Reg. No. :

M.E. DEGREE EXAMINATION, JUNE 2010

First Semester

Engineering Design

MA9214 — APPLIED MATHEMATICS FOR ENGINEERING DESIGN

(Common to M.E. Computer Aided Design and M.E. Product Design and

Development)

(Regulation 2009)

Time : Three hours Maximum : 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. If X and Y are independent random variables with variance 2 and 3. Find the

variance of 4Y3X + .

2. If joint pdf of the random variables ( )YX, is given by the ( )22

),( yxkxyeyxf +−= ,

0>x , 0>y . Find the value of k.

3. Convert the differential equation 02 =′+′′ yy into a difference equation.

4. Distinguish between implicit and explicit method of solving heat equation.

5. Write the law of transformation for the tensor ijklmA .

6. Define inner product of two tensors with an example.

7. Find the extremal of the functional ∫ ′1

0

)(

x

x

dxyF .

Question Paper Code: J7707 4

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J7707 2

8. State the transversality conditions and its usage.

9. Show that discrete Fourier transform is linear.

10. What is meant by Fast Fourier transform?

PART B — (5 × 16 = 80 Marks)

11. (a) (i) Let X and Y be two discrete random variables with joint probability

density function

==

+=

otherwise0

4,3,2,1and2,1,32),(

yxyx

yxf

Find )(and)(),(),( YVarXVarYEXE . (8)

(ii) The joint pdf of two-dimensional random variable is

><<

=−

otherwise0

0,20,2

1

),(yxxe

yxfy

Find the p.d.f. of YX + . (8)

Or

(b) (i) The joint pdf of )Y(X, is given by

10,20,8

),(2

2 ≤≤≤≤+= yxx

xyyxf

= 0 otherwise

Find 1)YP(X1),P(X ≤+> . (8)

(ii) If the joint pdf of YX, is given by

( ) <<<<+

=otherwise,0

10,10,

yxyxyxf

Find the correlation coefficient between X and Y. (8)

12. (a) Solve the equation xxututux

u

t

uπsin)0,(,0),1(),0(,

2

2

===∂

∂=

∂∂

by using

Crank Nicholson method. Take 36

1and3

1== kh and obtain solution upto

two time levels. (16)

Or

(b) Solve the equation )10(10 222 ++−=∇ yxu over the square with sides

yxyx ==== 3,0 with 0=u on the boundary. Take mesh length = 1. (16)

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J7707 3

13. (a) (i) Prove by using the quotient law, that the Kronecker delta ijδ is a

tensor. (8)

(ii) Show that Christoffel symbol of the first kind is not a tensor. (8)

Or

(b) (i) Given metric ( ) ( ) ( ) ( ) ( )( )2322222222 sin dxxxdxxxdds ′+′+′= find then

Christoffel symbols [ ]

1,2

23,32 and . (8)

(ii) Prove that [ ] [ ]ijkjikx

g

k

ij,, +=

∂

∂. (8)

14. (a) (i) Find the extremal of the functional ( )∫ +−′′=4

0

222 ,

π

dxxyyI under the

conditions 2

1

44,1)0(,0)0( =

′=

=′=

ππyyyy . (8)

(ii) Find the shortest distance between two points in a plane. (8)

Or

(b) Find the approximate solution of 12 −=∇ u in the square 11: ≤≤− xD ,

11 ≤≤− y , with 0=u on the boundary of D by using Kantorovich method.

(16)

15. (a) Find the discrete Fourier transform of }{ )(nx , given that

(i) )()(1,....2,1,0for,)( nxNnxNnanx n =+−==

(ii) 5,)( == − Nenx n

(iii) 4,1)2()1(,2)3()0( ==−==−= Nxxxx . (16)

Or

(b) Use a four point FFT to compute the Fourier transform of

} }{{ 4,3,2,1)( =nx . (16)

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