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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE  
End Semester Examination May 2019  
Course: B. Tech  
Subject Name: Engineering Mathematics-III  
Max Marks: 60 Date: 28-05-2019  
Instructions to the Students:  
Sem: III  
Subject Code: BTBSC301  
Duration: 3 Hr.  
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  
ꢏꢂ = 2 ꢪ  
ꢗ  
ꢕꢧꢨꢩꢄ  
ꢄ(ꢩ+ꢄ)  
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undefined  
C) Find the Fourier transform of() = ꢜ ꢠ2, ꢆꢁ |ꢠ|. Hence prove  
Understand  
Understand  
4
4
0
, ꢆꢁ |ꢠ| > ꢟ  
ꢰꢓꢔꢕꢰ−ꢕꢧꢨꢰ  
ꢱ  
3ꢡ  
that ꢒ  
ꢖ coꢣ 2 ꢏꢠ = 1ꢲ  
D) Find Fourier sine transform of 5ꢃ2 ꢍꢃꢰ  
Q. 4 Attempt the following.  
12  
4
A) Form the partial differential equation by eliminating arbitrary function ꢁ  
Synthesis  
Analysis  
2
2
2
from ꢁ(ꢠ ꢞ ꢛ ꢞ ꢴ, ꢠ ) = 0  
B) Solve ꢠꢴ(ꢴ2 ꢠꢛ)ꢵ ꢜ ꢛꢴ(ꢴ2 ꢠꢛ)ꢶ =4  
4
4
C) Find the temperature in a bar of length two units whose ends are kept at zero Application  
temperature and lateral surface insulated if the initial temperature is  
ꢡꢰ ꢳꢡꢰ  
ꢆꢇ ꢞ ꢝ ꢣꢤꢥ 2 .  
2
Q. 5 Attempt Any three.  
12  
4
A) If the function ꢁ(ꢴ) = (ꢠ2 ꢈꢠꢛ ꢞ ꢷꢛ2) ꢞ ꢆ(ꢊꢠ2 ꢏꢠꢛ ꢞ ꢛ2) is analytic, Understand  
find the values of the constants ꢈ, ꢷ, ꢊ and.  
B) If ꢁ(ꢴ) is an analytic function with constant modulus, show that ꢁ(ꢴ) is  
Understand  
4
constant.  
C) Find the bilinear transformation which maps the points = 0, ꢜꢆ, ꢜꢟ into  
the points = ꢆ, ꢟ,0.  
D) Prove that the function =(ꢠꢊꢋꢅꢛ ꢛꢅꢆꢇꢛ) satisfies the Laplace's  
Understand  
Synthesis  
4
4
equation. Also find the coresponding analytic function.  
Q. 6 Attempt ANY TWO of the following.  
12  
6
+4  
A)  
B)  
Evaluation  
Understand  
Evaluate ꢺ  
ꢏꢴ, where is the circle |ꢴ ꢜ ꢆ| =.  
+2ꢹ+ꢳ  
ꢧꢨꢹ  
6
Find the residues of ꢁ(ꢴ) = ꢹ  
at its poles inside the circle |ꢴ| =.  
ꢼꢽs ꢹ  
ꢧꢨꢡꢹ+ꢓꢔꢕꢡꢹꢗ  
(ꢹ−1)(ꢹ−2)  
C)  
Evaluation  
6
Evaluate ꢺ  
ꢏꢴ, where is the circle |ꢴ| =.  
** End ***  
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