Download JNTUK (Jawaharlal Nehru Technological University Kakinada) B.Tech Supplementary 2014 Feb-March R10 I Semester (1st Year 1st Sem) MATHEMATICAL METHODS Question Paper.

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find value of K if rank of A is 3, if A =

2

6

6

4

1 2 1 3

4 1 2 1

3 1 1 2

1 2 0 K

3

7

7

5

(b) Solve by Gauss elimination method 10x+y+z=12; 2x+10y+z=13; x+y+5z=7;

[7+8]

2. (a) Prove that the Eigen values of a triangular matrix are diagonal elements of

the matrix

(b) Find eigen vectors of B=2A

2

{ A + 3I when A =

8 4

2 2

[5+10]

3. Dene the nature of the quadratic form. Identify the nature of the quadratic form

x

2

1

+ 4x

2

2

+x

2

3

4x

1

x

2

+ 2x

1

x

3

4x

2

x

3

[15]

4. (a) Evaluate the real root of the equation x

2

9x + 1 = 0 by Bisection method

(b) Compute the real root of the equation x

3

x

2

1 = 0by the method of false

position. [8+7]

5. (a) Compute the approximate value of e

x

when x= 1.7489 from the following

table using the Gauss forward interpolation formula.

x 1.72 1.73 1.74 1.75 1.76 1.77 1.78

e

x

0.179066 0.177284 0.175520 0.173774 0.172045 0.170333 0.168638

(b) Find the Parabola passing through the points (0, 1), (1,3) and (3,5), Using

Lagrange's Interpolation formula. [8+7]

6. (a) Find the rst and second derivatives of the function tabulated below at the

point x = 1.5.

X 1.5 2.0 2.5 3.0 3.5 4.0

Y 3.375 7.0 13.625 24.0 38.875 59.0

(b) Evaluate

R

2:0

0:6

y dx using Trapizoidal, Simpsons 1/3 and Simpsons 3/8 rules.

X 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

y 1.23 1.58 2.03 4.32 6.25 8.38 10.23 12.45 [8+7]

1 of 2

FirstRanker.com - FirstRanker's Choice

Code No: R10107/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find value of K if rank of A is 3, if A =

2

6

6

4

1 2 1 3

4 1 2 1

3 1 1 2

1 2 0 K

3

7

7

5

(b) Solve by Gauss elimination method 10x+y+z=12; 2x+10y+z=13; x+y+5z=7;

[7+8]

2. (a) Prove that the Eigen values of a triangular matrix are diagonal elements of

the matrix

(b) Find eigen vectors of B=2A

2

{ A + 3I when A =

8 4

2 2

[5+10]

3. Dene the nature of the quadratic form. Identify the nature of the quadratic form

x

2

1

+ 4x

2

2

+x

2

3

4x

1

x

2

+ 2x

1

x

3

4x

2

x

3

[15]

4. (a) Evaluate the real root of the equation x

2

9x + 1 = 0 by Bisection method

(b) Compute the real root of the equation x

3

x

2

1 = 0by the method of false

position. [8+7]

5. (a) Compute the approximate value of e

x

when x= 1.7489 from the following

table using the Gauss forward interpolation formula.

x 1.72 1.73 1.74 1.75 1.76 1.77 1.78

e

x

0.179066 0.177284 0.175520 0.173774 0.172045 0.170333 0.168638

(b) Find the Parabola passing through the points (0, 1), (1,3) and (3,5), Using

Lagrange's Interpolation formula. [8+7]

6. (a) Find the rst and second derivatives of the function tabulated below at the

point x = 1.5.

X 1.5 2.0 2.5 3.0 3.5 4.0

Y 3.375 7.0 13.625 24.0 38.875 59.0

(b) Evaluate

R

2:0

0:6

y dx using Trapizoidal, Simpsons 1/3 and Simpsons 3/8 rules.

X 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

y 1.23 1.58 2.03 4.32 6.25 8.38 10.23 12.45 [8+7]

1 of 2

Code No: R10107/R10 Set No. 1

7. (a) Solve y

1

=3x+y/2, y(0)=1 by Taylor series method and hence nd y(0.1),

y(0.2)

(b) Solve the equation

dy

dx

= xy + 1 , y(0)=1 by Picard's method and hence nd

y(0.1) [8+7]

8. (a) Fit a least square parabola y= a+bx+cx

2

to the following data

x -3 -2 -1 0 1 2 3

y 4.63 2.11 0.67 0.09 0.63 2.15 4.58

(b) Fit a straight line of the form y= a+bx to the following data

x 1 2 4 5 6 8 9

y 2 5 7 10 12 15 19

[7+8]

?????

