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Code No: R2021011 R20 SET - 1II B. Tech I Semester Regular/Supplementary Examinations, January - 2023
MATHEMATICS - III
(Com to all branches, Except EEE &FE)
Time: 3 hours Max. Marks: 70
Answer any FIVE Questions, each Question from each unit
All Questions carry Equal Marks
~~~~~~~~~~~~~~~~~~~~~~~~~
UNIT-I
a) Apply Stoke’s theorem, to evaluate ∮ ( + + ) where C is the curve
of intersection of the sphere + + = and + = .
1 [7M]
b) Show that the vector ( − ) + ( − ) + ( − ) is irrotational and [7M]
find its scalar potential.
OR
a) Evaluate ∬ . where = 12 − 3 + 2 and S is the portion of the
2 [7M]
plane + + = 1 included in the first octant.
b) Prove that (
) = ( + 1)
. [7M]
UNIT-II
3 a) Solve the differential equation ! + 9 = $ using Laplace Transforms given [9M]
"
that (0) = 1, ′(0) = , ((/2) = 1.
b) Find the inverse Laplace transform of *+
,
[5M]
-.
OR
a) Solve using Laplace transforms y − 16y = 30 sin t , given that y(0) = 0,
(01)
y′(0) = −18, y″(π) = 0, y′′′(π) = −18
4 [7M]
b) Find ? @A" B
, CD FH
G
[7M]
E
, 0 ≤ ≤ ( L
UNIT-III
Obtain the Fourier series for the function I() = J
2(-x, ( ≤ ≤ 2(
5 a) [7M]
= ∑P
M B
and show that
N
QB (
B)
b) Using Fourier integral show that [7M]
2(b 2 − a 2 ) ∞ λ sin λ x
e− ax − e− bx =
π
0 ( λ 2
+ a 2
)( λ 2
+ b 2
)
d λ , (a, b > 0) .
OR
1 of 2
|'''|'|'|''|''||'|||
, Code No: R2021011 R20 SET - 1
IR
− 1 < < 0 L
Find the Fourier series of the function I() = J
+ 2 IR
0 < < 1
6 a) [7M]
.
And hence deduce that 1 − T + + − U +. . . . =
B B B M
V
b) 1 for 0 ≤ x ≤ π [7M]
Express f ( x) = as a Fourier sine integral and hence evaluate
0 for x > π
∞
1 − cos(πλ )
0 λ
sin xλ d λ .
UNIT-IV
7 a) Form the Partial differential equation from I(, + ) = 0 by elimination of [7M]
b) Solve (W + X ) = + .
arbitrary function.
[7M]
OR
8 a) Form the Partial differential equation from I( + + , + + ) = 0 by [7M]
elimination of arbitrary function
b) Solve ()W − ()X = ( − ) [7M]
UNIT-V
9 a) Solve YZ [ − 4 YZ [ + 4 YZ [ = 2 ( 3 + 2) [7M]
Y! Z Y[ Y] Y!Y]
b) By the method of separation of variables, find the solution of the P.D.E [7M]
2 Y" + 3 Y! = 3F, F(, 0) = 4^
! .
YE YE
OR
+ 2F = Y! ,
10 a) Solve, using method of separation of variables, the P D E YE Y E [7M]
Y]
given conditions are F = 0 = 1 + ^
T] when x=0 for all values of y.
YE
Y!
− 2 Y!Y] + Y] = ( 2 + 2)
b) Solve Y [ Y [ Y [ [7M]
Y!
2 of 2
|'''|'|'|''|''||'|||
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