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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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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