PTU B.Tech Mechanical Engineering 5th Semester May 2019 70601 MATHEMATICS III Question Papers

PTU Punjab Technical University B-Tech May 2019 Question Papers 5th Semester Mechanical Engineering (MECH)

Roll No.
Total No. of Pages : 02
Total No. of Questions : 09
B.Tech.(ME) (2011 Onwards) (Sem.?5)
MATHEMATICS-III
Subject Code : BTAM-500
M.Code : 70601
Time : 3 Hrs. Max. Marks : 60
INSTRUCTION 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) Expand x ? x2 in half range series in interval (0, ) upto first three terms.

b) Find Laplace transform of t3 e?3t
4s 15

c) Find the inverse Laplace Transform of

2

16s 25

d) Describe the conditions required for the Fourier expansion.

e) Express f (x) = x4 + 3x3 ? x2 + 5x ? 2 in terms of Legendre polynomials.

f) Evaluate
3
x j (x) dx
o



g) Form the partial differential equation z = (x + y) (x2 ? y2)
2
z
z

h) Solve
z 0 given that x = 0, z = ey and
1.
2
x

x


i) Define harmonic function

j) Check whether f (z) | xy | is analytic at origin or not?
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SECTION-B
1
2.
Obtain Fourier series to represent f (x)
2

( ? x) , 0 < x < 2.
4
3.
Solve the initial value problem
19
8
y ? 5y + 4y = e2t, y (0)
y (0)

12
3
4.
Find the frobenius series solution about x = 0 of equation

(1 ? x2) y ? 2xy + 6y = 0
5.
Find bilinear transformation which maps the points z = 1, i, ?1 onto the points w = i, 0, ?i.
Hence find

a) The image of | z | < 1

b) Invariant points of transformation.
6.
Solve (x2 ? yz)p + (y2 ? zx) q = z2 ? xy

SECTION-C
7.
State and prove convolution theorem. Apply convolution theorem to evaluate



s
1
L

2
2 2
(s a )


3
z
8.
Find the residue of f (z)
at its poles and hence evaluate
f (z) dx
4


(z 1) (z 2) (z 3)
C
where c is circle | z | = 2.5
2
2
u
u
9.
Solve the Laplace equation

0 subject to the conditions
2
2
x

y

sin n x


u (0, y) = u (x, y) = u (x, 0) = 0, u (x, a)
l


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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