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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) = ꢂ^{2} ꢃ^{−}^{3}^{ꢄ}ꢅꢆꢇℎꢈꢂ

Understand

Understand

B) Express ꢁ(ꢂ) in terms of Heaviside's unit step function and hence find its

4

Laplace transform where ꢁ(ꢂ) = ꢉ^{ꢊ}_{ꢅ }

ꢋꢅꢂ, 0 < ꢂ < ꢌ

ꢆꢇꢂ, ꢂ > ꢌ

Find ꢀ{ꢁ(ꢂ)}, where ꢁ(ꢂ) = ꢍ^{ꢄ} ∫_{ꢑ}^{ꢄ }

sin 3ꢎ

ꢏꢐ

C)

Understand

Evaluation

4

4

ꢎ

∞

ꢄ

D)

By using Laplace transform evaluate ∫ ꢃ

ꢖ ꢏꢂ

ꢑ

Q. 2 Attempt the following.

A)

12

4

Application

ꢕ^{ꢗ}+4)

Find ꢀ^{−}^{1}ꢙꢁ (ꢅ)ꢚ, where ꢁ (ꢅ) = ꢊꢋꢂ−1_{ ꢒ}^{ꢕ}^{+}_{2}^{3 }

B)

Application

Application

4

4

ꢖ

C) Using Laplace transform solve ꢛ^{′}^{′} ꢜ ꢝꢛ^{′} ꢞ ꢍꢛ = ꢟꢍꢃ^{−}^{2}^{ꢄ}; ꢛ(0) = ꢍ,

′

ꢛ

(0) = 6

Q. 3 Attempt any three.

12

4

A) _{E}_{x}_{p}_{r}_{e}_{s}_{s}_{ ꢁ}_{(}_{ꢂ}_{)}_{ =}_{ ꢉ}ꢟ, 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

ꢭ^{ꢗ }

1−ꢫ^{ꢬ }

ꢪ

ꢩ^{ꢗ }

∞

ꢕꢧꢨꢩꢄ

ꢄ(ꢩ^{ꢗ}+ꢄ^{ꢗ})

ꢡ

ꢏꢂ =_{ 2 }

∫

ꢮ

ꢑ

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Find the Fourier transform of ꢁ(ꢠ) = ꢯ^{ꢟ}^{ ꢜ}^{ ꢠ}^{2}^{, }

∞

ꢰꢓꢔꢕꢰ−ꢕꢧꢨꢰ

ꢰ^{ꢱ }

ꢰ

3ꢡ

that ∫ ꢒ

ꢖ coꢣ _{2} ꢏꢠ = ꢜ_{ 1}_{ꢲ }

ꢑ

Find Fourier sine transform of 5ꢃ^{−}^{2}^{ꢰ} ꢞ ꢍꢃ^{−}^{ꢳ}^{ꢰ }

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)ꢶ = ꢠ^{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

ꢅ

If the function ꢁ(ꢴ) = (ꢠ^{2} ꢞ ꢈꢠꢛ ꢞ ꢷꢛ^{2}) ꢞ ꢆ(ꢊꢠ^{2} ꢞ ꢏꢠꢛ ꢞ ꢛ^{2}) 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 = ꢃ^{ꢰ}(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

ꢧꢨꢡꢹ^{ꢗ}+ꢓꢔꢕꢡꢹ^{ꢗ }

(ꢹ−1)^{ꢗ}(ꢹ−2)

Evaluate ∮_{ꢺ }

ꢏꢴ, where ꢻ is the circle |ꢴ| = ꢝ.

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