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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 = 1 + , = 1 ꢅꢆꢇ cot = + 1 , prove that
ꢉ + ꢂ) + (ꢉ + ꢄ) = (ꢂ − ꢄ) sin(ꢆꢈ) ꢋꢌꢍꢎꢋ(ꢈ).
(
Show that all the roots of (ꢉ + 1)6 + (ꢉ 1)6 = 0 are given by ꢃ cot(
2ꢐꢑꢒ)ꢁ
,
C)
2
k=0,1,2,3,4,5.
D) If tan(ꢈ + ꢃ∅) = cos ꢂ + ꢃ sin ꢂ , prove that
I.
ꢈ = + 4.
2
II.
∅ = log tan ꢏ4 + 2.
2
Q.2 Solve Any Three of the following.
12
ꢙꢚꢛ ꢗ
A)
Solve = (1 + ꢉ)ꢎ sec ꢜ.
B) Solve ꢜꢇꢉ ꢉꢇꢜ + log ꢉ ꢇꢉ = 0.
ꢝ
Find the orthogonal trajectories of +
ꢝ
ꢑꢠ
C)
= 1 , ꢡℎꢎꢢꢎ 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.
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A)
ꢝ
ꢖꢗ
Solve + = ꢉꢎ sin.
ꢗ
ꢖꢗ
B)
C)
D)
Solve + ꢤ5ꢜ =2 + sin ꢉ +.
ꢗ
ꢖꢗ
Solve + + ꢤꢜ = .
2 ꢖꢗ
ꢖꢗ
ꢦꢉ + 5ꢜ =2 sin(log ꢉ).
Solve ꢉ
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
=
.
4ꢊꢪꢒ
2
B) Obtain the Fourier series for ꢨ(ꢉ) given by
2
1
1
+
, −ꢩ 0
, 0 ≤ ꢉ .
ꢒ ꢒ ꢒ
Hence deduce that + 3 + + ⋯ … … . = 8 .
(ꢉ) = {
2
2
4
ꢁ
C)
If ꢨ(ꢉ) = ꢤꢉ2 ꢃꢆ 0 ≤ ꢉ , show that ꢨ(ꢉ) = 3 ꢒ
cos(ꢆꢩꢉ) .
Q.5 Solve Any Three of the following.
A) Find the directional derivatives of =cos ꢜꢀ at (0,0,0) in the direction of the
tangent to the curve = sin ꢭ , ꢜ = cos ꢭ , = ꢅꢭ at = 4.
B) Find the cosine of the angle between the normals to the surfaces2ꢜ + = ꢦ
and log ꢀ2 = at the point of intersection ꢯ(−1,ꢤ,1).
C)
Find ꢋꢰꢢꢱ, where = ∇(ꢉ3 +3 +3 ꢦꢉꢜꢀ).
D) If ꢢ⃗ = ꢉꢳ + ꢜꢴ + ꢀꢵꢶ , find ꢢ⃗. ∇∅ ꢨꢌꢢ = ꢉ3 +3 +3 ꢦꢉꢜꢀ.
Q. 6 Solve Any Two of the following.
A)
Evaluate . ꢇꢢ⃗ where C is the square formed by the lines =
1 ꢅꢆꢇ = ±1, and = (ꢉ2 + ꢉꢜ)ꢳ + (ꢉ2 +2)ꢴ .
±
B)
C)
Verify the Green’s theorem for ꢸ(ꢉꢜ +2)ꢇꢉ +2ꢇꢜ} where C is bounded
by = and d =2.
Evaluate ꢸꢤꢉ2ꢜꢇꢜꢇꢀ2ꢇꢀꢇꢉ + ꢮꢉꢀ2ꢇꢉꢇꢜ} over the curved surface of
the cylinder2 +2 = 9 , bounded by = 0 and =.
*
** End ***
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