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Model Question Paper
First Semester M.Tech Degree Examination
Branch: Mechanical (Machine Design)
MMD 101: ADVANCED ENGINEERING MATHEMATICS
Time: 3 hours Max. Marks: 100
Instructions: Allquestions carry equalmarks
1. a) Express 1
0
)1( dxxxpnm
in terms of Gamma function and hence evaluate
1
0
1035)1( dxxx .
b) Evaluate 2/
0
tan
d .
c) Evaluate 1
0
)1(log dxx
x nm .
OR
2. a) Show that +
=1
1
0)()( dxxPxP nm if m n, and
12
2))((
1
1
2
+=
+
ndxxPn if m = n
b) Show that )]()([2
1)( 11
' xJxJxJ nnn + = .
c) Show that xx
xJ cos2
)(2
1
=
.
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3. a) Show that the expression A(i, j, k) is a tensor if its inner product with an arbitrary
tensorjl
kB is a tensor.
b) Show that the kronecker delta is a mixed tensor of order two.
c) Show that prr
q
q
p
x
x
x
x=
.
OR
4. (a) A convariant tensor has components xy, 2y-z2 , xz in rectangular co- ordinates.
Find its covariant components in spherical co-ordinates.(b) Show that the expression A(i,j,k) is a tensor if its inner product with anarbitrary tensor B k
jl is a tensor
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MMD 101 Advanced Engineering Mathematics
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7/28/2019 MMD 101
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5. a) Find the Fourier transform of
>
=
1,0
1,1)(
2
x
xxxF . Hence evaluate
dxx
x
xxx
2cos
sincos3 .
b) Solve the boundary value problem ,2
22
2
2
xua
tu
=
t > 0, x > 0 where
0,0)0,(,0,0)0,( >== xxuxxu t
0,0),(lim
,),0( =
= ttxux
ttu
OR
6. a) Using Laplace transform solve the integral equation =t
dsstsyty0
))((42
b) Find the integral equation corresponding to the boundary value problem0)1()0(,0)()( ===+ yyxyxy .
207. Convert the differential equation
2)0('1)0(,sin5)(2)('3)( ===+ yyxxyxyxy
OR
8. (a) Show that the integral equation ++x
dttytx0
1)()( is equalent to the
differential equation 0)0('1)0(,0)(3)('2)( === yyxyxxyxy
(b) Solve =dx
yd +
xt
dttytx0
2)()(2cos3 given y(0) = 1
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9. a) Prove that the Pfaffian differential equation 0. =drX is integrable if and only if0. =XcurlX
b) Reduce the equation 02 2
22
2
2
=
+
+
y
z
yx
z
x
zto canonical form and hence solve it.
OR
10. a) Show how you will use Schwarz-Christoffel transformation to map the semi-
infinite strip enclosed by the real axis and the lines 1=u of the w-plane into theupper half of the z-plane.
b) Discuss the transformation
z
zw1
+= .
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MMD 101 Advanced Engineering Mathematics