Printed Pages: 02 Sub Code: RAS203

Paper Id: 1 9 9 2 2 3 Roll No.

B. TECH
(SEM-II) THEORY EXAMINATION 2017-18
ENGINEERING MATHEMATICS - II
Time: 3 Hours Total Marks: 70
Note: Attempt all Sections. If require any missing data, then choose suitably.

SECTION A

1. Attempt all questions in brief. 2 x 7 = 14

(a) 
2 x
Determine the differential equation whose set of independent solutions is e , xe , x e .
x x

x
Solve: ( D  1) y  2e .
3
(b)
Prove that: Pn ( x)  (1) Pn ( x) .
n
(c)
s 8
(d) Find inverse Laplace transform of .
s  4s  5
2

(e) 
If L F ( t )  e 1 / s
s
 
, find L e  t F (3 t ) .

 
Solve: ( D  4 D   5) z  0 , where D  , D' 
2
(f) .
x y
(g) Classify the equation: z xx  2 x z xy  (1  y 2 ) z yy  0 .

SECTION B

2. Attempt any three of the following: 7 x 3 = 21

(a) Solve ( D 2  2 D  4) y  e x cos x  sin x cos 3 x.

2  3  x 2  3 cos x 
(b) Prove that: J 5 / 2 ( x)   2  sin x  .
x  x  x 
(c) Draw the graph and find the Laplace transform of the triangular wave function of period 2 given by
 t, 0t 
F (t )   .
2  t ,   t  2
 2t , 0  t  1
(d) Obtain half range cosine series for e x the function f (t )  
2(2  t ),1  t  2
u u
(e) Solve by method of separation of variables:   2u ; u ( x, 0)  10e  x  6e 4 x .
t x

SECTION C
3. Attempt any one part of the following: 7x1=7
(a) Solve the simultaneous differential equations:
d 2x dx d2y dy
2
 4  4 x  y and 2
 4  4 y  25x  16e t .
dt dt dt dt
(b) Use variation of parameter method to solve the differential equation x 2 y   xy   y  x 2 e x .
4. Attempt any one part of the following: 7x1=7
(a) State and prove Rodrigue’s formula for Legendre’s polynomial.
(b) Solve in series: 2 x(1  x) y   (1  x) y   3 y  0 .

5. Attempt any one part of the following: 7x1=7

2
s
(a) State convolution theorem and hence find inverse Laplace transform of .
( s  a )(s 2  b 2 )
2 2

d3y d2y dy
(b) Solve the following differential equation using Laplace transform 3
 3 2
 3  y  t 2et
dt dt dt
where y (0)  1, y (0)  0 and y (0)  2 .

6. Attempt any one part of the following: 7x1=7

 x,   x  0
(a) Obtain Fourier series for the function f ( x)   and hence show that
 x , 0 x 
1 1 1 2
   ......  .
12 3 2 5 2 8
2z 2z
(b) Solve the linear partial differential equation: 2  2  sin x cos 2 y .
x xy

7. Attempt any one part of the following: 7x1=7

(a) A string is stretched and fastened to two points l apart. Motion is started by displacing the string
x
in the form y  A sin from which it is released at time t  0 . Find the displacement of any
l
point at a distance x from one end at any time t.
(b) A rectangular plate with insulated surfaces is 8 cm wide and so long compared to its width that it
may be considered infinite in length without introducing an appreciable error. If the temperature
x
along one short edge y  0 is given by u ( x, 0)  100 sin , 0 x8
8
while the two long edges x  0 and x  8 as well as the other short edge are kept at 0  C . Find the
temperature u ( x, y ) at any point in steady state.
 KAS203
Printed Pages: 02 Sub Code: KAS203
Paper Id: 199243 Roll No.

B. TECH.
(SEM II) THEORY EXAMINATION 2018-19
MATHEMATICS-II
Time: 3 Hours Total Marks: 100
Note: Attempt all Sections. If require any missing data; then choose suitably.

