MA2161 MATHEMATICS – II ANNA UNIVERSITY QUESTION PAPER QUESTION BANK IMPORTANT QUESTIONS 2 MARKS AND 16 MARKS
2. Convert ( )
x
y xD D x
2
7 2 2 = + + into an equation with constant coefficients.
3. Define solenoidal and irrotational vector point functions.
4. State the Gauss divergence theorem.
5. Show that the families of curves θ α n rn sec = and θ β n rn ec cos = intersect
orthogonally where α and β are arbitrary constants.
6. Find the critical points of the transformation z w sin = .
7. State Cauchy’s integral theorem.
8. Find the residue of ( )
z z z
z
z f
2 2
4
2 3
2
+ +
+
= at 0 = z .
9. Find the Laplace transform of ( )
t
t
t f
2 1 +
= .
10. Find the Laplace transform of the unit step function ( ) a t u − .
PART B — (5 × 16 = 80 Marks)
11. (a) (i) Solve x e x y
dx
dy
dx
y d x 2 sin 8 4 4 2 2
2
2
= + − . (8)
(ii) Solve t y x
dt
dx
5 3 2 = − + ; t e y x
dt
dy 2 2 2 3 = + − . (8)
Or
(b) (i) Solve ( ) x x y
dx
dy
x
dx
y d
x log sin 4 2
2
2
2 = + − . (8)
(ii) Solve by the method of variation of parameters
x e y
dx
dy
dx
y d − = − − 25 3 2
2
2
. (8)
12. (a) (i) Find the values of the constants c b a , , so that
( ) ( ) ( )k y xz j cz x i bz axy F − + − + + = 2 2 3 3 3 may be irrotational. For
these values of c b a , , find also the scalar potential of F . (8)
(ii) Verify Stoke’s theorem for k xz j yz i xy F − − = 2 , where S is the
open surface of the rectangular parallelepiped formed by the planes
0 = x , 1 = x , 0 = y , 2 = y and 3 = z above the XoY plane. (8)
Or
(b) (i) Find the angle between the surfaces 11 2 2 2 = − − z y x and
18 = − + xz yz xy at the point ( ) 3 , 4 , 6 . (8)
(ii) Verify Gauss Divergence theorem for k yz j z i x F + + = 2 , where S is
the surface of the cube formed by the planes 1 − = x , 1 = x , 1 − = y ,
1 = y , 1 − = z and 1 = z . (8)
13. (a) (i) Show that an analytic function with
(1) constant real part is a constant and
(2) constant modulus is a constant. (8)
(ii) Discuss the transformation
z
w
1 = . (8)
Or
(b) (i) Prove that ( ) ( ) [ ] 2 2 2 1 log − + − = y x v is harmonic in every region
which does not include that point ( ) 2 , 1 . Find the corresponding
analytic function and real part. (8)
(ii) Find the bilinear map which maps the points 1 , , 1 − = i z onto the
points i i w − = , 0 , . Also find the image of the interior region of the
unit circle of the z plane. (8)
14. (a) (i) Evaluate ( ) ∫ + C z
dz
2 2 4
, where C is the circle 2 = − i z , by Cauchy’s
integral formula. (8)
(ii) Evaluate θ
θ
θ π
d
a a ∫ + −
2
0
2 cos 2 1
2 cos
, using contour integration, where
1 2 < a . (8)
Or
(b) (i) Show that
2
5
9 10
2
2 4
2 π
=
+ +
+ − ∫
∞
∞ −
dx
x x
x x
. (8)
(ii) Find the residues of ( )
( ) ( )2
2
2 1 + −
=
z z
z
z f at its isolated
singularities using Laurent series expansion. (8)
15. (a) (i) Evaluate ( )
+ +
+ + −
13 4
26 16 3
2
2
1
s s s
s s
L . (8)
(ii) Using Laplace transforms solve t e y y y 2 8 4 = + ′ − ′ ′ , ( ) 2 0 = y and
( ) 2 0 − = ′ y . (8)
Or
(b) (i) Use convolution theorem to find ( )
+
−
2 2 2
1
a s
s
L . (4)
(ii) Find the Laplace transform of ( )
p
kt
t f = , for p t < < 0 and
( ) ( ) t f t p f = + . (4)
(iii) Using Laplace transforms solve t t y y 2 2 + = ′ + ′ ′ , ( ) 4 0 = y and
( ) 2 0 − = ′ y . (8)
B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2009
Second Semester
Civil Engineering
MA 2161 — MATHEMATICS – II
(Regulation 2008)
(Common to all branches)
Time : Three hours Maximum : 100 Marks
Answer ALL Questions
PART A — (10 × 2 = 20 Marks)
1. Find the particular integral of ( ) x y D 2 sin 4 2 = + .2. Convert ( )
x
y xD D x
2
7 2 2 = + + into an equation with constant coefficients.
