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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE  
End Semester Examination – Summer 2019  
Course: B. Tech in All Branches  
Sem: II  
Subject Name: Engineering Mathematics II  
Subject Code: BTMA201  
Duration: 3 Hr.  
Max Marks:60  
Date:13/05/2019  
Instructions to the Students:  
1
2
3
4
.
.
. Assume suitable data wherever necessary and mention it clearly.  
. Figures to the right indicate full marks.  
Solve ANY FIVE questions out of the following.  
Use of non-programmable scientific calculators is allowed.  
Q. 1 Solve Any Three of the following.  
12  
ꢁ
2ꢁ  
A)  
If arg(ꢀ + 1) = 6 and arg(ꢀ − 1) = 3 , find z.  
B) If ꢂ = 1 + ꢃ , ꢄ = 1 − ꢃ ꢅꢆꢇ cot ꢈ = ꢉ + 1 , prove that  
ꢉ + ꢂ)ꢊ + (ꢉ + ꢄ)ꢊ = (ꢂ − ꢄ) sin(ꢆꢈ) ꢋꢌꢍꢎꢋꢊ(ꢈ).  
(
Show that all the roots of (ꢉ + 1)6 + (ꢉ − 1)6 = 0 are given by – ꢃ cot ꢏ(  
2ꢐꢑꢒ)ꢁ  
ꢓ ,  
C)  
ꢒ
2
k=0,1,2,3,4,5.  
D) If tan(ꢈ + ꢃ∅) = cos ꢂ + ꢃ sin ꢂ , prove that  
ꢊ
ꢁ
ꢁ
I.  
ꢈ = + 4.  
2
ꢒ
ꢁ
ꢕ
II.  
∅ = logꢔ tan ꢏ4 + 2ꢓ.  
2
Q.2 Solve Any Three of the following.  
12  
ꢖ
ꢗ
ꢙꢚꢛ ꢗ  
A)  
Solve ꢖꢘ − ꢒꢑꢘ = (1 + ꢉ)ꢎꢘ sec ꢜ.  
B) Solve ꢜꢇꢉ − ꢉꢇꢜ + log ꢉ ꢇꢉ = 0.  
ꢘꢝ  
Find the orthogonal trajectories of ꢞꢝ +  
ꢗꢝ  
ꢟꢝꢑꢠ  
C)  
= 1 , ꢡℎꢎꢢꢎ ꢣ is a parameter.  
D)  
A constant electromotive force E volts is applied to a circuit  
containing a constant resistance R ohm in series and a constant  
inductance L henries. If the initial current is zero , show that the  
current builds up to half its theoretical maximum in (L log2)/R sec.  
P.T.O.  
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Q.3) Solve Any Three of the following.  
12  
A)  
ꢖꢝ  
ꢗ
ꢖꢗ  
Solve ꢖ ꢝ − ꢤ ꢖꢘ + ꢜ = ꢉꢎꢘ sin ꢉ.  
ꢘ
ꢖꢝꢗ  
ꢖꢗ  
B)  
C)  
D)  
Solve ꢖ ꢝ − ꢥ ꢖꢘ + ꢤ5ꢜ = ꢎ2ꢘ + sin ꢉ + ꢉ.  
ꢘ
ꢖꢝꢗ  
ꢖꢗ  
ꢧ
Solve ꢖ ꢝ + ꢦ ꢖꢘ + ꢤꢜ = ꢎꢔ .  
ꢘ
2 ꢖꢝꢗ  
ꢖ
ꢖꢗ  
ꢝ − ꢦꢉ ꢖꢘ + 5ꢜ = ꢉ2 sin(log ꢉ).  
Solve ꢉ  
ꢘ
Q.4 Solve Any Two of the following.  
12  
12  
12  
A)  
Find the Fourier series for ꢨ(ꢉ) = √1 − cos ꢉ in the range (0,ꢤꢩ).Prove that  
ꢒ
ꢒ
∞
ꢊ
=
∑
.
ꢫꢒ 4ꢊꢝꢪꢒ  
2
B) Obtain the Fourier series for ꢨ(ꢉ) given by  
2
ꢘ
1
1
+
−
, −ꢩ ≤ ꢉ ≤ 0  
, 0 ≤ ꢉ ≤ ꢩ .  
ꢝ
ꢁ
ꢒ ꢒ ꢒ  
Hence deduce that ꢒꢝ + 3ꢝ + ꢬꢝ + ⋯ … … . = 8 .  
ꢁ
ꢨ
(ꢉ) = {  
2
ꢘ
ꢁ
2
4
ꢊꢝꢁ  
C)  
If ꢨ(ꢉ) = ꢤꢉ − ꢉ2 ꢃꢆ 0 ≤ ꢉ ≤ ꢤ , show that ꢨ(ꢉ) = 3 − ∑ꢊ∞ꢫꢒ  
ꢝ cos(ꢆꢩꢉ) .  
Q.5 Solve Any Three of the following.  
A) Find the directional derivatives of ∅ = ꢎꢤꢉ cos ꢜꢀ at (0,0,0) in the direction of the  
ꢁ
tangent to the curve ꢉ = ꢅ sin ꢭ , ꢜ = ꢅ cos ꢭ , ꢀ = ꢅꢭ at ꢭ = 4.  
B) Find the cosine of the angle between the normals to the surfaces ꢉ2ꢜ + ꢀ = ꢦ  
and ꢉ log ꢀ − ꢜ2 = ꢮ at the point of intersection ꢯ(−1,ꢤ,1).  
C)  
Find ꢋꢰꢢꢱ ꢲ⃗, where ꢲ⃗ = ∇(ꢉ3 + ꢜ3 + ꢀ3 − ꢦꢉꢜꢀ).  
D) If ꢢ⃗ = ꢉꢳ + ꢜꢴ + ꢀꢵꢶ , find ꢢ⃗. ∇∅ ꢨꢌꢢ ∅ = ꢉ3 + ꢜ3 + ꢀ3 − ꢦꢉꢜꢀ.  
Q. 6 Solve Any Two of the following.  
A)  
Evaluate ∮ ꢲ⃗ . ꢇꢢ⃗ where C is the square formed by the lines ꢜ =  
ꢷ
1 ꢅꢆꢇ ꢉ = ±1, and ꢲ⃗ = (ꢉ2 + ꢉꢜ)ꢳ + (ꢉ2 + ꢜ2)ꢴ .  
±
B)  
C)  
Verify the Green’s theorem for ∫ꢷ ꢸ(ꢉꢜ + ꢜ2)ꢇꢉ + ꢉ2ꢇꢜ} where C is bounded  
by ꢜ = ꢉ and dꢜ = ꢉ2.  
Evaluate ∬ꢹ ꢸꢤꢉ2ꢜꢇꢜꢇꢀ − ꢜ2ꢇꢀꢇꢉ + ꢮꢉꢀ2ꢇꢉꢇꢜ} over the curved surface of  
the cylinder ꢜ2 + ꢀ2 = 9 , bounded by ꢉ = 0 and ꢉ = ꢤ.  
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** End ***  
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