Reg. No. :

M.E. DEGREE EXAMINATION, JANUARY 2010

First Semester

Engineering Design

MA 9214 — APPLIED MATHEMATICS FOR ENGINEERING DESIGN

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

M.E. – Product Design and Development)

(Regulations 2009)

Time: Three hours Maximum: 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. Evaluate k, if the joint probability density function of a random variable

( ),X Y is given by ( ) ( )yxkxyxf −=, in ( ) 0,,,20 =<<−≤≤ yxfxyxx , elsewhere.

2. Find the conditional distribution of X given 2=Y from the bivariate

distribution of random variable ( ),X Y given in

Y

X

1

2

1 0.1 0.2

2 0.3 0.4

3. Classify the partial differential equation ( ) ( ) 0224 =++−+ yyxyxx xuuxux .

4. State the implicit formula for Crank-Nicholson scheme.

5. State Kronecker delta.

6. Define Christoffel symbol of first kind.

Question Paper Code: W7666 4

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7. State the Euler equation for ( )dxyyxf

x

x

∫ ′1

0

,, given that f is independent of y′ .

8. State the necessary condition for which ( )1

0

' ' '

1 2 1 2, , ,... , , ,...

x

n n

x

f x y y y y y y dx∫ to be

an extremum.

9. Define linear convolution.

10. Define the discrete Fourier transform pair.

PART B — (5 × 16 = 80 Marks)

11. (a) (i) Two dimensional random variable ( ),X Y has the joint probability

density function of ( ) ,10,8, <<<= yxxyyxf ( ) 0, =yxf elsewhere.

Find (1)

<∩<

4

1

2

1YXP , (2) the marginal distributions and

(3) verify whether X and Y are independent. (8)

(ii) Following table gives the data on rainfall and discharge in a certain

river. Obtain the line of regression of y on x and estimate y at 2=x .

(8)

Rain fall x in inches : 1.53 1.78 2.60 2.95 3.42

Discharge y (1000 cc) : 33.5 36.3 40.0 45.8 53.5

Or

(b) (i) The following table represents the joint probability distribution of

the discrete random variable ( ),X Y . Find all marginal

distributions, conditional distribution of X given 1=Y and

conditional distribution of Y given 1=X . (8)

X

Y

1 2 3

1 1/2 1/6 0

2 0 1/9 1/5

3 1/18 1/4 2/15

(ii) Obtain the coefficient of correlation between x and y from the given

data. (8)

x : 78 89 97 69 59 79 68 57

y : 125 137 156 112 107 138 123 108

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12. (a) (i) Find the solution of the parabolic equation txx uu 2= , when

( ) ( ) 0,4,0 == tutu and ( ) ( )xxxu −= 4100, taking 1=h . Find the values

of t upto 2=t . (8)

(ii) Solve the Laplace equation 222 8 yxu=∇ for the square mesh given

below with ( ) 0, =yxu on the boundary and mesh length = 1 (8)

Or

(b) (i) Give the values of ( )yxu , on the boundary of the square mesh

below. Solve 0=+ yyxx uu at the pivotal points by iterating till the

mesh values are correct to two decimal places. (8)

1000 1000 1000 1000

2000

2000

500

0

1000 500 0 0

(ii) The transverse displacement u at the point at a distance x from one

end at any time t of a vibrating string satisfies the equation

xxtt uu 4= with the boundary conditions 0=u at ,0=x 0>t , 0=u at

4=x , 0>t and the initial conditions ( )xxu −= 4 and 0=tu at 0=t ,

40 ≤≤ x . Solve this equation numerically for 20 ≤≤t , taking 0.1=h

and 5.0=k . (8)

13. (a) (i) Show that a symmetric tensor of the second order has only ( )12

1+nn

different components. (8)

u1

u2

u1

u1

u1

u2

u2

u3

u2

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(ii) Prove that [ ]gxij

ijlog

∂

∂=

. (8)

Or

(b) (i) A contravariant tensor has components 22, zyxy − , xz in

rectangular co-ordinates. Find its covariant components in

spherical co-ordinates. (8)

(ii) If ( ) ( ) ( ) ( )2222222sin φθθ drdrdrds ++= , find the values of

22

1

and

13

3. (8)

14. (a) (i) Find the path on which a particle in the absence of friction will slide

from one point to another in the shortest time under the action

gravity. (8)

(ii) Find an approximate solution of the Poisson equation 1−=∆Z in

the rectangle

≤≤−

≤≤−=

.

,

byb

axaD subject to the condition 0=z on the

contour using Kantrovitch method. (8)

Or

(b) (i) Determine the extremal of the functional

( )[ ] ( )∫ +′−′′=2

0

222

π

dxxyyxyv that satisfies the conditional ( ) 10 =y ,

( ) 00 =′y , 02

=

πy and 1

2−=

′ πy . (8)

(ii) Find an approximate solution of the nonlinear equation 2

2

3yy =′′

that satisfies conditions ( ) ,40 =y ( ) 11 =y using Ritz method. (8)

15. (a) (i) Find the linear convolution of (1) ( )nx and ( )ny (2) ( )nx and ( )nx ,

using circular convolution method, given that ( ) { }1,2,2,1=nx and

( ) { }1,2,3=ny . (8)

(ii) Find the DFS of the sequences (1) ( ) { }00,1,1=nx which is 4-periodic

and (2) ( )3

cosn

nxπ

= . (8)

Or

(b) (i) Find the DTFT of the following infinite sequences (1) ( ) ( )nnx δ=

and (2) ( ) −≤≤

=.,0

10,

otherwise

NnforAnx (8)

(ii) Find the IDFT of ( ){ }kX , given that (8)

(1) ( ) ( ) ( ){ }4;13,2,13,4 =+−= NiikX and

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(2) ( )( )

( )( )

=

==

−

−

.4,3,

2,1,0,

58.0

8.0

kfore

kforekX

ki

ki

π

π

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