3. Define solenoidal and irrotational vector point functions.
4. State the Gauss divergence theorem.
5. Show that the families of curves θ α n rn sec = and θ β n rn ec cos = intersect
orthogonally where α and β are arbitrary constants.
6. Find the critical points of the transformation z w sin = .
7. State Cauchy’s integral theorem.
8. Find the residue of ( )
z z z
z
z f
2 2
4
2 3
2
+ +
+
= at 0 = z .
9. Find the Laplace transform of ( )
t
t
t f
2 1 +
= .
10. Find the Laplace transform of the unit step function ( ) a t u − .
PART B — (5 × 16 = 80 Marks)
11. (a) (i) Solve x e x y
dx
dy
dx
y d x 2 sin 8 4 4 2 2
2
2
= + − . (8)
(ii) Solve t y x
dt
dx
5 3 2 = − + ; t e y x
dt
dy 2 2 2 3 = + − . (8)
Or
(b) (i) Solve ( ) x x y
dx
dy
x
dx
y d
x log sin 4 2
2
2
2 = + − . (8)
(ii) Solve by the method of variation of parameters
x e y
dx
dy
dx
y d − = − − 25 3 2
2
2
. (8)
12. (a) (i) Find the values of the constants c b a , , so that
( ) ( ) ( )k y xz j cz x i bz axy F − + − + + = 2 2 3 3 3 may be irrotational. For
these values of c b a , , find also the scalar potential of F . (8)
(ii) Verify Stoke’s theorem for k xz j yz i xy F − − = 2 , where S is the
open surface of the rectangular parallelepiped formed by the planes
0 = x , 1 = x , 0 = y , 2 = y and 3 = z above the XoY plane. (8)
Or
(b) (i) Find the angle between the surfaces 11 2 2 2 = − − z y x and
18 = − + xz yz xy at the point ( ) 3 , 4 , 6 . (8)
(ii) Verify Gauss Divergence theorem for k yz j z i x F + + = 2 , where S is
the surface of the cube formed by the planes 1 − = x , 1 = x , 1 − = y ,
1 = y , 1 − = z and 1 = z . (8)
13. (a) (i) Show that an analytic function with
(1) constant real part is a constant and
(2) constant modulus is a constant. (8)
(ii) Discuss the transformation
z
w
1 = . (8)
Or
(b) (i) Prove that ( ) ( ) [ ] 2 2 2 1 log − + − = y x v is harmonic in every region
which does not include that point ( ) 2 , 1 . Find the corresponding
analytic function and real part. (8)
(ii) Find the bilinear map which maps the points 1 , , 1 − = i z onto the
points i i w − = , 0 , . Also find the image of the interior region of the
unit circle of the z plane. (8)
14. (a) (i) Evaluate ( ) ∫ + C z
dz
2 2 4
, where C is the circle 2 = − i z , by Cauchy’s
integral formula. (8)
(ii) Evaluate θ
θ
θ π
d
a a ∫ + −
2
0
2 cos 2 1
2 cos
, using contour integration, where
1 2 < a . (8)
Or
(b) (i) Show that
2
5
9 10
2
2 4
2 π
=
+ +
+ − ∫
∞
∞ −
dx
x x
x x
. (8)
(ii) Find the residues of ( )
( ) ( )2
2
2 1 + −
=
z z
z
z f at its isolated
singularities using Laurent series expansion. (8)
15. (a) (i) Evaluate ( )
+ +
+ + −
13 4
26 16 3
2
2
1
s s s
s s
L . (8)
(ii) Using Laplace transforms solve t e y y y 2 8 4 = + ′ − ′ ′ , ( ) 2 0 = y and
( ) 2 0 − = ′ y . (8)
Or
(b) (i) Use convolution theorem to find ( )
+
−
2 2 2
1
a s
s
L . (4)
(ii) Find the Laplace transform of ( )
p
kt
t f = , for p t < < 0 and
( ) ( ) t f t p f = + . (4)
(iii) Using Laplace transforms solve t t y y 2 2 + = ′ + ′ ′ , ( ) 4 0 = y and
( ) 2 0 − = ′ y . (8)
