QUESTION BANK
PARTIAL DIFFERENTIATION
1. If u =
e xyz , show that
 3u
 1  3xyz  x 2 y 2 z 2 e xyz
xyz


2. If x = r cos , y = r sin 
 2r  2r   2r 

Pr ove that
.

x 2 y 2  xy 
2
3. Verify Euler’s theorem for
u=ax2 + 2hxy + by2
1
4. If u = sin-1
1
x4  y4
1
1
x6  y6
2
 2u
 2u
2  u
Evaluate x
 2 xy
y
x 2
xy
y 2
5. If x =  sin  , y   sin  , z  z
2
Show that
(u, y, z )

 ( x, y , z )
6. If x = eu cos , y = eu sin , show that
 (u, )
 e  2u
 ( x, y )
7. If u, v are functions of x and y then show that
(u,  ) ( x, y )
1
( x, y ) (u,  )
8. Find the approximate value of
(0.98)
2
 (2.01) 2  (1.94) 2

1
2
9. Expand ex cos y in powers of x and y as far as the terms of third degree.
10. Find the dimensions of rectangular box (without top) with a given volume so that the material
used is minimum.
11. In any ABC find the maximum value of cos A cos B cos C by Lag range’s method.
12. Find the volume of greatest rectangular parallelepiped that can be inscribed inside the ellipsoid
x2 y2 z 2


1
a2 b2 c2
13. The temp T and any pt (x, y, z) in space is T = 400 xyz2. Find highest temperature on surface of
unit sphere x2 + y2 + z2 = 1.
14. Evaluate
  xy( x  y)dx dy over the region bounded by the line y = x and the curve y = x .
2
15. Find by double integration the area lying inside the circle r = a sin  and out side the cardiod
r=a(1 – cos ).
16. Change the order of integration
a
a2  y2
a
0
  f ( x, y) dx dy
17. Evaluate

xy  y 2 dx dy where s is a  with vertices (0,0), (10,1) and (1,1).
s
18. Evaluate
1
1 x 2
1 x 2  y 2
0
0
0
 
19. Evaluate
 xyz dz dy dx
dx dy dz
   ( x  y  z  1)
3
where R is bounded by the planes x = 0, y = 0, z = 0 and x+y+z=1
R
20. Change the order of integration in the integral
a
2
a2  y2
0
y
  x dx dy and then evaluate it.
21. Express as a single integral
a
2
x
a
a2  x2
0
0
a
2
0
  x dy dx  
 x dy dx and evaluate it.
Functions of complex variables
1.
Prove that  = e x ( x sin y  y cos y ) is harmonic.
2.
Evaluate
3.
Verify CAUCHY’S theorem for the function – 3z2 + i z – 4
4.
Show that the transformation w 
5.
4u + 3 = 0.
Find the bilinear transformation which maps
 zdz
from z=0 to z = 4 + 2i along the curve c: z = t2 + i t
c
2z  3
maps the circle x2 + y2 – 4x = 0 onto the straight line
z 3
a. the points z = 1, i,-1, into the pts w = 0, 1 
b. the points z=0, -1, , into the pts w = -1, -2, i
i
6.
Evaluate
 sin
h 5 z dz
0
7.
Evaluate
e2z
e ( z  1) 4 dz , where c : |z| = 3
8.
Evaluate
e iz
e z 3 dz , where c : |z| = 1
9.
Find the region of convergence of

 n !2
n
n 1
10.
Expand f(z) =
11.
Evaluate
1
in a LAURENT SERIES for |z| < 3
z 3
2
d
 3  2 cos   sin 
0
2
12.
Show that
cos 3

 5  4 cos d  12
0

sin x

dx 
x
2
0

13.
Show that
14.
Find the fixed points of the Transform w 
15.
Prove that

cos h ax
 cos h x dx 
0
16.

