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Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 1
SUBJECT NAME : Engineering Mathematics - II
SUBJECT CODE : MA2161
MATERIAL NAME : University Questions
MATERIAL CODE : JM08AM1004
Name of the Student: Branch:
Unit – I (Ordinary Differential Equation)
Type – I to VI
1. Solve the equation 2 24 cos 2D y x x . (M/J 2009),(N/D 2011)
2. Solve the equation 23 2 2cos 2 3 2
xD D y x e . (N/D 2009)
3. Solve 2 316 cosD y x . (N/D 2010)
4. Solve : 2 23 2 sinD D y x x . (M/J 2011)
5. Solve the equation 25 4 sin 2
xD D y e x
. (A/M 2011),(ND 2012)
6. Solve the equation 24 3 sin
xD D y e x
. (M/J 2010)
7. Solve: 24 3 cos 2
xD D y e x . (M/J 2012)
8. Solve 2 24 3 6 sin sin 2
xD D y e x x
. (N/D 2011)
Method of Variation of Parameters
1. Solve 2 2tanD a y ax by the method of variation of parameters. (M/J 2009)
2. Solve, 2
2
2tan
d ya y ax
dx by method of variation of parameters. (M/J 2011)
3. Apply method of variation of parameters to solve 24 cot 2D y x .
(N/D 2009),(N/D 2011)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 2
4. Solve 2 2secD a y ax using the method of variation of parameters.(M/J 2012)
5. Solve 2
2cos
d yy ecx
dx by the method of variation of parameters.
(A/M 2011),(ND 2012)
6. Solve 21 sinD y x x by the method of variation of parameters. (M/J 2010)
7. Using variation of parameters, solve 22 3 25
xD D y e
. (N/D 2011)
Cauchy and Legendre Equations
1. Solve the equation 2 23 5 cos logx D xD y x x . (M/J 2009)
2. Solve 2 2 23 4 cos logx D xD y x x . (N/D 2010)
3. Solve 2 2 24 sin logx D xD y x x . (M/J 2012),(N/D 2009)
4. Solve 2 2 22 4 2logx D xD y x x . (M/J 2010)
5. Solve the equation2
2 2
1 12logd y dy x
dx x dx x . (N/D 2012)
6. Solve 2
2 2
23 4 ln
d y dyx x y x x
dx dx . (N/D 2011)
7. Solve 2
2
2(1 ) (1 ) 2sin log(1 )
d y dyx x y x
dx dx . (A/M 2011)
8. Solve: d y dy
x x y xdx dx
2
2
2(1 ) (1 ) 4cos log(1 ) . (N/D 2011)
Simultaneous Differential Equations
1. Solve sin , cosdx dy
y t x tdt dt
given 2x and 0y at 0t . (M/J 2009)
2. Solve 2 sin 2 ,dx
y tdt
2 cos 2dy
x tdt
. (M/J 2012),(N/D 2009)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 3
3. Solve 2 sin ,dx
y tdt
2 cosdy
x tdt
given 1x , 0y at 0t . (N/D 2010)
4. Solve dx
y tdt
and 2dyx t
dt . (A/M 2011)
5. Solve dx
y tdt
and 2dyx t
dt given x y (0) (0) 2 . (N/D 2011)
6. Solve tdxy e
dt ,
dyx t
dt . (N/D 2012)
7. Solve 22 3 2 ,
tdxx y e
dt 3 2 0.