SECTION A

1. Attempt all questions in brief. 2 x 10 = 20

QNo. Question Marks CO
a. Find the P.I of + 4𝑦 = 𝑠𝑖𝑛2𝑥 2 1
b. Solve simultaneous equations = 3𝑦 , = 3𝑥 2 1
c. Find the volume of solid generated by revolving the circle 𝑥 + 𝑦 = 25 2 2
about 𝑦-axis.

A
d.  5 2 2

PT
Evaluate Γ  −  . where Γ is gamma function
 2
e.
U
Find the Fourier constant 𝑎 of 𝑓(𝑥) = 𝑥 , −𝜋 ≤ 𝑥 ≤ 𝜋 2 3
G
f. 2 3

.2
Discuss the convergence of sequence 𝑎 = .
AR

62
g. Show that complex function 𝑓(𝑧) = 𝑧 is analytic. 2 4
M

h. Define Conformal mapping. 2 4

5.
KU

11
i. Evaluate (𝑥 − 𝑖𝑦)𝑑𝑧 along the path 𝑦 = 𝑥. 2 5
j. Find residue of 𝑓(𝑧) = at 𝑧 = 0
5.
2 5
Y

( )
|4
JA

49
VI

SECTION B
9:
R

2. Attempt any three of the following:

:5
D

QNo. Question Marks CO

08

a. Use Frobenius method to solve 9𝑥(1 − 𝑥) − 12 + 4𝑦 = 0 10 1

9
01

b. Apply Dirichlet integral to find the volume of an octant of the sphere 10 2

𝑥 + 𝑦 + 𝑧 = 25.
-2

c. 𝑥 0<𝑥<2 10 3
ay

Find half range sine series of 𝑓(𝑥) =

4−𝑥 2<𝑥 <4
M

d. Show that 𝑢 = 𝑥 − 6𝑥 𝑦 + 𝑦 is harmonic function. Find complex 10 4

5-

function 𝑓(𝑧) whose 𝑢 is a real part.

e. 10 5
|1

1
Expand f ( z ) = in regions
( z − 1)( z − 2)
(i) 1 < |𝑧| < 2 (ii) 2 < |𝑧|

SECTION C
3. Attempt any one part of the following:
QNo. Question Marks CO
a. Solve + 𝑦 = 𝑡𝑎𝑛𝑥 by method of variation of parameter. 10 1

Page 1 of 2
DR VIJAY KUMAR GUPTA | 15-May-2019 08:59:49 | 45.115.62.2
 KAS203
b. Solve 𝑥 − 2(𝑥 + 𝑥) + (𝑥 + 2𝑥 + 2)𝑦 = 0 by Normal Form. 10 1

4. Attempt any one part of the following:

QNo. Question Marks CO
a. ΓmΓn 10 2
Prove that β (m, n) = where Γ is gamma function
Γ ( m + n)
b. ∞ 1 π 10 2
Use Beta and Gamma function to solve 
0 1+ x 4
dx 0
2
cot θ dθ

5. Attempt any one part of the following:

QNo. Question Marks CO
a. Find the Fourier series of 𝑓(𝑥) = 𝑥𝑠𝑖𝑛𝑥 , −𝜋 ≤ 𝑥 ≤ 𝜋 10 3
b. State D’ Alembert’s test. Test the series 1 + + + ……+ + 10 3
⋯ … … ….

A
6. Attempt any one part of the following:

PT
QNo. Question Marks CO
a.
U
𝑤ℎ𝑒𝑛 𝑧 ≠ 0, 𝑓(𝑧) = 0 𝑤ℎ𝑒𝑛 𝑧 = 0. Prove 10 4
( )
G
Let 𝑓(𝑧) =

.2
AR

that Cauchy Riemann satisfies at 𝑧 = 0 but function is not differentiable

62
at 𝑧 = 0.
M

5.
b. Find Mobius transformation that maps points 𝑧 = 0, −𝑖, 2𝑖 into the 10 4
KU

11
points 𝑤 = 5𝑖, ∞, − respectively.

