . . i, ?s lrrotational.
2)Sfiow that the vector f =
\jx2 + y2 +z2'
A
+ sin x)k, then prove that
F - (sin y + z cosx)i + (x cosy + sin z) j + (y cosz
curl F = 0. Find <j> such that F = V<j>.
4) Find the equation of the tangent plane to the surface z = x2 - y2 at (2, -1, 3).
(2x5=10)
IV. Answer any two questions :
Devaluate : jF.dr, where F = x2i + (2xz + y)j + zk along the line C from (0, 0, 0) ' c
to (2, 3, 4).
2) Evaluate jj ydx dy; where R is the region bounded
by the parabolae y2 = 4ax
R
a 4 a*-x2 /a2-x2-y2
|
and x2 = 4ay. |
 |
dzdy dx
Evaluate : f f f 7--,.
J / 2 2 2 2
oo o ya-x-y-zl
AN 1319 .4.
V. Answer any two questions :
(2x5=10)
1) Evaluate JJ [yzi + zyj + xykj.nds, where S is the surface of the sphere s
x2 + y2 + z2 - 1 in the first Octant.
2) Using Stokes theorem, evaluate
f [(x + y)dx + (2x - z)dy + (y + z)dz], where C is the boundary of the triangle
whose vertices are (2, 0, 0), (0, 3, 0) and (0, 0, 6).
3) Verify Green's theorem for <j(x2 + y2)dx - xy2dy
taken around the boundary
1
of the square whose vertices are (0, 0); (1, 0); (1, 1) and (0, 1). 4) State and prove Stokes theorem.
(4x5=20)
VI. Answer any four questions :
z-ll 71
1) Show that arg
- = represents a circle.
z + 1 J 3
frove that the necessary condition for a function if(z) = u(x, y) + iv(x, y) to be
, . . du dv du -dv
analytic is = and ---
dx dy dy dx
, * x-y
he imaginary part is =-=-
x + y
3) Find the analytic function f(z) = u + iv, given that
4) Prove that the real and imaginary parts of an analytic function are harmonic. Is the converse true ? Justify.
iJvaluate J(3xy + iy2 )dz along the straight line C joining z = i and z = 2 i.
z -1
dz, where C is |z i| =
2.
c (z+1)2(z-2)
7) Discuss the transformation W = Z2.
8) State and prove Liouvilles theorem.
-5- ! AN - 1319
i
I. cdtoradcta20 ; (20x2=40)
1) r(t) ~ e_t i + log(t2 + l)j - tant k, e?ddt = 0ste)wdd derfoedrdodb&ccoO.
I
2) A = t2i - tj - (2t + l)k dbD B = (2t -1) i + j - tk 'sdd, (a x b)oc&&8o3oO.
3) od> odb<D sstaeF&odW KcSers&rteb (1,1,1) dd, dd rfaed tpeccb derd3b odb&oso. I
4) r - 4 cos tin- cos 2tj + cos21 k eodb dde<d wdd, edd xfefcd c|d5ffs&?r
2
5) (j) = xyz, esdd, V2(|) odbSo&ctfoo. i
? * f 71 : '
6) (j) = x + cos y + z e?dd, grad (j) [ 0, , 1 d<odb&>coo.
J
7) (1, 2,0) octa i + j + k odbcdD(|) = xy + jyz + zxrfQaia3
wdootfdsb tfodbSoccoo. - ' j <
i. "
i
8) F = (ax + 3y + 4z)j + (x - 2y + 3z) j + (3x + 2y - z)k*j.ot>*wto)oDrfor
Qcisdd a ddoi>;> iocso2cQoDDO.
=<. ,
9) Coio x = t2 + t3;yt2+1, z = et! d~l <t <1 des3pd I xdx + ydy + zdz
odbSoo2oo. i
I
i
i Vx > I
10) odDSoacODO: j j(x2+y2)dydx. j
X !
iii ;
11) odbMoOoO: J J J (x + y + z) dxdydz. !
ooo ' :
2__;
12) A = ti - t2j + (t - l)k s)d B = 2t2i + 6tk ydd, JA.Bjdt odb2oSo3oO.
o
AN - 1319 -6
13) rf erodalfteF&, JJr.rids = 3V >odbatoeo&. <ao2oc>db
s
odtode dboi|d d3?d sjfc%>o.
14) fyc? doeo&dab erodoirDeft, <j*xdy-ydx <aowodo C ou do>2|d
c
ddesdccood too;fe)ftdbd codb ;ft>eo.
15)
df v dtJ
16) f(t) = e-t sin t i + e_t cost j, wdd j
dt c&ab odoSoox>o.
( 3 Z+1
v1 - zi y
lim
z-m
F&OtoQoM.
17)
18) f (z) = z estuaorib eOA.
19) fz) = z-zmfC-R
oDodo dOea&.
eA
_dz cdb odbSoQosoo.
20) u = x3 - 3xy2 + 3x2 - 3y2 + 1 arosftaeFDaf t?Adcdb
23) C <aoioodb0 <5*od2i* ddrtodbddedd, J3:
c
2z 1
24) W =-atosroodd Ad ttoribrfcfcdb odoSoQcft}0.
t
(2x5=10)
A A A
1) r = ti +1 j + t k &oocb ddesj esdd t = 1 a&f db& scd 0020 Dddb
odb&CCDQ.
r r jr]
PF
eodb nJ3i
'ri.
2) r - f (u) toodb tssarf ddea? add, Scbzb x
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Attachment: |
| Earning: Approval pending. |
