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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) = ꢂ2 ꢃ−3ꢄꢅꢆꢇℎꢈꢂ  
Understand  
Understand  
B) Express ꢁ(ꢂ) in terms of Heaviside's unit step function and hence find its  
4
Laplace transform where ꢁ(ꢂ) = ꢉꢊꢅ  
ꢋꢅꢂ, 0 < ꢂ < ꢌ  
ꢆꢇꢂ, ꢂ > ꢌ  
Find ꢀ{ꢁ(ꢂ)}, where ꢁ(ꢂ) = ꢍꢄ ∫ꢑꢄ  
sin 3ꢎ  
ꢏꢐ  
C)  
Understand  
Evaluation  
4
4
ꢎ
∞
−ꢄ ꢒ1−ꢓꢔꢕ2ꢄ  
ꢄ
D)  
By using Laplace transform evaluate ∫ ꢃ  
ꢖ ꢏꢂ  
ꢑ
Q. 2 Attempt the following.  
A)  
12  
4
ꢕꢗ  
Using convolution theorem find ꢀ−1 ꢉ(  
Application  
ꢗꢘ  
ꢕꢗ+4)  
Find ꢀ−1ꢙꢁ (ꢅ)ꢚ, where ꢁ (ꢅ) = ꢊꢋꢂ−1 ꢒꢕ+23  
B)  
Application  
Application  
4
4
ꢖ
C) Using Laplace transform solve ꢛ′′ ꢜ ꢝꢛ′ ꢞ ꢍꢛ = ꢟꢍꢃ−2ꢄ; ꢛ(0) = ꢍ,  
′
ꢛ
(0) = 6  
Q. 3 Attempt any three.  
12  
4
A) Express ꢁ(ꢂ) = ꢉꢟ, 0 ≤ ꢠ ≤ ꢌ  
Evaluation  
Application  
as a Fourier sine integral and hence  
0
, ꢠ > ꢌ  
∞
1−ꢓꢔꢕꢡꢢ  
ꢡ
ꢣꢤꢥ ꢌꢦ ꢏꢦ = 4.  
deduce that ∫ꢑ  
ꢢ
B) Using Parseval's identity for cosine transform, prove that  
4
ꢭꢗ  
1−ꢫꢬ  
ꢪ
ꢩꢗ  
∞
ꢕꢧꢨꢩꢄ  
ꢄ(ꢩꢗ+ꢄꢗ)  
ꢡ
ꢏꢂ = 2  
∫
ꢮ
ꢑ
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Find the Fourier transform of ꢁ(ꢠ) = ꢯꢟ ꢜ ꢠ2,  
0, ꢆꢁꢆꢁ|ꢠ|ꢠ||>≤ꢟꢟ. Hence prove  
∞
ꢰꢓꢔꢕꢰ−ꢕꢧꢨꢰ  
ꢰꢱ  
ꢰ
3ꢡ  
that ∫ ꢒ  
ꢖ coꢣ 2 ꢏꢠ = ꢜ 1ꢲ  
ꢑ
Find Fourier sine transform of 5ꢃ−2ꢰ ꢞ ꢍꢃ−ꢳꢰ  
Form the partial differential equation by eliminating arbitrary function ꢁ  
2
2
2
from ꢁ(ꢠ ꢞ ꢛ ꢞ ꢴ, ꢠ ꢞ ꢛ ꢞ ꢴ ) = 0  
Solve ꢠꢴ(ꢴ2 ꢞ ꢠꢛ)ꢵ ꢜ ꢛꢴ(ꢴ2 ꢞ ꢠꢛ)ꢶ = ꢠ4  
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
ꢅ
If the function ꢁ(ꢴ) = (ꢠ2 ꢞ ꢈꢠꢛ ꢞ ꢷꢛ2) ꢞ ꢆ(ꢊꢠ2 ꢞ ꢏꢠꢛ ꢞ ꢛ2) is analytic,  
find the values of the constants ꢈ, ꢷ, ꢊ and ꢏ.  
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 ꢐ = ꢃꢰ(ꢠꢊꢋꢅꢛ ꢜ ꢛꢅꢆꢇꢛ) satisfies the Laplace's  
equation. Also find the coresponding analytic function.  
ꢹ
ꢹꢗ+2ꢹ+ꢳ  
+4  
Evaluate ∮ꢺ  
ꢏꢴ, where ꢻ is the circle |ꢴ ꢞ ꢟ ꢜ ꢆ| = ꢍ.  
ꢕ
ꢧꢨꢹ  
Find the residues of ꢁ(ꢴ) = ꢹ  
at its poles inside the circle |ꢴ| = ꢍ.  
ꢼꢽs ꢹ  
ꢕ
ꢧꢨꢡꢹꢗ+ꢓꢔꢕꢡꢹꢗ  
(ꢹ−1)ꢗ(ꢹ−2)  
Evaluate ∮ꢺ  
ꢏꢴ, where ꢻ is the circle |ꢴ| = ꢝ.  
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