5.
Y

|4
JA

7. Attempt any one part of the following:

49
VI

QNo. Question Marks CO

9:
R

a. sin z 10 5
Using Cauchy Integral formula evaluate  (z dz where c is
:5
D

2
c + 25) 2
08

circle |𝑧| = 8
b. 10 5
9

Apply residue theorem to evaluate

01

( )( )
-2
ay
M
5-
|1

Page 2 of 2
DR VIJAY KUMAR GUPTA | 15-May-2019 08:59:49 | 45.115.62.2
 Printed Page: 1 of 2
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 following 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 1
origins?
Q.1(b) State the criterion for linearly independent solutions of the homogeneous linear nth order 1
differential equation.
Q.1(c) 𝟏 𝒅𝒙 2
Evaluate: 𝟎 .
𝒍𝒐𝒈𝒙

Q.1(d) Find the volume of the solid obtained by rotating the ellipse 𝑥 9𝑦 9 about the 𝑥-axis. 2

Q.1(e) 𝟏 𝟏 3
Test the series ∑𝒏 𝟏 𝒏 𝐬𝐢𝐧 𝒏 .

Q.1(f) Find the constant term when 𝑓 𝑥 1 |𝑥| is expanded in Fourier series in the interval (-3, 3). 3
90

2
13
Q.1(g) Show that 𝑓 𝑧 𝑧 2𝑧̅ is not analytic anywhere in the complex plane. 4
_2

2.
P2

Q.1(h) Find the image of |𝑧 4

24
2𝑖|=2 under the mapping 𝑤 .
2E

5.
Q.1(i) Expand 𝑓 𝑧 𝑒 in a Laurent series about the point 𝑧 2. 5
.5
P2

17
Q.1(j) Discuss the nature of singularity of 𝑎𝑡 𝑧 𝑎 𝑎𝑛𝑑 𝑧 ∞. 5
Q

|1
5 0

SECTION -B Attempt any three of the following questions Marks (3×10=30) CO

0:

Q.2(a) 1
:0

Solve: 3𝑥 𝑒 , 4 3𝑦 sin 2𝑡.

09

Q.2(b) 𝒙𝒑 𝟏 𝝅 2
Assuming Γ𝑛 Γ 1 𝑛 𝜋 𝑐𝑜𝑠𝑒𝑐 𝑛𝜋, 0 𝑛 1, show that 𝟎
𝒅𝒙 ;0 𝑝 1.
𝟏 𝒙 𝐬𝐢𝐧 𝒏𝝅
2

Q.2(c) 𝒙 𝒙𝟐 𝒙𝟑 𝒙𝟒 3
02

Test the series ⋯⋯⋯

𝟏.𝟐 𝟑.𝟒 𝟓.𝟔 𝟕.𝟖
-2

Q.2(d) 𝒆𝒚 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒙 4

If 𝑓 𝑧 𝑢 𝑖𝑣 is an analytic function, find 𝑓 𝑧 in term of 𝑧 if 𝒖 𝒗 when
08

𝐜𝐨𝐬𝐡 𝒚 𝐜𝐨𝐬 𝒙
𝑓 .
6-

Q.2(e) 5
|1

Evaluate by contour integration: 𝑒 cos 𝑛𝜃 sin 𝜃 𝑑𝜃 ; 𝑛𝜖𝐼 .

QP22EP2_290 | 16-08-2022 09:00:50 | 117.55.242.132

 Printed Page: 2 of 2
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

SECTION -C Attempt any one of the following questions Marks (1×10=10) CO

Q.3(a) Use the variation of parameter method to solve the differential equation 1
𝐷 1 𝑦 2 1 𝑒
Q.3(b) Solve: 1 𝑥 1 𝑥 𝑦 4 cos log 1 𝑥 . 1

SECTION -C Attempt any one of the following questions Marks (1×10=10) CO

Q.4(a) The arc of the cardioid 𝑟 𝑎 1 cos 𝜃 included between 𝜃 is rotated about the 2
line . Find the area of surface generated.
Q.4(b) Evaluate ∭ 𝑥𝑦𝑧 sin 𝑥 𝑦 𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧 , the integral being extended to all positive values of 2
the variables subject to the condition 𝑦 𝑧 .

