Bangalore University 2008 B.Sc Mathematics III Maths VII - - Question Paper
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II1IHII111 AN -1320
III Year B.ATB.Sc. Examination, November/December 2008 (1999-2000 and Onwards Scheme)
MATHEMATICS (Paper - VH)
Time: 3 Hours Max. Marks: 100
Instructions: 1) Answer all questions.
2) Answer should be written completely either in English or in Kannada.
I. Answer any 20 questions : (2x20=40)
+ 1 = n pn
271
<2)'l2valaute J sin 0d0.
2 2!
<3)Evaluate f sin 0 cos 0 d0 .
4)JShow that ["n = J log
r i x 11 x i <a
*-5TFind the Fourier transform of f (x) = {
' [0 | x| >a
6) Define inverse Fourier cosine and inverse sine transform of a function.
JjyPvovei that Fef (x) J == f (a + a) where f (a) = F [f (x)].
J&jrTfsL and b are any two constants and f(x) and g(x) are any two functions, S.T.
F[a f(x) + b g(x)] = a F[f(x)] + b F [g(x)]
TForm the PDE by the method of elimination of arbitrary function z = f(x2 - y2). Ip)-Blm5nate the arbitrary constants a and b from the equation z = (x - a)2 + (y - b)2 and form the PDE.
11) Solve Pey = qex.
12) Solve P2q3 = 1.
13)Solve the variational problem under the given conditions
AN - 1315 -2-
7l/2
L2)Evaluate Jcos xx.
-tt/2
13) Find the area of the astroid x2//3 + y23 = a23.
14) Show that the circumference of the circle of radius r is 2nr by integration.
n. Answer any five of the following :
2x
l) Find the n* derivative of + _ 2
y = eras,n x show that (I-x2)yn+2 ~(2n + l)xy+, -(n'1 + m'!)yn =0
3) State and prove Eulers theorem for a homogeneous function of degree n in the variables x and y.
a2u
ax2
+ 2xy---y =- = sin 4u - sin2u
axay ay2
If u = x (1 - y) and v = xy, frnd J = and verify JJ' -1.
3(x,y) <3(u, v) I
6) Show that rn = an cosnG, rn = bn sinnG intersect orthogonally.
7) Find the pedal equation of y2 = 4a (x + a).
r
1 +
8) With the usual notations prove that p =-
*JL
dx2
-3- AN - 1315
III. Answer any three of the following : (3x5=15)
1) Find the evolute of the curve x - a cosG and y = b sin0 .
2 2 x y
2) Find the envelope of + 1 where a and b are parameters and a+b= c.
a b j
Ind the asymptotes of (y x) (y - 2x)2 + (y + 3x) (y - 2x) + (2x + y - 1) = 0.
Find the position and nature of the double points of x3 + x2 + y2 x 4y + 3 = 0.
5) Trace the curve r2 - a2 cos 20.
IV. Answer any three of the following : (3x5=15) 1) Obtain the reduction formula In = Jsec11 xdx.
nr Vl COS0 . 2 8V2
_2VS!tqw that J --r-sm ' 0d0 = -.
- 0J l + cos0 3
1 xa -1
3) Evaluate f-dx where a.is a parameter.
0logeX
A
4) Find the surface area of the hemisphere of radius *a\
5) Find the perimeter of x = a cos3 0, y - a sin3 0.
dLrssod
I. (10x2=20)
1) e3x'suAc <ao2oodd n-sfci) tfodbSoSceoa
(y\ a2f 52f
vazri ~ <aoc> 3taeo*>. oxoy ay ox
V > J
0(x, y)
3) x = r cos.0, y = r sin0, wdd tfoc&SoSotoO.
- S(r> 0)
AN - 1315
~V) dz dz 2 3
4) z = (l-2xy + y ) sjoartx ~ Y ~ Y z oab &ae*>.
du
5) u = x2 + xy + y2 y = sinx tjcrorf oosbSooto.
6) ctoccto 3f)rbras5>ft, p = r sin <j> .ooab ;&<?&.
7) r3 = a2 p odtesJcSe tOoctod s>ed dtfocb&ao&o.
8) x2 + y2 ~ 2gx + (g2 - c2) = 0 dd<?<55odb cdtoesf cdb ?foc3bSoQco. <2 g ccbo suda
9) x - 3x2 - 5xy + 6y + 2 = 0 ti$d?33rt
?fcdD2ocD0.
bfioccoO. I \
<aodo 3>e&.
10) x3 + x2 + y2 - x - 4v
 |
*2
*5
& |
|
6) Show that r
' i*7 Ti*' |
fiaDOt) Ak)Qft.
Jsh>sb9&.
7) Find the pedal equation of y2
|
I |
|
a: .-v <D
dx2 , |  |
Y2ucaad>3Dp*x |
8) With the usual notations prove that p ~
-5- AN - 1315
6) rn = an cos n0, rn = bn sin n0 <addb dd?<s3rteb oo<aodb
7) y2 = 4a (x + a) ddosdrt stoetfdresfcb tfodb&QcfloO.
8) y = f (x) oeo s&d&airt ctaoob rtoSe&S v&rbrasroA p =
|
/ 1 \2
i+W |
3/
/2 |
|
\dxy _
*7 |
-<>od} TOQ |
III. oiraroddo dx/sd? emOA:
1) x = a cosO, y = b sin0 ddea3o> <adeWcfcoc>&>aoaoO.
2) a dbsb b steb a + b = c tsosn1, = 1 oto sftde<stfrte* rbo&rf
7 <-d Z> a s> z>
*
casf dd/aesf odb&So3oo.
3) (y - x) (y - 2x)2 + (y + 3x) (y - 2x) + (2x + y - 1) = 0 adea5oi> otiod do
SoQoDO. I
4) x3 + x2 + y2 - x - 4y + 3 = 0 dde<s3oi> D,&odbr& dDsfr epadrtsfcb rfodb &>&o3oO.
5) r2 = a2 cos 20 oto ddeai>3b
IV. odtocrodcta dxrsdb :
1) In = Jsecn xdx ddod&ri/sifodbkaoLDO.
"f Vl-COS0 . 2q q 8V2
2) h
1 xa 1
3) |-dx fodb&aosoo, S&V toodb 3350 <odb ofcixraQriuaflri.
0lo8eX
4) sbecj <S)Acrarsteb odD&aoDoa rteetfdV <aoc> rfdOAd.
5) x = a cos3 0, y = a sin3 0 sd&stiocb tfodb&aofoD.
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