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Subject Code: KAS203T
0Roll No: 0 0 0 0 0 0 0 0 0 0 0 0 0
BTECH
(SEM II) THEORY EXAMINATION 2021-22
ENGINEERING MATHEMATICS-II
Time:3 Hours Total Marks:100
Notes-
Attempt all sections and assume any missing data.
Appropriate marks are allotted to each question, answer accordingly.
SECTION -A Attempt all of followin
g
question in brief Marks (10×2=20) CO
Q.1(a) Find the differential equation which represents the family of straight lines passing through the
origins?
1
Q.1(b) State the criterion for linearly independent solutions of the homogeneous linear nth order
differential equation.
1
Q.1(c)
Evaluate:𝒅𝒙
𝒍𝒐𝒈𝒙
𝟏
𝟎 . 2
Q.1(d)
Find the volume of the solid obtained by rotating the ellipse 𝑥9𝑦9 about the 𝑥-axis. 2
Q.1(e)
Test the series ∑𝟏
𝒏
𝒏𝟏 𝐬𝐢𝐧𝟏
𝒏 . 3
Q.1(f)
Find the constant term when
𝑓
𝑥1|𝑥| is expanded in Fourier series in the interval (-3, 3). 3
Q.1(g)
Show that
𝑓
𝑧𝑧2𝑧
is not analytic anywhere in the complex plane. 4
Q.1(h)
Find the image of |𝑧2𝑖|=2 under the mapping 𝑤
. 4
Q.1(i)
Expand
𝑓
𝑧𝑒
in a Laurent series about the point 𝑧2 . 5
Q.1(j)
Discuss the nature of singularity of
𝑎𝑡 𝑧𝑎 𝑎𝑛𝑑 𝑧∞ . 5
SECTION -B Attempt any three of the following questions Marks (3×10=30) CO
Q.2(a) Solve:
3𝑥𝑒 ,
4
3𝑦sin2𝑡. 1
Q.2(b) Assuming Γ𝑛 Γ1𝑛𝜋 𝑐𝑜𝑠𝑒𝑐 𝑛𝜋,0𝑛1, show that 𝒙𝒑𝟏
𝟏
𝒙
𝟎𝒅𝒙𝝅
𝐬
𝐢𝐧
𝒏𝝅
; 0𝑝1 . 2
Q.2(c)
Test the series 𝒙
𝟏.𝟐𝒙𝟐
𝟑.𝟒𝒙𝟑
𝟓.𝟔𝒙𝟒
𝟕.𝟖⋯⋯⋯ 3
Q.2(d)
If
𝑓
𝑧𝑢𝑖𝑣 is an analytic function, find
𝑓
𝑧 in term of 𝑧 if 𝒖𝒗𝒆𝒚𝐜𝐨𝐬𝒙 𝐬𝐢𝐧𝒙
𝐜𝐨𝐬𝐡𝒚𝐜𝐨𝐬𝒙 when
𝑓
.
4
Q.2(e)
Evaluate by contour integration: 𝑒
cos𝑛𝜃sin𝜃 𝑑𝜃 ;𝑛𝜖𝐼 . 5