dyx y
dt (M/J 2010)
8. Solve 2 3dx
x y tdt
and 23 2
tdyx y e
dt . (N/D 2011)
9. Solve for x from the equations 2 23
tD x y e , 2
3t
Dx Dy e . (M/J 2011)
Unit – II (Vector Calculus)
Simple problems on vector calculus
1. Find the directional derivative of 22xy z at the point 1, 1,3 in the direction of
2 2i j k . (M/J 2009)
2. Prove that 3 2 26 3 3F xy z i x z j xz y k is irrotational vector and
find the scalar potential such that F . (M/J 2010)
3. Show that 2 2 22 2 2 2F y xz i xy z j x z y z k is irrotational and
hence find its scalar potential. (M/J 2012)
4. Show that 2 2 22 2 2F xy z i x yz j y zx k is irrotational and
find its scalar potential. (N/D 2012)
5. Find the angle between the normals to the surface 3 24xy z at the
points 1, 1,2 and 4,1, 1 . (M/J 2009)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 4
6. Find the angle between the normals to the surface 2xy z at the points 1,4,2 and
3, 3,3 . (A/M 2011)
7. Find the work done in moving a particle in the force field given by
23 (2 )F x i xz y j zk along the straight line from 0,0,0 to 2,1,3 .
(M/J 2012)
8. If r is the position vector of the point , ,x y z , Prove that 2 2( 1)
n nr n n r
.
(N/D 2010)
9. Determine ( )f r , where r xi yj zk , if ( )f r r is solenoidal and irrotational.
(N/D 2011)
10. If F is a vector point function, prove that 1 1 2curl curlF F F
.
(N/D 2011)(AUT)
11. Prove that curl div div u v v u u v u v v u . (N/D 2009)
12. Evaluate 2 2 2
C
x xy dx x y dy where C is the square bounded by the
lines 0, 1, 0 and 1x x y y . (N/D 2009),(N/D 2011)
13. Evaluate s
F n ds where 22F xyi yz j xzk and S is the surface of the
parallelepiped bounded by 0, 0, 0, 2, 1x y z x y and 3z . (M/J 2011)
Green’s Theorem
1. Verify Green’s theorem in a plane for 2 23 8 4 6
C
x y dx y xy dy , Where C is
the boundary of the region defined by the lines 0, 0x y and 1x y .
(N/D 2010) ,(A/M 2011),(M/J 2011), (M/J 2012)
2. Verify Green’s theorem for 2 23 8 4 6
C
x y dx y xy dy where C is the boundary
of the region defined by 2 2, x y y x . (M/J 2010)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 5
3. Verify Green’s theorem for 2 22V x y i xyj taken around the rectangle
bounded by the lines , 0x a y and y b . (N/D 2012)
Stoke’s Theorem
1. Verify Stoke’s theorem for 2F xyi yzj zxk where S is the open surface of the
rectangular parallelepiped formed by the planes 0, 1, 0, 2x x y y and 3z
above the XY plane. (M/J 2009)
2. Verify Stoke’s thorem for the vector ( )F y z i yzj xzk , where S is the surface
bounded by the planes 0, 0, 0, 1, 1, 1x y z x y z and C is the square
boundary on the xoy -plane. (N/D 2011)
3. Verify Stoke’s theorem when 2 2 22F xy x i x y j and C is the boundary of
the region enclosed by the parabolas 2y x and 2
x y . (N/D 2009)
4. Evaluate sin cos sinC
zdx xdy ydz by using Stoke’s theorem, where C is the
boundary of the rectangle defined by 0 , 0 1, 3x y z . (N/D 2009)
5. Using Stokes theorem, evaluate C
F dr , where 2 2( )F y i x j x z k and ‘C’ is
the boundary of the triangle with vertices at 0,0,0 , 1,0,0 , 1,1,0 .(M/J 2012)
6. Using Stoke’s theorem prove that curl grand 0 . (M/J 2011)
Gauss Divergence Theorem
1. Verify Gauss divergence theorem for 2 2 2F x i y j z k where S is the surface of
the cuboid formed by the planes 0, , 0, , 0x x a y y b z and z c .(M/J 2009)
2. Verify Gauss Divergence theorem for 24F xzi y j yzk over the cube bounded
by 0, 1, 0, 1, 0, 1x x y y z z . (N/D 2010),(A/M 2011),(N/D 2012)
3. Verify Gauss – divergence theorem for the vector function
3 22 2f x yz i x yj k over the cube bounded by 0, 0, 0x y z and
, ,x a y a z a . (M/J 2010),(N/D 2011)
4. Verify Gauss’s theorem for 2 2 2F x yz i y zx j z xy k over the
rectangular parallelepiped formed by 0 1,0 1x y and 0 1z .