2 of 2

FirstRanker.com - FirstRanker's Choice

Code No: R10107/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find value of K if rank of A is 3, if A =

2

6

6

4

1 2 1 3

4 1 2 1

3 1 1 2

1 2 0 K

3

7

7

5

(b) Solve by Gauss elimination method 10x+y+z=12; 2x+10y+z=13; x+y+5z=7;

[7+8]

2. (a) Prove that the Eigen values of a triangular matrix are diagonal elements of

the matrix

(b) Find eigen vectors of B=2A

2

{ A + 3I when A =

8 4

2 2

[5+10]

3. Dene the nature of the quadratic form. Identify the nature of the quadratic form

x

2

1

+ 4x

2

2

+x

2

3

4x

1

x

2

+ 2x

1

x

3

4x

2

x

3

[15]

4. (a) Evaluate the real root of the equation x

2

9x + 1 = 0 by Bisection method

(b) Compute the real root of the equation x

3

x

2

1 = 0by the method of false

position. [8+7]

5. (a) Compute the approximate value of e

x

when x= 1.7489 from the following

table using the Gauss forward interpolation formula.

x 1.72 1.73 1.74 1.75 1.76 1.77 1.78

e

x

0.179066 0.177284 0.175520 0.173774 0.172045 0.170333 0.168638

(b) Find the Parabola passing through the points (0, 1), (1,3) and (3,5), Using

Lagrange's Interpolation formula. [8+7]

6. (a) Find the rst and second derivatives of the function tabulated below at the

point x = 1.5.

X 1.5 2.0 2.5 3.0 3.5 4.0

Y 3.375 7.0 13.625 24.0 38.875 59.0

(b) Evaluate

R

2:0

0:6

y dx using Trapizoidal, Simpsons 1/3 and Simpsons 3/8 rules.

X 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

y 1.23 1.58 2.03 4.32 6.25 8.38 10.23 12.45 [8+7]

1 of 2

Code No: R10107/R10 Set No. 1

7. (a) Solve y

1

=3x+y/2, y(0)=1 by Taylor series method and hence nd y(0.1),

y(0.2)

(b) Solve the equation

dy

dx

= xy + 1 , y(0)=1 by Picard's method and hence nd

y(0.1) [8+7]

8. (a) Fit a least square parabola y= a+bx+cx

2

to the following data

x -3 -2 -1 0 1 2 3

y 4.63 2.11 0.67 0.09 0.63 2.15 4.58

(b) Fit a straight line of the form y= a+bx to the following data

x 1 2 4 5 6 8 9

y 2 5 7 10 12 15 19

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank of A =

2

4

2 1 3

0 1 2

4 0 2

1

2

6

3

5

using Normal Form

(b) Solve by Gauss seidal method x+4y+15z=24, x+12y+z=26, 10x+y-2z=10

[7+8]

2. (a) Find Eigen Vectors of

5 4

1 2

(b) If is an Eigen value of A then prove that

jAj

is an Eigen value of Adj. A

[7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Using Newton- Raphson's method compute

p

41 correct to four decimal places.

(b) Find a real root of the equatione

x

=x+2in the interval [1, 1.4] using bisection

method. [8+7]

5. (a) Apply Gauss backward interpolation formula to nd y when x = 26 form the

following table:

x 20 24 28 32

Y 2854 3162 3544 3992

(b) Using Lagrange's interpolation formula, nd the value of y when x = 2 from

the following data:

x 1 3 4 6

y 4 40 85 259

[8+7]

6. (a) Find the value of f

0

(x) at x=0.01 from the following table using Bessel's

formula.

x 0.01 0.02 0.03 0.04 0.05 0.06

f(x) 0.1023 0.1047 0.1071 0.1096 0.1122 0.1148

(b) Find the area bounded by the curve y = e

x

2

2

, x - axis between x = 0 and x

= 3 by using Simpson's 3/8 rule. [8+7]

1 of 2

FirstRanker.com - FirstRanker's Choice

Code No: R10107/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find value of K if rank of A is 3, if A =

2

6

6

4

1 2 1 3

4 1 2 1

3 1 1 2

1 2 0 K

3

7

7

5

(b) Solve by Gauss elimination method 10x+y+z=12; 2x+10y+z=13; x+y+5z=7;

[7+8]

2. (a) Prove that the Eigen values of a triangular matrix are diagonal elements of

the matrix

(b) Find eigen vectors of B=2A

2

{ A + 3I when A =

8 4

2 2

[5+10]

