181202(MA 2161) – MATHEMATICS - II Model question paper

1
MISRIMAL NAVAJEE MUNOTH JAIN ENGINEERING COLLEGE
THORAPAKKAM, CHENNAI-600097
MODEL PAPER
BE/B.TECH DEGREE EXAMINATION-2009
SECOND SEMESTER
MA 2161 – MATHEMATICS - II
(Common to All braches)
Time: Three hours Maximum: 100 Marks
Part - A (102 = 20 Marks)
1. Find the particular integral of D2  4D  5 y  e2x cos x .
2. Solve the equation
2
2
2 d y 4 dy 4y e x
dx dx
    .
3. Prove that divcurl A  0

.
4. Prove that F   y2 cos x  z3 i  2y sin x  4 j  3xz2 k
   
is irrotational.
5. Determine whether the function 2xy  i(x2  y2 ) is analytic.
6. Prove that the real and imaginary parts of an analytic function are harmonic.
7. State Cauchy’s Residue Theorem.
8. Evaluate
 3 C 1
z dz
z   where C is z  2 using Cauchy’s integral formula.
9. State the conditions under which Laplace transform of a function f t  exists.
10. Find L te4t sin 3t .
Part - B (516 = 80 Marks)
11. (a) (i) Solve  
2
2
2 x d y 4x dy 6y sin log x .
dx dx
  
(ii) Solve by the method of variation of parameters
2
2 d y 4y sec 2x
dx
 
OR
(b) (i) Solve the Legendre’s linear differential equation
 x  22 D2  x  2D 1 y  3x  4
(ii) Solve the system of simultaneous differential equations,
2
dx 3x 8y, dy x 3y, x(0) 6, y(0) 2
dt dt
       
12. (a) (i) Find the directional derivative of   x2 yz  4xz2  xyz at (1, 2, 3) in the
direction of the vector 2i  j  k.
  
(ii) Verify Green’s Theorem in a plane for  2 (1 ) ( 3 3 ) 
C
 x  y dx  y  x dy where
C is the square bounded by x   a, y   a
OR
(b) (i) Verify Gauss divergence theorem for F  yi  x j  z2 k
   
over the cylindrical
region bounded by x2  y2  9, z  0 and z  2 .
(ii) Evaluate   2 )   
C
  x  y dx  x  z dy  y  z dz  , where C is the boundary of
the triangle with vertices 2,0,0,0,3,0&0,0,6 using Stoke’s theorem.
13. (a) (i) Prove that an analytic function with constant modulus is constant.
(ii) Determine the analytic function whose real part is sin 2 .
cosh 2 cos 2
x
y  x
OR
(b) (i) Obtain the bilinear transformation which maps the points z 1, i, 1 into
the points w  0,1, .
(ii) Prove that
1
w z
z

maps the upper half of the z-plane onto the upper half
of the w-plane.
14. (a) (i) Using Cauchy integral formula evaluate 4 3
( 1)( 2) C
z dz
z z z

   , where C is the
circle 3
2
z  .
(ii) Find the Laurent’s Series expansion of the function   
1
2 3
z
z z

 
, valid in
the region 2  z  3.
OR
(b) (i) Prove that 2 1 a 2
2
a 2a cos 1
2 d
0 

  
 

, given a2 < 1.
3
(ii) Using contour integration evaluate
2
2 2
0 ( 9)( 4)
x dx
x x

  
15. (a) (i) Find the Laplace transform of 4
0
cos3 sin 5
t
t e t t t dt
t
      
(ii) Find the Laplace transform of the periodic function
( ) 0
2 2
f t t t a
a t a t a
  
  
and f (t  2a)  f (t)
OR
(b) (i) State Convolution Theorem. Hence find the Inverse Laplace transform of
  2 
1
s 1 s  9
.
(ii) Using Laplace transform solve
2
2
d y 3dy 2y 4
dx dx
   given that
y 0  2, y0  3
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