SECTION -C Attempt any one of the following questions Marks (1×10=10) CO

90

2
Q.5(a) Test for convergence of the series ⋯⋯⋯ 3

13
! ! !
_2

Q.5(b) 1 , 𝜋 𝑥 0 3

2.
P2

Obtain Fourier series for the function 𝑓 𝑥 .

24
1 , 0 𝑥 𝜋
2E

5.
Hence deduce that ⋯⋯⋯ .
.5
P2

17
Q

|1

SECTION -C Attempt any one of the following questions Marks (1×10=10) CO

0

Q.6(a) Prove that 𝑤 maps the upper half of the z-plane onto upper half of the w-plane. What is 4
5

the image of the circle |𝑧| 1 under this transformation?

0:

Find a bilinear transformation which maps the points 𝑖, 𝑖, 1 of the 𝑧 plane into 0, 1, ∞ of the 4
:0

Q.6(b)
𝑤 𝑝𝑙𝑎𝑛𝑒 respectively.
09
2
02

SECTION -C Attempt any one of the following questions Marks (1×10=10) CO

-2

Q.7(a) Evaluate ∮ 𝑑𝑧 , 𝑤ℎ𝑒𝑟𝑒 𝑐 𝑖𝑠 𝑖 |𝑧| 𝑖𝑖 |𝑧 1| 𝑖𝑖𝑖 |𝑧| 2. 5

08

Q.7(b) Find the Taylor’s and Laurent’s series which represent the function when 𝑖 |𝑧| 2 5
6-
|1

𝑖𝑖 2 |𝑧| 3 𝑖𝑖𝑖 |𝑧| 3.

QP22EP2_290 | 16-08-2022 09:00:50 | 117.55.242.132

Printed Pages:04 Sub Code: BAS203

Paper Id: 238023 Roll No.

B. Tech.
(SEM II) THEORY EXAMINATION 2022-23
ENGINEERING MATHEMATICS-II
Time: 3 Hours Total Marks: 70
le;% 03 ?k.Vs iw.kkZad% 70
Note:
1. Attempt all Sections. If require any missing data; then choose suitably.
2. The question paper may be answered in Hindi Language, English Language or
in the mixed language of Hindi and English, as per convenience.
uksV% 1- lHkh iz”uks dk mRrj nhft,A fdlh iz”u esa] vko”;d MsVk dk mYys[k u gksus dh
fLFkfr esa mi;qDr MsVk Lor% ekudj iz”u dks gy djsaA
2- iz”uksa dk mRrj nsus gsrq lqfo/kkuqlkj fgUnh Hkk’kk] vaxzsth Hkk’kk vFkok fganh ,oa vaxzsth
dh fefJr Hkk’kk dk iz;ksx fd;k tk ldrk gSA
SECTION A

1. Attempt all questions in brief. 2 x 7 = 14

fuEu lHkh iz”uksa dk la{ksi esa mRrj nhft,A
(a) d90
Solve: ( D 3  2 D 2  3D ) y  e x , D 

2
.
dx

13
_2
हल क िजये:

2.
d
P2

( D 3  2 D 2  3D ) y  e x , D 

24
.
dx
3E

5.
.5
(b) Explain the first shifting property of the Laplace transform with example.
P2

ला लास प रवतन के थम थानांतरण गुण को उदाहरण सिहत समझाइये।

17
Q

(c) Discuss the convergence of sequence {un}, where u n  sin (1 / n).

|1
अनु म {un} के अिभसरण पर चचा कर, जहां u n  sin (1 / n).
1

(d) Show that the function f ( z ) | z | 2 is not analytic at origin.

:2

दखाएँ क फ़ं शन f ( z ) | z | 2 मूल प से िव ेषणा मक नह है।

46

(e) e1 / z
:

Classify the singularity of f ( z )  .