(N/D 2011)(AUT)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 6
Unit – III (Analytic Function)
Harmonic Function & Analytic Function
1. Verify that the families of curves 1
u c and 2
v c cut orthogonally, when 3u iv z .
(N/D 2009)
2. Prove that cosy
u e x and sin
xv e y
satisfy Laplace equations, but that u iv is
not an analytic function of z . (M/J 2011)
3. When the function ( )f z u iv is analytic, prove that the curves constantu and
constantv are orthogonal. (N/D 2009)
4. Show that the families of curves secn
r a n and cosn
r b ecn cut orthogonally.
(M/J 2011)
5. Show that 2 21log
2u x y is harmonic. Determine its analytic function. Find also
its conjugate. (A/M 2011)
6. Prove that 2 2u x y and
2 2
yv
x y
are harmonic but u iv is not regular.
(N/D 2010)
7. Prove that every analytic function w u iv can be expressed as a function z alone,
not as a function of z . (M/J 2010),(M/J 2012)
8. Find the analytic function ( )f z P iQ , if sin 2
cosh 2 cos 2
xP Q
y x
.(M/J 2009)
9. Determine the analytic function whose real part is sin 2
cosh 2 cos 2
x
y x. (N/D 2012)
10. If ( )w f z is analytic, prove thatdw w w
idz x y
. (A/M 2011)
11. Find the analytic function u iv , if 2 24u x y x xy y . Also find the
conjugate harmonic function v . (N/D 2009)
12. Find the analytic function w u iv when yv e y x x x
2cos 2 sin 2 and find u .
(N/D 2011)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 7
13. Prove that ( cos sin )x
u e x y y y is harmonic and hence find the analytic function
( )f z u iv . (N/D 2010)
14. If ( )f z is a regular function of z , prove that 2 2
2 2
2 2( ) 4 ( )f z f z
x y
.
(M/J 2009), (A/M 2011)
15. If ( )f z is an analytic function of z , prove that 2 2
2 2log ( ) 0f z
x y
. (M/J 2012)
16. If ( )f z is analytic function of z in any domain, prove that
2 22 22
2 2( ) ( ) ( )
p pf z p f z f z
x y
. (N/D 2011)(AUT)
Conformal Mapping
1. Find the image of the half plane x c , when 0c under the transformation 1
wz
.
Show the regions graphically. (M/J 2009),(N/D 2012)
2. Find the image of the circle 1 1z in the complex plane under the mapping 1
wz
.
(N/D 2009)
3. Find the image of the hyperbola 2 21x y under the transformation
1w
z .
(M/J 2010),(M/J 2012),(N/D 2012)
4. Find the image of 2z under the mapping (1) 3 2w z i (2) 3w z .
(A/M 2011)
5. Prove that the transformation1
zw
z
maps the upper half of z - plane on to the
upper half of w - plane. What is the image of 1z under this transformation?
(M/J 2010),(N/D 2012)
6. Show that the map wz
1
maps the totality of circles and straight lines as circles or
straight lines. (N/D 2011)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 8
7. Prove that the transformation wz
1
maps the family of circles and straight lines into
the family of circles or straight lines. (N/D 2011)
8. Show that the transformation1
wz
transforms, in general, circles and straight lines
into circles and straight lines that are transformed into straight lines and circles
respectively. (N/D 2011)(AUT)
Bilinear Transformation
1. Find the bilinear transformation which maps the points 0, , 1z i into w – plane
,1,0w i respectively. (M/J 2009)
2. Find the bilinear transformation which maps the points 0,1,z into
,1,w i i respectively. (M/J 2010),(M/J 2012)
3. Find the bilinear transformation that maps the points , , 0z i onto 0, ,w i respectively. (N/D 2012)
4. Find the bilinear transformation which maps the points 1, , 1z i into the points
, 0,w i i . Hence find the image of 1z . (M/J 2011)
5. Find the bilinear transformation that transforms 1, i and 1 of the z – plane onto
0, 1 and of the w – plane. Also show that the transformation maps interior of
the unit circle of the z – plane on to upper half of the w – plane. (N/D 2010)
6. Find the Bilinear transformation that maps the points1 , , 2i i i of the z - plane
into the points 0,1, i of the w - plane. (N/D 2011)
Unit – IV (Complex Integration)
C.I.F and C.R.T
1. Evaluate
21 2c
zdz
z z where c is the circle
12
2z using Cauchy’s integral
formula. (M/J 2009),(N/D 2009),(M/J 2012)
2. Evaluate
22
( 1)
2 4C
zdz
z z
where C is 1 2z i using Cauchy’s integral formula.
