PTU B.Tech Robotics Engineering 3rd Semester May 2019 63001 MATHEMATICS III Question Papers

PTU Punjab Technical University B-Tech May 2019 Question Papers 3rd Semester Robotics-ECE-EIE

Roll No.
Total No. of Pages : 02
Total No. of Questions : 09
B.Tech.(Automation & Robotics) (2011 & Onwards) (Sem.?3)
MATHEMATICS ? III
Subject Code : BTAR-301
M.Code : 63001
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1.
SECTION-A is COMPULSORY consisting of TEN questions carrying T WO marks
each.
2.
SECTION-B contains FIVE questions carrying FIVE marks each and students
have to attempt any FOUR questions.
3.
SECTION-C contains T HREE questions carrying T EN marks each and students
have to attempt any T WO questions.





SECTION?A
1.
Write briefly :

a) What is the laplace transform of unit impulse function?
t
e sin t

b) Find Laplace transform of

t

s 1

c) Find the inverse Laplace Transform of log



s 1

d) Show that Pn (1) = 1
x

e) Prove that Jn (x)
[J
(x) J
(x)]
1
1
2
n
n
n



f) Define analytic function.
2z 3

g) Show that the transformation w
maps the circle x2 + y2 ? 4x = 0 into straight
z 4
line 4u + 3 = 0

h) For conformal transformation w = z2 prove that angle of rotation at z = 1 + i is /4.
z sin z

i) Find the nature and location of singularities of

2
z
sin z

j) Find the sum of residues of f (z)
at its poles inside the circle | z | = 2.
z cos z
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SECTION?B
3

1
2.
Evaluate L
t



t
5s 3
3.
Find the inverse Laplace transform of

2
(s 1) (s 2s 5)
4.
Solve in series the equation
9x (1 ? x)y ? 12y + 4y = 0
5.
If u ? v = (x ? y) (x2 + 4xy + y2) and f (z) = u + iv is an analytic function of z = x + iy, find
f (z) in terms of z.
6.
Verify Cauchy's theorem by integrating eiz along the boundary of triangle with the
vertices at the points 1 + i, ?1 + i, ?1 ? i.


SECTION?C
7.
Define Harmonic function. Show that the function
u = e?2xy sin ( x2 ? y2)

is harmonic. Find the conjugate function v and express u + iv as an analytic function of z.

,
x
0 x 1
8.
Obtain the Fourier series for f (x)

(2 x), 1 x 2

1
9.
Expand
in region
[(z 1)(z 2)]

| z | < 1, 1 < | z | < 2, | z | > 2, 0 < | z ?1| < 1




NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any
page of Answer Sheet will lead to UMC against the Student.

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