 a 
2 cos  
 2 
2z  5
z4
, |a| < 1
z 2 dz
Prove that
2 i c z 2  4
1
1
17.
Evaluate
 ze
2 zx
dz
0
18. If f(z) is a regular function of z.
2
 2
2 
2
Prove that  2  2  f ( z )  4 f ' ( z )
y 
 x
19.
z8  z 4  2
Locate the singularities of
( z  1)3 (3z  z ) 2
20.
Find the Laurent series of following f(z) with centre 0.
(i) z-5 sin z (ii)
21.
z2 e1/z
Obtain Laurent’s series
hence evaluate
z
2
C
1  cos z
z3
(iii)
for the function f(z) =
1
at the isolated singularity and
z sin hz
2
dz
; where C is the circle |z – 1| = 2
sin hz
VECTOR CALCULUS
1.
2.
Define scalar point function and vector point function.
Find the angle between the tangents to the curve
3.
Define directional derivative. Find the directional derivative of the scalar function  = (x2+y2+z2)-1/2
at a point (3,1,2) in the direction of the vector (yz, zx, xy)
4.
For what values of b and c will F = (y2 + 2cxz)i + y (bx + cz) j + (y2 + cx2) k is irrotational. Find
5.
6.
the scalar  s . t F =  .
Define (i) Line integral, (ii) Circulation
Sate and prove Green’s Lemma in the plane.
7.
Suppose F is the force field as F  x 3i  yj  zk . Find the work done by F along the line from
8.
(1,2,3) to (3,5,7).
State stoke’s circulation theorem and Guass divergence theorem.
9.
A vector function F is defined as F  xy2 i  yz 2 j  zx 2 k




r  t 2i  2t j  t 3k at the points t = 1 and t = 2






Evaluate
Evaluate




 xz dydz  x
2
2



 F. ds over the surface given by x
S
10.

2

+ y2 + z2 = 1

y  z 3 dzdx  (2 xy  y 2 z )dxdy
S
Where S is the surface enclosing the region bounded by the hemisphere x2 + y2 + z2 =4 above the
XY plane.
11.
Find the value of the surface integral
 (2 x
S
2
y dy dz  y 2 dz dx  y x z 2 dx dy )
Where S is the curved surface of the cylinder y2 + z2 = 9 bounded by the planes x = 0, x = 2.
12.
Use Green’s Leema to evaluate
 (2 xdy  3 ydx) around the square with vertices (0,2), (2,0),
(–2,0), (0,–2).
13.
Find
 
F
 . n ds where
S




F  (2 x  3z )i  ( xz  y ) j  ( y 2  2 z )k and S is the surface of the sphere having centre at
14.
(3,-1,2) and radius 3.
A particle moves along the curve x = 4 cos t y = 4 sin t
Find the velocity and acceleration at time t = 0 and t =
15.
16.
17.
z=6t

2
, Find also the magnitudes of the
velocity and acceleration of the particle at any time t.
Find the greatest rate of increase of  = xyz2 at the point (1, 0, 3).
Find the angle between the surfaces x2 + y2 + z2 = 9 and x2 + y2 – z – 3 = 0 at the point (2, –1, 2).
Compute the directional derivative of x2 + y2 + 4xz at (1, –2, 2) is the direction of the vector
2iˆ  j  k
18.
Define Solenoidal vector and Irrotational vector.
19.
If F  xy iˆ  yz ˆj  zx kˆ , then prove that 2 F  0
20.
Find the D.D. of the scalar function   ( x 2  y 2  z 2 ) 1 / 2 at the point (3, 1, 2) in the direction



of the vector (yz, zx, xy).
Laplace Transform
Q1.
(a)
Find the inverse Laplace transform of
s
s  4a 4
4
(b)
Solve the initial value problem (IVP)
y’’ + ay’ – 2a2y = 0
y(o) = 6
y’(o) = 0

Q2.
sin t

dt 
t
2
0

(a)
Using L.T. Show that
(b)
Using convolution, find the inverse of
s
(c)
s2
2
 w2