(A/M 2011)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 9
3. Evaluate2
4
2 5c
zdz
z z
, where C is the circle 1 2z i , using Cauchy’s integral
formula. (N/D 2010),(N/D 2011),(N/D 2012)
4. Using Cauchy’s integral formula evaluate2
1c
zdz
z , whereC is the circle 1z i .
(M/J 2011)
5. Using Cauchy’s integral formula, evaluate4 3
( 1)( 2)C
zdz
z z z
, Where ‘C ’ is the
circle3
2z . (M/J 2010)
6. Evaluate using Cauchy’s residue theorem,
C
z zdz
z z
2 2
sin cos
( 1)( 2), where C: z 3 .
(N/D 2011)
7. Evaluate2
1
( 1) ( 2)C
zdz
z z
, where C is the circle 2z i using Cauchy’s
residue theorem.
Contour Integral of Types – I ,II &III
1. Evaluate 2
02 cos
d
using contour integration.
(N/D 2009), (M/J 2010), (N/D 2009) ,(A/M 2011)
2. Evaluate 2
0
0cos
da b
a b
, using contour integration. (N/D 2011)
3. Evaluate 2 2
0
sin, 0
cosd a b
a b
. (N/D 2012)
4. Evaluate 2
2
0
, 0 11 2 sin
dx
x x
. (M/J 2009)
5. Evaluate, by contour integration, 2
2
01 2 sin
d
a a
, 0 1a . (M/J 2011)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 10
6. Evaluate
2
4 2
2
10 9
x xdx
x x
using contour integration. (M/J 2010),(A/M 2011)
7. Evaluate
2
2 2 2 2
x dx
x a x b
, using contour integration, where 0a b .
(M/J 2009)
8. Evaluate 2 2
1 4
dx
x x
using contour integration. (N/D 2010)
9. Evaluate using contour integration
xdx
x
2
22
1. (N/D 2011)
10. Evaluate
3
2 20
dx
x a
, 0a using contour integration. (N/D 2009)
11. Evaluate2 2
0
cos mxdx
x a
, using contour integration. (M/J 2012)
12. Evaluate 2 2 2 2
cos x dx
x a x b
using contour integration, where 0a b .
(N/D 2011)
Taylor’s and Laurent’s Series
1. Expand 2
1( )
( 2)( 3)
zf z
z z
as a Laurent’s series in the region 2 3z .
(A/M 2011),(M/J 2011),(N/D 2011)
2. Find the Laurent’s series of
2
2
1( )
5 6
zf z
z z valid in 2 3z .(M/J 2009)
3. Evaluate
1( )
1 3f z
z z
in Laurent series valid for the regions 3z and
1 3z . (N/D 2009),(M/J 2012)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 11
4. Find the Laurants’s series expansion of
1( )
1f z
z z
valid in the regions
1 1, 1 1 2z z and 1 2z . (N/D 2011)
5. Find the Laurent’s series of 7 2
( )( 1)( 2)
zf z
z z z
in 1 1 3z . (M/J 2010)
6. Find the residues of
2
2 2( )
1 2
zf z
z z
at its isolated singularities using
Laurent’s series expansions. Also state the valid region. (N/D 2010),(N/D 2012)
Unit – V (Laplace Transform)
Laplace Transform of Periodic Function
1. Find the Laplace transform of, for 0
( )2 , for 2
t t af t
a t a t a
, ( 2 ) ( )f t a f t .
