Bangalore University 2008 B.Sc Mathematics I Year Maths II - - Question Paper
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I Year B.A./B.Sc. Examination, November/December 2008 (97-98 & Onwards Scheme) MATHEMATICS (Paper - II)
Max. Marks: 75
Instructions :1) Answer all questions.
2) Answers are to be written in English or in Kannada completely.
I. Answer any ten of the following : ('TfFind the nth derivative of e3x sin2x.
<znff(x,y)= verify
f x = r cos0, y =? r sin0, find
-X
Tf z = (l-2xy + y2) .showthat* ,
5J"If u = x2 + xy + y2 and y = sinx find .
6) With the usual notation, prove that p = r sin <|).
7) Find the radius of curvature at any point (p, r) to the curve r1 = a2 p.
KFmd the envelope of the family of curves x2 + y2 - 2gx + (g2 - c2) = 0, where g is a parameter.
-9)rmd the asymptotes parallel to y-axis for the curve x2y - 3x2 - 5xy + 6y + 2 = 0.
the singular points on the curve x3 + x2 + y2 - x - 4y + 3 = 0.
11) Show that the origin is a cusp for the curve (x2 + y2) x - 2ay2 = 0.
<\pV
P.T.O.
1
linillllllUIIllllllB
Show that extremum of the functional . J yi + y'2dx is a line.
o
r i+y
s condition for the variational problem I = J y-dx .
17) A particle starting from rest and executing simple harmonic motion has period
6 seconds and travels 12 mts in 2 seconds. Find the amplitude and max velocity of the particle.
18) Using projectile equation show that range on the horizontal plane in the motion
2 2 u sin a
of a projectile is
g
19) A particle is projected with velocity lOOft/sec from the foot of a plane making an angle 45 with the plane. Find the inclination of he plane if the particle strikes at right angles.
20) If the angular velocity of a point moving in a plane curve be constant about a fixed origin. Show that its transverse acceleration varies as radial velocity.
d0
21) A particle describes a circle r = 2a sin 0. Show that the resultant speed is 2a1 .
22) Define central orbit. Write the expressions for the velocity at any point of the central orbit.
23) Write the expressions for tangent and normal acceleration.
24) Determine the law of force for a particle describing central orbit whose pedal equation is pr = a2.
II. Answer any two questions : (2x5=10)
IlilWIUlU .3- AN -1320
3) Show that f VtanG d0 =
0
f s 2 J?) Evaluate J cos 0 sin 0 d0.
0
HI. Answer any two questions : (2x5=10)
1) Find the complex Fourier transform of f(x) = e~a x where a is a positive constant.
is self reciprocal in respect of complex fourier
transform.
2) Prove, if a is any real constant then
1) F[f(x + a)]=e 10taf(a)
2)E em f(x)j = f (a + a)
yfny Employing Parsevals identity to the function f(x) defined by
2 2\ l-x |xj< 1 T(xcosx-sinx) \
.Hence, find --t-dx.
0 x >1 I v6
v 1*10 X
Using Parsevals identity for Fourier cosine tranforms show that
dx - 71/
0 fa'
IV. Answer any two questions. (2x5=10)
1) Solve zp + yq = x.
2) Using Charpits Method Solve P+qjy = (lz.
3) Reduce the equation r + 2s +1 = 0 to cannonical form.
 |
a2u |
= 4 2 subjected to the condition
1) u(0, t) = 0, u (1, t) = 0 V t
2) u(x, 0) = x - x2, 0< x < 1. Solve.
<
L enQXid: (2x20=40)
[7+1 n | n od ?3
2n
J sin 0d0 *>&&.
J sin 2 0 cos2 0 d0 toQ&O.
0
V |
|
|
log |
|
|
|
|
_ |
dy aod> toqa.
' f I x I I x |<a j
f(x) = | I | cftjeOodbodrcSodbcdrfor&aocoo. i
5
6
7
8
9
10
11
12
13
14
15
fcdba) sgpe&ofto* Z&& issfiff dodFtfo>ck&?*>.
f (a) F [f (x)] tsdd ijef (x)J = f (a + a) ,aodb &aee*>.
a dbab b &d ddrteaAdb f (x) dasb g (x) rteb x enrto Add,
F [a f (x) + b g (x)] = a F [f (x)] + b F [g (x)] <aod} j
t?z - f(x2 - y2)od i&> tsdr tfodbSoQoooo.
z = (x - a)2 + (y - b)2 <aod Aporf a ds& b .atosbrtdb o&tf t?dr odD&QcpO. Pey = qe* '
p2q3 -1 Wcssa*.
y (0) = 3;y(l) = 6 I = J 12xy + y'2 j dx dd doarad tfodb&@o!DO.
