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ANNA UNVERSITY
June / July 2012 Examination
(Common to all colleges over Tamil Nadu)
MA2161 Mathematics II – Important Questions
Unit I
1. Solve (D2+2D+1)y = 2e-x+cos x+x2+1
2. Solve the simultaneous equation __
__ + 2_ − 3 = 5 , __
__ − 3_ + 2 = 0 given that x(o)=0 ,
y(0) = -1.
3. Solve __
__ + 2_ − 3 = _, __
__ − 3_ + 2 = ___
4. Solve (2x+3)2 y ‘’ – (2x+3) y’
-12y = 6x
5. Solve (D2 – 4D +3)y = sin
3x cos 2x
6. Solve (3x+2)2 y” + 3(3x+2) y’ –
36y = 3x2 + 4x + 1
7. Solve _(_ + 1)_ __ + (_ + 1)_ + 1_ = 4 cos(log(_ +
1))
Unit II
1. Verify Green’s theorem for _ (3__ − 8 _ # )!_ + (4 − 6_ )! , where C is the boundary
of the region defined by y=x2 , x=y2
2. Determine f( r) so that the vector f ( r) r is both solenoidal and
irrotational
3. Verify Gauss divergence theorem for $% = (4_&)'%− _(%+ &)
over the cube bounced by
x=0 , x=1 , y=0,y=1,z=0,z=1.
4. Verify Stroke’s theorem for $% = _&'%+ &__(%+ __ )*% where S is the open surface of the
cube formed by the planes x= - a , x=a , y= -a, y= a ,z=-a ,z=a
5. Show that $% = (6_ + &+)'%+ (3__ − ,)(%+
(3_&_ − ,))*% is irrotational vector and find the
vector potential function p such that $% = ∇.
Unit III
1. If f(x) is a regular function of z. Prove that ∇__/(&)__ = 4_/′(&)__
2. Find the bilnear transformation that maps points 0,1,∞ onto i , 1 ,-i
3. If f(x) = u+iv is a regular function, prove that ∇__/(&)__ = 4_/′(&)__
4. Using Laplace transformation method, solve _1_
__1 + 9, = 18t , given that
y(0) = 0 and 34
_ 5=
0.
5. Find the image of the circle [z] = ƛ under the transformation w=5z.
Unit IV
1. Using Cauchy ‘s integral formula evaluate _ 6 _6
(678)(67_) where C is the circle [z-2] = .
2. Evaluate _ _∅
8+:;<=> ∅
_4
? using contour integration.
3. Find the Laurent’s series expansion for the function /(&) = @67_
6 (67_)(6:8) in the region 1<
[z+1]<3 .="." span="span">
4. Find the Taylor’s series to represent 6178
(6:_)(6:+) in |z| <2 span="span">
5. Prove that _ _1
(_17A1)(_1:B1) !_ = 4
A:B F , C > E > 0
7F
Unit V
1. Solve using Laplace Transform _1_
__1 + 4 __
__ + 4 = __7_ given y(0) = 0 and y’(0) =1
2. Using convolution theorem find the inverse Laplace transform of _
(G:8)(G1:H)
3. Find the Laplace Transform of ___ _ IJK+_
_ !_ _
?
4. Find the Laplace transform of /(_) = /(_) = L _, 0 < _ < C
2C − _, C
< _ < 2C with /(_ + 2C) = /(_)