2
t ,   t  
  t ,   t  2
Plot the 2 - periodic functions, given by f(t) = 
Its laplace transform.
Q3.
Q4.
Q5.
(a)
s 2  2s  3
Find the inverse L.T. of
s( s  3)( s  2)
(b)
Find the L.T. of exponential function f(t) = ea+
(a)
Find the inverse L.T. of
(b)
find the L.T. of t2 U (t–3)
(a)
s 2  2s  3
Find the inverse L.T. of
s( s  3)( s  2)
(b)
Find the L.T. of exponential function f (t )  e at
(c)
Solve the simultaneous equation
s4
s ( s  1)( s 2  4)
(b 2  3) x  4 y  0
x  ( D 2  1) y  0
for
x y
Q6.
t>0, give that
dy
dx
 0 and
 2at
dt
dt
when t = 0
e  t sin t
t
(a)
Find the L.T. of
(b)
Solve, using L.T. technique, the difference equation
and find
d2y
dy
 3  2 y  4 x  e 3s
2
dx
dx
Where y(0) = 1
Q7.
&
y’(0) = –1
(a)
Find
(i)
(b)
Solve the following boundary value problem using the L.T. y" (t )  9 y (t )  cos 2t
 s3 
L1  4
4 
s  a 
(ii)


s
L 2
2 2 
 (s  a ) 
 
y (0)  1, y   1
2
Q8
Q9.
(a)
Find
(i)
2  t 2 , 0  t  2

L f (t ), where f(t) – 6, 2  t  3
2t  5, 3  t  

(ii)
 3 s 1

L  2 2
e 3s 
 s ( s  4)

1
(b)
State and prove the second shifting theorem of L.T.
(a)
Solve
y"ty' y  1
y(0) = 1
y’(0) = 2 using L.T.
(b)
Solve, y” + y’– 2y = 2 sin t cos t, if 0 < t < 2
&
3 sin2t – cos 2t, if t > 2;
y(0) = 1, y’(0), y’(0) = 0
(c)
Using convolution theorem, find the value of
 1 
L1 

s s  4 
Q10
(a)
 e 4 3 s 
5/ 2 
 (s  4) 
Evaluate L1 
(b)
t 2 for 0  t  1
4t for t  1
A function f(t) is given by f(t) = 
Express it in the terms of unit step function and find its L.T.
Q11.
Q12.
Evaluate the following
(a)
 sin at 
L
 . Does
 t 
(b)
s  1

L1  log

s 

(c)
Lsin 2t U (t   )
(d)

 2
L1  tan 1  2
s

(e)

 s  3 
L1  cot 1 
 
 2 

(a)
Using L. T. show that
 cos at 
L
 exist?
 t 

 

 s2  b 
cos at  cos bt
1
dt

log
 2
0
2 
t
2s
s  a 
1
(b)
For the periodic function f(t) of period 4, defined by
3t ,
f (t )  
6,
(c)
0t 2
2t 4
find L[f(t)]
Using L.T. solve the following yn – 3y’+ 2y = 4e2t under conditions
y(0) = -3 and y’(0) = 5
Q14.
(a)
Find the L.T. of f(t) defined as f(t) = |t – 1| + |t + 1|, t  0
(b)
Solve d 2 y(dx 2  4 y  U ( x  2)


Where U is unit step function, y(0) = 0, y’(0) = 1
Q15.
(a)
Find the L.T. of the function
0t c
c  t  2c
t ,
(2c  t )' ,
f(t) = 
(b)
t


Show that L  e'
0
sin t  1
dt   cot 1 ( s  1)
t
 s
 cos t 
.
t . Hence find L

t


Q16
Find the Laplace transform of sin
Q17
Given L 2
Q18
Find the Laplace transform of
Q19
Express the following function interms of unit step function and find its Laplace transform.



t  1

, show that
  s 3 2
 1  1
  1 .
L
 t  s 2
t
sin at
dt .
t
0

0, 0  t  1

f (t )  t  1, 1  t  2
1, 2  t

Q20
If f(t) be a periodic function with period T, then prove that
T
1
L f (t ) 
e st f (t )dt
 sT 
1 e 0
Q21
Define unit step function and impulse function.
Q22
State and prove convolution theorem. Use this theorem to evaluate


s2
, a  b
L1  2
2
2
2 
 ( s  a )( s  b ) 
Q23
Solve the equation by the Laplace transform:
t
dy
 2 y   ydt  sin t , y(0)  0, y' (0)  1
dt
0