3. Dene the nature of the quadratic form. Identify the nature of the quadratic form

x

2

1

+ 4x

2

2

+x

2

3

4x

1

x

2

+ 2x

1

x

3

4x

2

x

3

[15]

4. (a) Evaluate the real root of the equation x

2

9x + 1 = 0 by Bisection method

(b) Compute the real root of the equation x

3

x

2

1 = 0by the method of false

position. [8+7]

5. (a) Compute the approximate value of e

x

when x= 1.7489 from the following

table using the Gauss forward interpolation formula.

x 1.72 1.73 1.74 1.75 1.76 1.77 1.78

e

x

0.179066 0.177284 0.175520 0.173774 0.172045 0.170333 0.168638

(b) Find the Parabola passing through the points (0, 1), (1,3) and (3,5), Using

Lagrange's Interpolation formula. [8+7]

6. (a) Find the rst and second derivatives of the function tabulated below at the

point x = 1.5.

X 1.5 2.0 2.5 3.0 3.5 4.0

Y 3.375 7.0 13.625 24.0 38.875 59.0

(b) Evaluate

R

2:0

0:6

y dx using Trapizoidal, Simpsons 1/3 and Simpsons 3/8 rules.

X 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

y 1.23 1.58 2.03 4.32 6.25 8.38 10.23 12.45 [8+7]

1 of 2

Code No: R10107/R10 Set No. 1

7. (a) Solve y

1

=3x+y/2, y(0)=1 by Taylor series method and hence nd y(0.1),

y(0.2)

(b) Solve the equation

dy

dx

= xy + 1 , y(0)=1 by Picard's method and hence nd

y(0.1) [8+7]

8. (a) Fit a least square parabola y= a+bx+cx

2

to the following data

x -3 -2 -1 0 1 2 3

y 4.63 2.11 0.67 0.09 0.63 2.15 4.58

(b) Fit a straight line of the form y= a+bx to the following data

x 1 2 4 5 6 8 9

y 2 5 7 10 12 15 19

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank of A =

2

4

2 1 3

0 1 2

4 0 2

1

2

6

3

5

using Normal Form

(b) Solve by Gauss seidal method x+4y+15z=24, x+12y+z=26, 10x+y-2z=10

[7+8]

2. (a) Find Eigen Vectors of

5 4

1 2

(b) If is an Eigen value of A then prove that

jAj

is an Eigen value of Adj. A

[7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Using Newton- Raphson's method compute

p

41 correct to four decimal places.

(b) Find a real root of the equatione

x

=x+2in the interval [1, 1.4] using bisection

method. [8+7]

5. (a) Apply Gauss backward interpolation formula to nd y when x = 26 form the

following table:

x 20 24 28 32

Y 2854 3162 3544 3992

(b) Using Lagrange's interpolation formula, nd the value of y when x = 2 from

the following data:

x 1 3 4 6

y 4 40 85 259

[8+7]

6. (a) Find the value of f

0

(x) at x=0.01 from the following table using Bessel's

formula.

x 0.01 0.02 0.03 0.04 0.05 0.06

f(x) 0.1023 0.1047 0.1071 0.1096 0.1122 0.1148

(b) Find the area bounded by the curve y = e

x

2

2

, x - axis between x = 0 and x

= 3 by using Simpson's 3/8 rule. [8+7]

1 of 2

Code No: R10107/R10 Set No. 2

7. (a) Solve y

1

=x-y, y(0)=1 by modied Euler's method and nd y(0.1), y(0.2)

(b) Apply third order R-K method to nd y(0.25) where y

1

=1+xy, y(0)=1 [8+7]

8. (a) Fit a power curve y=ax

b

to the following data

x 5 6 7 8 9 10

y 133 55 23 7 2 2

(b) Fit a curve of the type y= a+bx+cx

2

to the following data

x 0 1 2 3 4 5 6

y 14 18 23 29 36 40 46

[7+8]

?????

2 of 2

FirstRanker.com - FirstRanker's Choice

Code No: R10107/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find value of K if rank of A is 3, if A =

2

6

6

4

1 2 1 3

4 1 2 1

3 1 1 2

1 2 0 K

3

7

7

5

(b) Solve by Gauss elimination method 10x+y+z=12; 2x+10y+z=13; x+y+5z=7;

[7+8]

2. (a) Prove that the Eigen values of a triangular matrix are diagonal elements of

the matrix

(b) Find eigen vectors of B=2A

2

{ A + 3I when A =

8 4

2 2

[5+10]

3. Dene the nature of the quadratic form. Identify the nature of the quadratic form

x

2

1

+ 4x

2

2

+x

2

3

4x

1

x

2

+ 2x

1

x

3

4x

2

x

3

[15]

