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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE
End Semester Examination May 2019
Course: B. Tech
Sem: III
Subject Name: Engineering Mathematics-III
Subject Code: BTBSC301
Duration: 3 Hr.
Max Marks: 60
Date: 28-05-2019
Instructions to the Students:
1
2
. Solve ANY FIVE questions out of the following.
. The level question/expected answer as per OBE or the Course Outcome (CO) on
which the question is based is mentioned in ( ) in front of the question.
. Use of non-programmable scientific calculators is allowed.
3
4
. Assume suitable data wherever necessary and mention it clearly.
(
Level/CO) Marks
Q. 1 Attempt any three.
12
4
A) Find ꢀ{ꢁ(ꢂ)}, where ꢁ(ꢂ) =23ꢅꢆꢇℎꢈꢂ
Understand
Understand
B) Express ꢁ(ꢂ) in terms of Heaviside's unit step function and hence find its
4
Laplace transform where ꢁ(ꢂ) =ꢅ
ꢋꢅꢂ, 0 < ꢂ < ꢌ
ꢆꢇꢂ, ꢂ > ꢌ
Find ꢀ{ꢁ(ꢂ)}, where ꢁ(ꢂ) =ꢄ
sin 3ꢎ
ꢏꢐ
C)
Understand
Evaluation
4
4
1−ꢓꢔꢕ2ꢄ
D)
By using Laplace transform evaluate ꢃ
ꢖ ꢏꢂ
Q. 2 Attempt the following.
A)
12
4
ꢗ
Using convolution theorem find−1(
Application
ꢘ
+4)
Find 1ꢙꢁ (ꢅ)ꢚ, where (ꢅ) = ꢊꢋꢂ−1+23
B)
Application
Application
4
4
C) Using Laplace transform solve ꢝꢛ ꢍꢛ = ꢟꢍꢃ2; ꢛ(0) = ꢍ,
(0) = 6
Q. 3 Attempt any three.
12
4
A) Express() =ꢟ, 0 ≤ ꢠ ꢌ
Evaluation
Application
as a Fourier sine integral and hence
0
, ꢠ > ꢌ
1−ꢓꢔꢕꢡꢢ
ꢣꢤꢥ ꢌꢦ ꢏꢦ = 4.
deduce thatꢑ
B) Using Parseval's identity for cosine transform, prove that
4
ꢗ
1−ꢫꢬ
ꢗ
ꢕꢧꢨꢩꢄ
ꢄ(ꢩ+ꢄ)
ꢏꢂ = 2
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Find the Fourier transform of ꢁ(ꢠ) =2,
0, ||||>. Hence prove
ꢰꢓꢔꢕꢰ−ꢕꢧꢨꢰ
ꢱ
3ꢡ
that ꢒ
ꢖ coꢣ 2 ꢏꢠ = 1ꢲ
Find Fourier sine transform of 5ꢃ2 ꢍꢃꢰ
Form the partial differential equation by eliminating arbitrary function ꢁ
2
2
2
from ꢁ(ꢠ ꢞ ꢛ ꢞ ꢴ, ꢠ ) = 0
Solve ꢠꢴ(ꢴ2 ꢠꢛ)ꢵ ꢜ ꢛꢴ(ꢴ2 ꢠꢛ)ꢶ =4
Find the temperature in a bar of length two units whose ends are kept at zero
temperature and lateral surface insulated if the initial temperature is
ꢡꢰ ꢳꢡꢰ
ꢆꢇ ꢞ ꢝ ꢣꢤꢥ 2 .
2
If the function ꢁ(ꢴ) = (ꢠ2 ꢈꢠꢛ ꢞ ꢷꢛ2) ꢞ ꢆ(ꢊꢠ2 ꢏꢠꢛ ꢞ ꢛ2) is analytic,
find the values of the constants ꢈ, ꢷ, ꢊ and.
If ꢁ(ꢴ) is an analytic function with constant modulus, show that ꢁ(ꢴ) is
constant.
Find the bilinear transformation which maps the points = 0, ꢜꢆ, ꢜꢟ into
the points = ꢆ, ꢟ,0.
Prove that the function =(ꢠꢊꢋꢅꢛ ꢛꢅꢆꢇꢛ) satisfies the Laplace's
equation. Also find the coresponding analytic function.
+2ꢹ+ꢳ
+4
Evaluate ꢺ
ꢏꢴ, where is the circle |ꢴ ꢜ ꢆ| =.
ꢧꢨꢹ
Find the residues of ꢁ(ꢴ) = ꢹ
at its poles inside the circle |ꢴ| =.
ꢼꢽs ꢹ
ꢧꢨꢡꢹ+ꢓꢔꢕꢡꢹꢗ
(ꢹ−1)(ꢹ−2)
Evaluate ꢺ
ꢏꢴ, where is the circle |ꢴ| =.
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