Cse-III-Engineering Mathematics - III [10mat31]-Question Paper

SUBJECT:MATHEMATICS SUBJECTCODE:10MAT31

DEPT OF MATHS/SJBIT Page 1

Third Semester B.E. Degree Examination, Dec - 2011Engineering mathematics -III

UNIT-I

1. a) Obtain the Fourier expansion of

, 0( )

2 , 2

x xf x

x x

Deduce that ......5

1

3

11

8 22

2

b) Obtain the sine half range series of

1 10

4 23 1

14 2

x in xf x

x in x

1. c) . Express y as a Fourier series upto the third harmonic given the following values :

UNIT-II

2. a) If 21 1

0 1

x xf x

x

Find the fourier transform of f x and hence find the value of 3cos sin cos 2x x x x

dxx

x 0 1 2 3 4 5

y 4 8 15 7 6 2



2. b) Find the Fourier cosine transforms of 211f x x

2. c) Solve the integral equation 0

1 ,0 1cos

0 , 1f d

and hence evaluate

2

20

sin tdt

t

UNIT-III

3. a) Solve the two dimensional Laplaces equation uxx+uyy=0 by the method of separation of variables

b) Solve the heat equation 2

2

u u

t x

with the boundary conditions u(0,t)=0,u(l,t)and u(x,0) =3sin

x

. c) Solve DAlemberts solution of the one dimensional wave equation

UNIT-IV

4. a) Fit an exponential of the form bxy a e by the method of least squares for the following data

No .of petals 5 6 7 8 9 10No .of flowers 133 55 23 7 2 2

b) Use the graphical method to minimize subject to the constraints

4 ) 2 3 int ,

3 2 11 2 5 19

3 4 25 , , 0

c Maximize p x y z subject to theconstra s

x y z x y z

x y z x y z

21 1020 xxZ .0,,6034,303,402 21212121 xxxxxxxx



PART BUNIT-V

5. a) Find the root of the equation xex = cos x using Regula falsi method correct to three decimal places.

. b) Solve the following system of equations by relaxation method.

10 2 3 205

2 10 2 154

2 10 120

x y z

x y z

x y z

5. c) Find the dominant eigen value and the corresponding eigen vector of the matrix

UNIT-VI

6.a) From the following data find the number of students who have obtained 45 marks. Also find the number of students who have scored between 41 and 45 marks.

Marks 0 - 40 41 - 50 51 - 60 61 -70 71 - 80

No. of students 31 42 51 35 31

b) Using lagranges formula find the interpolating polynomial approximate to the function

described in the following table

X 0 1 2 5

y 2 3 12 147



c) A curve is drawn to pass through the points given by the following tableX 1 1.5 2 2.5 3 3.5 4Y 2 2.4 2.7 2.8 3 2.6 2.1

Use Weddles rule, estimate the area bounded by the curve the x-axis and the lines x=1, x=4

UNIT-VII

7. a) Solve Laplaces uxx+uyy=0 for the following square mesh with boundary values

b) .Solve the wave equation 2 2

2 24

u u

t x

subject to u=(0,t) =0,u(4,t)=0,ut(x,0)=0 and

u(x,0)=x(4-x) by taking h=1,k=0.5 upto four steps

c) Solve numerically the equation 2

2

u u

t x

subject to the conditions

0, 0 1, , 0 ,0 sin ,0 1.u t u t t and u x x x Carryout computations for two levels taking 13h and 136k

UNIT-VIII

8. a) (i) Find the Z-transforms of cos2 4

n

(ii) Find the Z-transform of 21n

b) Find the inverse Z-transform of 2

3 2

4 2

5 8 4

z z

z z z

c) Solve ,0296 1012 yywithyyy nnnn using Z-transform


Third Semester B.E. Degree Examination, Dec - 2011

Engineering mathematics -III

UNIT-I

1. a) Obtain the Fourier expansion of

Deduce that

b) Obtain the sine half range series of

1. c) . Express y as a Fourier series upto the third harmonic given the following values :

x

0

1

2

3

4

5

y

4

8

15

7

6

2

UNIT-II

2. a) If

Find the fourier transform of and hence find the value of

2. b) Find the Fourier cosine transforms of

2. c) Solve the integral equation and hence evaluate

UNIT-III

3. a) Solve the two dimensional Laplaces equation uxx+uyy=0 by the method of separation of variables

b) Solve the heat equation with the boundary conditions u(0,t)=0,u(l,t)and u(x,0) =3sin x

. c) Solve DAlemberts solution of the one dimensional wave equation

UNIT-IV

4. a) Fit an exponential of the form by the method of least squares for the following data

No .of petals

5

6

7

8

9

10

No .of flowers

133

55

23

7

2

2

b) Use the graphical method to minimize subject to the constraints

PART B

UNIT-V

5. a) Find the root of the equation xex = cos x using Regula falsi method correct to three decimal places.

. b) Solve the following system of equations by relaxation method.

5. c) Find the dominant eigen value and the corresponding eigen vector of the matrix

UNIT-VI

6.a) From the following data find the number of students who have obtained 45 marks. Also find the number of students who have scored between 41 and 45 marks.

Marks

0 - 40

41 - 50

51 - 60

61 -70

71 - 80

No. of students

31

42

51

35

31

b) Using lagranges formula find the interpolating polynomial approximate to the function described in the following table

X

0

1

2

5

y

2

3

12

147

c) A curve is drawn to pass through the points given by the following table

X

1

1.5

2

2.5

3

3.5

4

Y

2

2.4

2.7

2.8

3

2.6

2.1

Use Weddles rule, estimate the area bounded by the curve the x-axis and the lines x=1, x=4

UNIT-VII

7. a)

Solve Laplaces uxx+uyy=0 for the following square mesh with boundary values

b) .Solve the wave equation subject to u=(0,t) =0,u(4,t)=0,ut(x,0)=0 and u(x,0)=x(4-x) by taking h=1,k=0.5 upto four steps

c) Solve numerically the equation subject to the conditions

Carryout computations for two levels

taking and

UNIT-VIII

8. a) (i) Find the Z-transforms of

(ii) Find the Z-transform of

b) Find the inverse Z-transform of

c) Solve using Z-transform

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Cse-III-Engineering Mathematics - III [10mat31]-Question Paper

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