08

z
e1 / z
3

f (z)  क एकलता का वग करण क िजए

02

z
(f) 1
-2

Find the inverse Laplace transform of F ( s)  .

s  2s  2
2
07

1
F (s)  का ु म ला लास पांतरण ात क िजए।
1-

s  2s  2
2
|3

(g) 2z  6
Find the invariant points of the transformation w  .
z7
2z  6
ांसफॉमशन w  के अप रवतनीय बदु ात क िजए।
z7

QP23EP2_290 | 31-07-2023 08:46:21 | 117.55.242.132

 SECTION B

2. Attempt any three of the following: 7 x 3 = 21

fuEu esa ls fdlh rhu iz”uksa dk mRrj nhft,A
(a) Solve the following differential equation:
िन िलिखत अवकल समीकरण को हल कर:
2
2 d y dy
x 2
 2 x  12 y  x 3 log x.
dx dx
(b) Find the Laplace transform of the function f ( x)  x 3 sin x . Hence, prove that


e
x 3
x sin xdx  0.
0

f ( x)  x 3 sin x फ़ं शन का ला लास पांतरण ात क िजए। िस कर क



e x 3 sin xdx  0 .
x

0
(c) Test the convergence of following series:
िन िलिखत ृंखला के अिभसरण का परी ण कर:
1 x x2
   ..., Where x is a real number.
1.2.3 4.5.6 7.8.9
1 x x2
   ..., जहाँ x एक वा तिवक सं या है।
1.2.3 4.5.6 7.8.9
90

2
(d) x 3 y 5 ( x  iy )

13
Show that the function f(z) defined by f ( z )  , z  0, f (0)  0 is
_2
x 6  y 10

2.
not analytic at the origin even though it satisfies Cauchy-Riemann equations at
P2

24
the origin.
3E

x 3 y 5 ( x  iy )

5.
दखाएँ क f ( z )  , z  0, f (0)  0 ारा प रभािषत फ़ं शन f(z) मूल

.5
x 6  y 10
P2

बदु पर िव ेषणा मक नह है, य िप यह मूल बदु पर कॉची-रीमैन समीकरण को संतु 17

Q

करता हो।
|1
(e) sin 2 z
Using Cauchy-integral formula, evaluate  dz , where C is a
1

C ( z  3)( z  1)
2
:2

rectangle with vertices at 3  i,  2  i.

46

sin 2 z
कॉची-इं टी ल सू का उपयोग करके  dz का मू यांकन कर। जहाँ पर C,
:
08

C
( z  3)( z  1) 2
3  i,  2  i शीष वाला एक आयत है।
3
02
-2

SECTION C
07

3. Attempt any one part of the following: 7x1=7

1-

fuEu esa ls fdlh ,d iz”u dk mRrj nhft,A

|3

(a) Solve the following differential equation by the variation of parameters:

ाचल प रवतन िविध ारा िन िलिखत अवकल समीकरण को हल कर:
d2y
 y  cosec x.
dx 2

(b) Solve the differential equation by the changing the independent variable:
वतं चर को बदलकर अवकल समीकरण को हल कर:

QP23EP2_290 | 31-07-2023 08:46:21 | 117.55.242.132

 d 2 y dy
x   4 x 3 y  8 x 3 sin x 2 .
dx 2 dx

4. Attempt any one part of the following: 7x1=7

fuEu esa ls fdlh ,d iz”u dk mRrj nhft,A
(a) State convolution theorem of the Laplace transforms. Hence, find inverse
1
Laplace transform of 2 .
s ( s  1) 2
1
ला लास ांसफॉम के convolution theorem िलिखए। 2 . का ु म ला लास
s ( s  1) 2
पांतरण ात क िजएA
(b) Using Laplace transform, solve the following differential equation:
ला लास ांसफॉम का उपयोग करके , िन िलिखत अवकल समीकरण को हल कर:
d2y
 y  6 cos 2 x, y (0)  3 & y ' (0)  1.
dx 2

5. Attempt any one part of the following: 7x1=7

fuEu esa ls fdlh ,d iz”u dk mRrj nhft,A
(a) Find a Fourier series to represent f ( x)  x  x 2 ,  x   . Hence, show that
1 1 1 1 2 90

2
    ...  .