4. (a) Evaluate the real root of the equation x

2

9x + 1 = 0 by Bisection method

(b) Compute the real root of the equation x

3

x

2

1 = 0by the method of false

position. [8+7]

5. (a) Compute the approximate value of e

x

when x= 1.7489 from the following

table using the Gauss forward interpolation formula.

x 1.72 1.73 1.74 1.75 1.76 1.77 1.78

e

x

0.179066 0.177284 0.175520 0.173774 0.172045 0.170333 0.168638

(b) Find the Parabola passing through the points (0, 1), (1,3) and (3,5), Using

Lagrange's Interpolation formula. [8+7]

6. (a) Find the rst and second derivatives of the function tabulated below at the

point x = 1.5.

X 1.5 2.0 2.5 3.0 3.5 4.0

Y 3.375 7.0 13.625 24.0 38.875 59.0

(b) Evaluate

R

2:0

0:6

y dx using Trapizoidal, Simpsons 1/3 and Simpsons 3/8 rules.

X 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

y 1.23 1.58 2.03 4.32 6.25 8.38 10.23 12.45 [8+7]

1 of 2

Code No: R10107/R10 Set No. 1

7. (a) Solve y

1

=3x+y/2, y(0)=1 by Taylor series method and hence nd y(0.1),

y(0.2)

(b) Solve the equation

dy

dx

= xy + 1 , y(0)=1 by Picard's method and hence nd

y(0.1) [8+7]

8. (a) Fit a least square parabola y= a+bx+cx

2

to the following data

x -3 -2 -1 0 1 2 3

y 4.63 2.11 0.67 0.09 0.63 2.15 4.58

(b) Fit a straight line of the form y= a+bx to the following data

x 1 2 4 5 6 8 9

y 2 5 7 10 12 15 19

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank of A =

2

4

2 1 3

0 1 2

4 0 2

1

2

6

3

5

using Normal Form

(b) Solve by Gauss seidal method x+4y+15z=24, x+12y+z=26, 10x+y-2z=10

[7+8]

2. (a) Find Eigen Vectors of

5 4

1 2

(b) If is an Eigen value of A then prove that

jAj

is an Eigen value of Adj. A

[7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Using Newton- Raphson's method compute

p

41 correct to four decimal places.

(b) Find a real root of the equatione

x

=x+2in the interval [1, 1.4] using bisection

method. [8+7]

5. (a) Apply Gauss backward interpolation formula to nd y when x = 26 form the

following table:

x 20 24 28 32

Y 2854 3162 3544 3992

(b) Using Lagrange's interpolation formula, nd the value of y when x = 2 from

the following data:

x 1 3 4 6

y 4 40 85 259

[8+7]

6. (a) Find the value of f

0

(x) at x=0.01 from the following table using Bessel's

formula.

x 0.01 0.02 0.03 0.04 0.05 0.06

f(x) 0.1023 0.1047 0.1071 0.1096 0.1122 0.1148

(b) Find the area bounded by the curve y = e

x

2

2

, x - axis between x = 0 and x

= 3 by using Simpson's 3/8 rule. [8+7]

1 of 2

Code No: R10107/R10 Set No. 2

7. (a) Solve y

1

=x-y, y(0)=1 by modied Euler's method and nd y(0.1), y(0.2)

(b) Apply third order R-K method to nd y(0.25) where y

1

=1+xy, y(0)=1 [8+7]

8. (a) Fit a power curve y=ax

b

to the following data

x 5 6 7 8 9 10

y 133 55 23 7 2 2

(b) Fit a curve of the type y= a+bx+cx

2

to the following data

x 0 1 2 3 4 5 6

y 14 18 23 29 36 40 46

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 3

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank using Normal Form A =

2

6

6

4

1

2

3

6

2

4

2

8

3

3

1

7

0

2

3

5

3

7

7

5

(b) Solve Homogeneous equations x

1

+2x

2

+3x

3

=0 , 2x

1

+3x

2

+x

3

=0,

4x

1

+5x

2

+4x

3

=0 , X

1

+x

2

-2x

3

=0 [7+8]

2. (a)Find Eigen values and Eigen vectors of

8 4

2 2

(b) If is an Eigen value of A then prove that

1

is an Eigen value of A

1

if it

exists [7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Find out square root of 25 given x

0

=2, x

1=

7 using Bisection method

(b) Solve the equation x

3

+ 2x

2

+ 10x = 20by iteration method [8+7]

5. (a) Use gauss forward interpolation formula to estimate f(32), given f(25) =

0.2707, f(30) = 0.3027, f(35) = 0.3386, f(40) = 0.3794.