(M/J 2009),(N/D 2009),(A/M 2011)
2. Find the Laplace transform of the following triangular wave function given by
, 0( )
2 , 2
t tf t
t t
and ( 2 ) ( )f t f t . (M/J 2010),(M/J 2012)
3. Find the Laplace transform of , 0 1
( )0, 1 2
t tf t
t
and ( 2) ( )f t f t for 0t .
(N/D 2011)(AUT)
4. Find the Laplace transform of square wave function defined by
1, in 0( )
1, in 2
t af t
a t a
with period 2a . (N/D 2009)
5. Find the Laplace transform of
( ) , 0
, 2
f t t a
a t a
and ( 2 ) ( )f t a f t for all t . (N/D 2010)
6. Find the Laplace transform of a square wave function given by
aE t
f ta
E t a
for 02
( )
for 2
, and f t a f t ( ) ( ) . (N/D 2011)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 12
7. Find the Laplace transform of the Half wave rectifier
sin , 0 /( )
0, / 2 /
t tf t
t
and ( 2 / ) ( )f t f t for all t .(N/D 2012)
Initial and Final Value Theorem& Other Simple Problems
1. Find the Laplace transform of 2cos 3
tte t
. (M/J 2009)
2. Verify initial and final value theorems for ( ) 1 (sin cos )t
f t e t t .
(M/J 2010),(N/D 2010),(M/J 2012)
3. Find cos cosat bt
Lt
. (A/M 2011),(N/D 2012)
4. Find the Laplace transform ofat bt
e e
t
. (M/J 2012)
5. Find the Laplace transform of 4
0
sin 3
t
te t t dt
. (M/J 2009)
6. Evaluate tte t dt
2
0
cos using Laplace transforms. (N/D 2011),(M/J 2012)
7. Find the inverse Laplace transform of 2
1
1 4s s . (M/J 2009)
8. Find 2 2
1
2 2
1ln
s aL
s s b
. (N/D 2011)(AUT)
Laplace Transform Using Convolution Theorem
1. Using Convolution theorem
1 1L
s a s b
. (A/M 2011)
2. Apply convolution theorem to evaluate
1
22 2
sL
s a
. (M/J 2010),(M/J 2012)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 13
3. Find
2
1
22
4
sL
s
using convolution theorem. (N/D 2012)
4. Find the inverse Laplace transform of
2
2 2 2 2
s
s a s b using convolution theorem.
(N/D 2010),(M/J 2011)
5. Using convolution theorem find the inverse Laplace transform of 2
1
1 1s s .
(N/D 2009),(N/D 2011)(AUT)
6. Find
Ls s
1
2
1
4using convolution theorem. (N/D 2011)
Solving Differential Equation By Laplace Transform
1. Solve 2
23 2 2
d x dxx
dt dt , given 0x and 5
dx
dt for 0t using Laplace
transform method. (A/M 2011),(N/D 2012)
2. Solve the equation 9 cos 2 , (0) 1y y t y and 12
y
using Laplace
transform. (M/J 2009)
3. Solve the differential equation 2
2sin 2 ;
d yy t
dt (0) 0, (0) 0y y by using Laplace
transform method. (N/D 2009)
4. Using Laplace transform solve the differential equation 3 4 2t
y y y e with
(0) 1 (0)y y . (M/J 2010),(N/D 2010)
5. Solve the differential equation 2
23 2
td y dyy e
dt dt
with (0) 1y and (0) 0y ,
using Laplace transform. (M/J 2012)
6. Solve 23 2 4 , (0) 3, (0) 5
ty y y e y y , using Laplace transform.
(N/D 2011)(AUT)
Engineering Mathematics 2013
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 14
7. Solve, by Laplace transform method, the equation 2
22 5 sin
td y dyy e t
dt dt
,
(0) 0, (0) 1y y . (M/J 2011)
8. Solve d y dy
y tdt dt
2
24 4 sin , if
dy
dt 0 and y 2 when 0t using Laplace
transforms. (N/D 2011)
----All the Best----