13
12 2 2 32 4 2 12
_2
clear

2.
f ( x)  x  x 2 ,  x   . को करने के िलए फू रयर ृंखला ात क िजये। तथा
P2

24
1 1 1 1 2
3E

दशाइए क     ...  .

5.
12 2 2 32 4 2 12

.5
P2

(b) 17
Find the half range cosine series for the function f ( x )  ( x  1) 2 in the interval
Q

|1
(0,1). Hence, prove that
1 1 1 1 2
1

    ...  .
:2

12 32 5 2 7 2 8
46

clear
अंतराल (0,1) म फ़ं शन f ( x )  ( x  1) 2 के िलए हाफ रज कोसाइन ृंखला ात कर।
:
08

1 1 1 1 2
तथा िस कर क     ...  .
12 32 5 2 7 2
3

8
02
-2

6. Attempt any one part of the following: 7x1=7

07

fuEu esa ls fdlh ,d iz”u dk mRrj nhft,A

1-

(a) Determine an analytic function f(z)=u+iv in terms of z whose real part u(x,y) is
e x ( x cos y  y sin y ) and f(1)=e.
|3

z के पद के प म एक िव ेषणा मक फ़ं शन f(z)=u+iv िनधा रत क िजये

िजसका वा तिवक भाग u(x,y)= e x ( x cos y  y sin y ) है और f(1)=e है।

(b) Find the bilinear transformation which maps the points z  0,  1, i onto
w  i, 0, . Also, find the image of the unit circle |z|=1.
ऐसा ि रे खीय प रवतन ात क िजये जो बदु z  0,  1, i को w  i, 0, . , पर मैप
करता है। इकाई वृ |z|=1 क इमेज भी ात क िजये।

QP23EP2_290 | 31-07-2023 08:46:21 | 117.55.242.132

7. Attempt any one part of the following: 7x1=7
fuEu esa ls fdlh ,d iz”u dk mRrj nhft,A
(a) 7z  2
Expand f ( z )  in the following regions:
z  z 2  2z
3

7z  2
िन िलिखत े म f ( z )  3 का िव तार क िजयेA
z  z 2  2z
(i ) 0 | z | 1 (ii ) 1 | z | 2 (iii ) | z | 2.

(b) 
a d
Using contour integration, evaluate the real integral a
0
2
 sin 2 
, a  0.


a d
contour integration का उपयोग करके , वा तिवक समाकलन a
0
2
 sin 2 
, a  0. का

आकलन कर।

90

2
13
_2

2.
P2

24
3E

5.
.5
P2

17
Q

|1
1
:2
: 46
08
3
02
-2
07
1-
|3

QP23EP2_290 | 31-07-2023 08:46:21 | 117.55.242.132

 Printed Page: 1 of 2
Subject Code: BAS203

0Roll No: 0 0 0 0 0 0 0 0 0 0 0 0 0

BTECH
(SEM II) THEORY EXAMINATION 2023-24
ENGINEERING MATHEMATICS-II
TIME: 3 HRS M.MARKS: 70

Note: 1. Attempt all Sections. If require any missing data; then choose suitably.

SECTION A

1. Attempt all questions in brief. 2 x 7 = 14

Q no. Question Marks CO
a. 𝑑2𝑦 2 1
Find Particular integral of 2 + 4y = sin 2𝑥.
𝑑𝑥
b. Find the complementary function of (D2+a2)y = 0 2 1
c. Find the Laplace transform of f(t) = t4 e2t. 2 2
d. Find the constant term if the function f(x) = x+x2 is expanded in Fourier 2 3
series defined in (-1, 1).
e. 𝒛𝟐 2 4
Find the Residue of (𝒛−𝟏)(𝒛−𝟐)𝟐 at z = 2.
f. 𝑒 2𝑧 2 5
∫𝑐 dz where c is the circle |𝑧| = 2
(𝑧+1)5