(b) Find the interpolating polynomial f(x) from the table given below.

x 0 1 4 5

f(x) 4 3 24 39

[8+7]

6. (a) Using the table below, nd f

0

(0)

x 0 2 3 4 7 9

f(x) 4 26 58 110 460 920

(b) Evaluate

R

1

0

p

1 +x

3

dx taking h = 0.1 using Simpson's 3/8

th

rule. [8+7]

7. (a) Solve y

1

=x+y subject to the condition y(0)=1 by Taylor series method and

hence nd y(0.2), y(0.4)

(b) Solve y

1

=x-y, y(0)=1 by Picard's method and hence nd y at x=0.2 [8+7]

1 of 2

FirstRanker.com - FirstRanker's Choice

Code No: R10107/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find value of K if rank of A is 3, if A =

2

6

6

4

1 2 1 3

4 1 2 1

3 1 1 2

1 2 0 K

3

7

7

5

(b) Solve by Gauss elimination method 10x+y+z=12; 2x+10y+z=13; x+y+5z=7;

[7+8]

2. (a) Prove that the Eigen values of a triangular matrix are diagonal elements of

the matrix

(b) Find eigen vectors of B=2A

2

{ A + 3I when A =

8 4

2 2

[5+10]

3. Dene the nature of the quadratic form. Identify the nature of the quadratic form

x

2

1

+ 4x

2

2

+x

2

3

4x

1

x

2

+ 2x

1

x

3

4x

2

x

3

[15]

4. (a) Evaluate the real root of the equation x

2

9x + 1 = 0 by Bisection method

(b) Compute the real root of the equation x

3

x

2

1 = 0by the method of false

position. [8+7]

5. (a) Compute the approximate value of e

x

when x= 1.7489 from the following

table using the Gauss forward interpolation formula.

x 1.72 1.73 1.74 1.75 1.76 1.77 1.78

e

x

0.179066 0.177284 0.175520 0.173774 0.172045 0.170333 0.168638

(b) Find the Parabola passing through the points (0, 1), (1,3) and (3,5), Using

Lagrange's Interpolation formula. [8+7]

6. (a) Find the rst and second derivatives of the function tabulated below at the

point x = 1.5.

X 1.5 2.0 2.5 3.0 3.5 4.0

Y 3.375 7.0 13.625 24.0 38.875 59.0

(b) Evaluate

R

2:0

0:6

y dx using Trapizoidal, Simpsons 1/3 and Simpsons 3/8 rules.

X 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

y 1.23 1.58 2.03 4.32 6.25 8.38 10.23 12.45 [8+7]

1 of 2

Code No: R10107/R10 Set No. 1

7. (a) Solve y

1

=3x+y/2, y(0)=1 by Taylor series method and hence nd y(0.1),

y(0.2)

(b) Solve the equation

dy

dx

= xy + 1 , y(0)=1 by Picard's method and hence nd

y(0.1) [8+7]

8. (a) Fit a least square parabola y= a+bx+cx

2

to the following data

x -3 -2 -1 0 1 2 3

y 4.63 2.11 0.67 0.09 0.63 2.15 4.58

(b) Fit a straight line of the form y= a+bx to the following data

x 1 2 4 5 6 8 9

y 2 5 7 10 12 15 19

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank of A =

2

4

2 1 3

0 1 2

4 0 2

1

2

6

3

5

using Normal Form

(b) Solve by Gauss seidal method x+4y+15z=24, x+12y+z=26, 10x+y-2z=10

[7+8]

2. (a) Find Eigen Vectors of

5 4

1 2

(b) If is an Eigen value of A then prove that

jAj

is an Eigen value of Adj. A

[7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Using Newton- Raphson's method compute

p

41 correct to four decimal places.

(b) Find a real root of the equatione

x

=x+2in the interval [1, 1.4] using bisection

method. [8+7]

5. (a) Apply Gauss backward interpolation formula to nd y when x = 26 form the

following table:

x 20 24 28 32

Y 2854 3162 3544 3992

(b) Using Lagrange's interpolation formula, nd the value of y when x = 2 from

the following data:

x 1 3 4 6

y 4 40 85 259

[8+7]

6. (a) Find the value of f

0

(x) at x=0.01 from the following table using Bessel's

formula.

x 0.01 0.02 0.03 0.04 0.05 0.06

f(x) 0.1023 0.1047 0.1071 0.1096 0.1122 0.1148

(b) Find the area bounded by the curve y = e

x

2

2

, x - axis between x = 0 and x

= 3 by using Simpson's 3/8 rule. [8+7]