J /l + y'2dx <3dd dddtodp doarod ttodb deatabodb I
X2 i _,_ 2 r /I+y
wdosdrfobdridbft I = J if-dx o>db sas? staeQ.
x2f *
16) J 1 dx wdoadsirtod ridb?|oi3 dddtodp doarody = Ax3 + B <aodb XjV x J
17) ioSofiood atodu aBsdbrc riSodbddb 6 aS/soOd 2|prW>) 12 aoe&o6' dtoddskoxbd. rt03|dtod rt03|dertd&odbk&o3c>o. I
2 . 2
18) z&bgd sfooetfdrsd ;te>odbDod dbod d/sd u sma. cDodD
19) odb&100alo8rttfD3riddQod45 OOSJSron tfC&rt <3<?2SK>Orf odb&>Qo>0.
O j
20) xfe> d de<dojbd?&dbd ttoc><De$ &rae)?cdb derfe$ Ad tfod:<D<sO&& ddo>c& aoteoDdd t?dd dtfderfej} zodcsdocbo ecteA ecdo dertdo eodoartdodb 3tae&.
d0
21) odb sg r = 2a sin 0 cfcaa&ri. isoderfe 2 a aodh.
22) eoeocbdorfccjbsds&eoecobdosScci)catode&od>Dri<)derid*&De&ta> todotoO.
23) fcodo&F db& drcr dertoedr T&oetfdra sodcoQ.
24) Pr=a2 ?&3$dfs| ddo6, afoxtfdrad jstec&Qod eoxb wo odbdo3 zoo 3dbsfeb tfodb SoStfDO. *
n. <ad:& enOfc: (2x5-10)
*
ml
1) (3(m,n) =
m + n
f1 x2 , f1 dx n J it \ ~ dx J
if\ \/ A\
2) 0 Ji-X4j 0 Vl7x4 4 aodsW
72
3) J VtanB d0 = ~5= <aodb*>*>.
o
V2 5 2
4) J cos 0 sin 0 d0 &Q&
AN - 1320
V. Answer any two questions : (2x5=10)
1) Prove that Geodesic on a plane is a straight line.
<2)Show that general solution of Euler's equation for the integral J \ + y'2 dx is
a y
(x - B)2 + y2 = R2.
3) Show.that sphere is the solid fig of revolution which for a surface area has maximum volume.
*2 I--
4)8ftow that extremal of the functional J y yl + yx dx and subjected to the
*2
*2 j-2
constraint j yfl + y' dx -lisa catenary.
VI. Answer any four questions : (4x5=20)
1) A particle executes simple harmonic Motion such that is has speed v when the
acceleration is a and speed u when the acceleration is p. Show that distance
2 2
. ... u - V
between two positions is-.
a + P
2) At the end of three sucessive seconds, the distance of a point moying with simple harmonic motion from its mean position measured in the same direction
are 1,5,5. Show that the period of complete osciliation is where cos 0 = .
3) If the particle is projected on a plane, prove that the path traced is parasola,
4) A particle is projected from a point on the ground with a velocity v at an angle a to the horizontal such that its horizontal range is twice the greatest height attained.
4v2
Show that the horizontal range is-.
8 . . . I.
5) A heavy particle of weight W, attached to a fixed point by a light inextensible string describes a circle in a verticle plane. The tension in the string has the values mw and nw respectively when the patale is at the heighest point and lower point in the path show that n == m + 6. [
6) A point P describes with a costant angular velocity about, O , the equiangular spiral r - ae0, O being the pole of the Spiral. Obtain the radial and transverse acceleration.
7) Derive the equation for velocity at any point of central orbit. j
8) If the central orbit is r = a tan e, S.T. magnitude of acceleration towards the centre of force is lAi3 (3+2a2u2). Find the velocity in terms of r.
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