2
g. Define Laurent’s series. 2 5

13
90

2.
_2

24
SECTION B
P2

5.
2. Attempt any three of the following: 7 x 3 = 21
4E

.5
Q no. Question Marks CO
17
P2

a. 𝑑2𝑦 𝑑𝑦 7 1
Using variation of parameter method, solve x2
|1
+2x -12y = 0.
Q

𝑑𝑥 2 𝑑𝑥
AM

b. Use convolution theorem to find the inverse Laplace transform of 7 2

𝟏
𝟐 𝟐 𝟐 .
(𝒔 +𝒂 )
2

c. 2 6 14 7 3
Test the convergence of the series 1+ x + x2 + x3 +…
:3

5 9 17
Show that the function u = ½ log (x2 + y2) is harmonic .Find its 7
01

d. 4
harmonic conjugate.
9:

e. Evaluate the following integral using Cauchy Integral formula 7 5

24

4−3𝑧 3
∫𝑐 dz, where C is circle |𝑧| =
𝑧(𝑧−1)(𝑧−2) 2
20
g-

SECTION C
Au

3. Attempt any one part of the following: 7x1=7

6-

Q no. Question Marks CO

|0

a. Solve the following differential equation 7 1

( D2 -4D +4)y = 8x2 e2x sin 2𝑥.
b. Solve simultaneous differential equation : 7 1
𝒅
D2x-4Dx+4x = y, D2y+4Dy +4y= 25x+16et, where D = .
𝒅𝒕

1|Page
QP24EP2_290 | 06-Aug-2024 9:01:32 AM | 117.55.242.132
 Printed Page: 2 of 2
Subject Code: BAS203

0Roll No: 0 0 0 0 0 0 0 0 0 0 0 0 0

BTECH
(SEM II) THEORY EXAMINATION 2023-24
ENGINEERING MATHEMATICS-II
TIME: 3 HRS M.MARKS: 70

4. Attempt any one part of the following: 7x1=7

Q no. Question Marks CO
a. 1−𝑐𝑜𝑠𝑡 7 2
Find the Laplace transform of f(t) = 2 .
𝑡
b. Using Laplace transformation solve the following differential 7 2
equation
y”+ 4y’+ 4y = 6𝑒 −𝑡 , if y(0) = -2, y’(0) = 8

5. Attempt any one part of the following: 7x1=7

Q no. Question Marks CO
a. Find the half range Fourier sine series f(x) defined over the range 0<x<4 7 3
𝐱,𝟎 < 𝐱 < 𝟐
as f(x) = {
𝟒 − 𝐱, 𝟐 < 𝐱 < 𝟒
b. Test for the convergence of the series 7 3
𝑥 1.3 2 1.3.5 3
1+ + 𝑥 + 𝑥 +......,x>0

2
2 2.4 2.4.6

13
90

2.
_2

6. Attempt any one part of the following: 7x1=7

24
P2

Q no. Question Marks CO

5.
a. Show that ex (x cosy – y siny) is a harmonic function. Find the 7 4
4E

.5
analytic function for which ex (x cosy – y siny) is imaginary part.
17
P2

b. Define analytic function and show that f(z) = z |𝒛| is not analytic 7 4
|1
anywhere.
Q

AM

7. Attempt any one part of the following: 7x1=7

2

Q no. Question Marks CO

:3

𝑧
a. Expand f(z) = is Laurent series valid for 7 5
01

(𝑧−1)(2−𝑧)
𝑎)|𝑧 − 1| > 1 and b) 0 < |𝑧 − 2| < 1
9:

b. 𝒆𝒛
Evaluate ∫ (𝒛−𝟏)(𝒛−𝟒) dz where C is the circle |z| = 2 by using Cauchy’s 7 5
24

integral formula.
20
g-
Au
6-
|0

2|Page
QP24EP2_290 | 06-Aug-2024 9:01:32 AM | 117.55.242.132