1 of 2

Code No: R10107/R10 Set No. 2

7. (a) Solve y

1

=x-y, y(0)=1 by modied Euler's method and nd y(0.1), y(0.2)

(b) Apply third order R-K method to nd y(0.25) where y

1

=1+xy, y(0)=1 [8+7]

8. (a) Fit a power curve y=ax

b

to the following data

x 5 6 7 8 9 10

y 133 55 23 7 2 2

(b) Fit a curve of the type y= a+bx+cx

2

to the following data

x 0 1 2 3 4 5 6

y 14 18 23 29 36 40 46

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 3

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank using Normal Form A =

2

6

6

4

1

2

3

6

2

4

2

8

3

3

1

7

0

2

3

5

3

7

7

5

(b) Solve Homogeneous equations x

1

+2x

2

+3x

3

=0 , 2x

1

+3x

2

+x

3

=0,

4x

1

+5x

2

+4x

3

=0 , X

1

+x

2

-2x

3

=0 [7+8]

2. (a)Find Eigen values and Eigen vectors of

8 4

2 2

(b) If is an Eigen value of A then prove that

1

is an Eigen value of A

1

if it

exists [7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Find out square root of 25 given x

0

=2, x

1=

7 using Bisection method

(b) Solve the equation x

3

+ 2x

2

+ 10x = 20by iteration method [8+7]

5. (a) Use gauss forward interpolation formula to estimate f(32), given f(25) =

0.2707, f(30) = 0.3027, f(35) = 0.3386, f(40) = 0.3794.

(b) Find the interpolating polynomial f(x) from the table given below.

x 0 1 4 5

f(x) 4 3 24 39

[8+7]

6. (a) Using the table below, nd f

0

(0)

x 0 2 3 4 7 9

f(x) 4 26 58 110 460 920

(b) Evaluate

R

1

0

p

1 +x

3

dx taking h = 0.1 using Simpson's 3/8

th

rule. [8+7]

7. (a) Solve y

1

=x+y subject to the condition y(0)=1 by Taylor series method and

hence nd y(0.2), y(0.4)

(b) Solve y

1

=x-y, y(0)=1 by Picard's method and hence nd y at x=0.2 [8+7]

1 of 2

Code No: R10107/R10 Set No. 3

8. (a) Fit a curve of the type y= a+bx+cx

2

to the following data

x 10 15 20 25 30 35

y 35.3 32.4 29.2 26.1 23.2 20.5

(b) Fit a curve of the type y=ab

x

to the following data by the method of least

squares

x 1 2 5 10 20 30 40 50

Y 98.2 91.7 81.3 64 36.4 32.6 7.1 11.3 [7+8]

?????

2 of 2

FirstRanker.com - FirstRanker's Choice

Code No: R10107/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find value of K if rank of A is 3, if A =

2

6

6

4

1 2 1 3

4 1 2 1

3 1 1 2

1 2 0 K

3

7

7

5

(b) Solve by Gauss elimination method 10x+y+z=12; 2x+10y+z=13; x+y+5z=7;

[7+8]

2. (a) Prove that the Eigen values of a triangular matrix are diagonal elements of

the matrix

(b) Find eigen vectors of B=2A

2

{ A + 3I when A =

8 4

2 2

[5+10]

3. Dene the nature of the quadratic form. Identify the nature of the quadratic form

x

2

1

+ 4x

2

2

+x

2

3

4x

1

x

2

+ 2x

1

x

3

4x

2

x

3

[15]

4. (a) Evaluate the real root of the equation x

2

9x + 1 = 0 by Bisection method

(b) Compute the real root of the equation x

3

x

2

1 = 0by the method of false

position. [8+7]

5. (a) Compute the approximate value of e

x

when x= 1.7489 from the following

table using the Gauss forward interpolation formula.

x 1.72 1.73 1.74 1.75 1.76 1.77 1.78

e

x

0.179066 0.177284 0.175520 0.173774 0.172045 0.170333 0.168638

(b) Find the Parabola passing through the points (0, 1), (1,3) and (3,5), Using

Lagrange's Interpolation formula. [8+7]

6. (a) Find the rst and second derivatives of the function tabulated below at the

point x = 1.5.

X 1.5 2.0 2.5 3.0 3.5 4.0

Y 3.375 7.0 13.625 24.0 38.875 59.0

(b) Evaluate

R

2:0

0:6

y dx using Trapizoidal, Simpsons 1/3 and Simpsons 3/8 rules.

X 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

y 1.23 1.58 2.03 4.32 6.25 8.38 10.23 12.45 [8+7]

1 of 2

Code No: R10107/R10 Set No. 1

7. (a) Solve y

1

=3x+y/2, y(0)=1 by Taylor series method and hence nd y(0.1),

y(0.2)

(b) Solve the equation

dy

dx

= xy + 1 , y(0)=1 by Picard's method and hence nd

y(0.1) [8+7]

8. (a) Fit a least square parabola y= a+bx+cx

2

to the following data

x -3 -2 -1 0 1 2 3

y 4.63 2.11 0.67 0.09 0.63 2.15 4.58

(b) Fit a straight line of the form y= a+bx to the following data

x 1 2 4 5 6 8 9

y 2 5 7 10 12 15 19

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank of A =

2

4

2 1 3

0 1 2

4 0 2

1

2

6

3

5

using Normal Form

(b) Solve by Gauss seidal method x+4y+15z=24, x+12y+z=26, 10x+y-2z=10

[7+8]

2. (a) Find Eigen Vectors of

5 4

1 2

(b) If is an Eigen value of A then prove that

jAj

is an Eigen value of Adj. A

[7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Using Newton- Raphson's method compute

p

41 correct to four decimal places.

(b) Find a real root of the equatione

x

=x+2in the interval [1, 1.4] using bisection

method. [8+7]

5. (a) Apply Gauss backward interpolation formula to nd y when x = 26 form the

following table:

x 20 24 28 32

Y 2854 3162 3544 3992

(b) Using Lagrange's interpolation formula, nd the value of y when x = 2 from

the following data:

x 1 3 4 6

y 4 40 85 259

[8+7]

6. (a) Find the value of f

0

(x) at x=0.01 from the following table using Bessel's

formula.

x 0.01 0.02 0.03 0.04 0.05 0.06

f(x) 0.1023 0.1047 0.1071 0.1096 0.1122 0.1148

(b) Find the area bounded by the curve y = e

x

2

2

, x - axis between x = 0 and x

= 3 by using Simpson's 3/8 rule. [8+7]

1 of 2

Code No: R10107/R10 Set No. 2

7. (a) Solve y

1

=x-y, y(0)=1 by modied Euler's method and nd y(0.1), y(0.2)

(b) Apply third order R-K method to nd y(0.25) where y

1

=1+xy, y(0)=1 [8+7]

8. (a) Fit a power curve y=ax

b

to the following data

x 5 6 7 8 9 10

y 133 55 23 7 2 2

(b) Fit a curve of the type y= a+bx+cx

2

to the following data

x 0 1 2 3 4 5 6

y 14 18 23 29 36 40 46

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 3

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank using Normal Form A =

2

6

6

4

1

2

3

6

2

4

2

8

3

3

1

7

0

2

3

5

3

7

7

5

(b) Solve Homogeneous equations x

1

+2x

2

+3x

3

=0 , 2x

1

+3x

2

+x

3

=0,

4x

1

+5x

2

+4x

3

=0 , X

1

+x

2

-2x

3

=0 [7+8]

2. (a)Find Eigen values and Eigen vectors of

8 4

2 2

(b) If is an Eigen value of A then prove that

1

is an Eigen value of A

1

if it

exists [7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Find out square root of 25 given x

0

=2, x

1=

7 using Bisection method

(b) Solve the equation x

3

+ 2x

2

+ 10x = 20by iteration method [8+7]

5. (a) Use gauss forward interpolation formula to estimate f(32), given f(25) =

0.2707, f(30) = 0.3027, f(35) = 0.3386, f(40) = 0.3794.

(b) Find the interpolating polynomial f(x) from the table given below.

x 0 1 4 5

f(x) 4 3 24 39

[8+7]

6. (a) Using the table below, nd f

0

(0)

x 0 2 3 4 7 9

f(x) 4 26 58 110 460 920

(b) Evaluate

R

1

0

p

1 +x

3

dx taking h = 0.1 using Simpson's 3/8

th

rule. [8+7]

7. (a) Solve y

1

=x+y subject to the condition y(0)=1 by Taylor series method and

hence nd y(0.2), y(0.4)

(b) Solve y

1

=x-y, y(0)=1 by Picard's method and hence nd y at x=0.2 [8+7]

1 of 2

Code No: R10107/R10 Set No. 3

8. (a) Fit a curve of the type y= a+bx+cx

2

to the following data

x 10 15 20 25 30 35

y 35.3 32.4 29.2 26.1 23.2 20.5

(b) Fit a curve of the type y=ab

x

to the following data by the method of least

squares

x 1 2 5 10 20 30 40 50

Y 98.2 91.7 81.3 64 36.4 32.6 7.1 11.3 [7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 4

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank of matrix using Normal form A =

2

4

1 2 3

2 2 1

3 0 4

2

3

1

3

5

(b) Solve system of equations, if consistent 2x-y-z=2 , x+2y+z=2, 4x-7y-5z=2

[7+8]

2. Verify Cayley - Hamilton theorem and nd A

1

if A =

2

4

2 1 2

1 2 1

1 1 2

3

5

[15]

3. Reduce the quadratic form to canonical from by an orthogonal reduction and state

the nature of the quadratic form 5x

2

+ 26y

2

+ 6xy + 4yz + 14zx. Also nd its rank

signature and index. [15]

4. (a) Using Newton-Raphson's method nd the square root of a number and hence

nd the square root of 24.

(b) Find a real root of the equation x=e

x

, using Bisection method [8+7]

5. (a) Apply Gauss's forward formula to nd f(x) at x = 3.5 from the table below.

X 2 3 4 5

F(x) 2.626 3.454 4.784 6.986

(b) Find sin 45

0

using Gauss's backward interpolation formula given that sin 20

0

= 0.342, sin 30

0

= 0.502, sin 40

0

=0.642, sin 50

0

= 0.766, sin 60

0

=0.866, sin

70

0

= 0.939, sin 80

0

= 0.984. [8+7]

6. (a) Given the following table. Find f

0

(1) and f

00

(3)

x 0 2 4 6 8

f(x) 7 13 43 145 367

(b) Find approximate value of

R

1:04

1

f(x)dxusing the following table.

x 1 1.01 1.02 1.03 1.04

f(x) 3.953 4.066 4.182 4.300 4.421

[8+7]

7. (a) Given that

dy

dx

=

(1+x

2

)y

2

2

, y(0)=1, y(0.1)=1.06, y(0.2)=1.12, y(0.3)=1.21 then

evaluate y(0.4) by Milne`s predictor corrector method

1 of 2

FirstRanker.com - FirstRanker's Choice

Code No: R10107/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find value of K if rank of A is 3, if A =

2

6

6

4

1 2 1 3

4 1 2 1

3 1 1 2

1 2 0 K

3

7

7

5

(b) Solve by Gauss elimination method 10x+y+z=12; 2x+10y+z=13; x+y+5z=7;

[7+8]

2. (a) Prove that the Eigen values of a triangular matrix are diagonal elements of

the matrix

(b) Find eigen vectors of B=2A

2

{ A + 3I when A =

8 4

2 2

[5+10]

3. Dene the nature of the quadratic form. Identify the nature of the quadratic form

x

2

1

+ 4x

2

2

+x

2

3

4x

1

x

2

+ 2x

1

x

3

4x

2

x

3

[15]

4. (a) Evaluate the real root of the equation x

2

9x + 1 = 0 by Bisection method

(b) Compute the real root of the equation x

3

x

2

1 = 0by the method of false

position. [8+7]

5. (a) Compute the approximate value of e

x

when x= 1.7489 from the following

table using the Gauss forward interpolation formula.

x 1.72 1.73 1.74 1.75 1.76 1.77 1.78

e

x

0.179066 0.177284 0.175520 0.173774 0.172045 0.170333 0.168638

(b) Find the Parabola passing through the points (0, 1), (1,3) and (3,5), Using

Lagrange's Interpolation formula. [8+7]

6. (a) Find the rst and second derivatives of the function tabulated below at the

point x = 1.5.

X 1.5 2.0 2.5 3.0 3.5 4.0

Y 3.375 7.0 13.625 24.0 38.875 59.0

(b) Evaluate

R

2:0

0:6

y dx using Trapizoidal, Simpsons 1/3 and Simpsons 3/8 rules.

X 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

y 1.23 1.58 2.03 4.32 6.25 8.38 10.23 12.45 [8+7]

1 of 2

Code No: R10107/R10 Set No. 1

7. (a) Solve y

1

=3x+y/2, y(0)=1 by Taylor series method and hence nd y(0.1),

y(0.2)

(b) Solve the equation

dy

dx

= xy + 1 , y(0)=1 by Picard's method and hence nd

y(0.1) [8+7]

8. (a) Fit a least square parabola y= a+bx+cx

2

to the following data

x -3 -2 -1 0 1 2 3

y 4.63 2.11 0.67 0.09 0.63 2.15 4.58

(b) Fit a straight line of the form y= a+bx to the following data

x 1 2 4 5 6 8 9

y 2 5 7 10 12 15 19

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank of A =

2

4

2 1 3

0 1 2

4 0 2

1

2

6

3

5

using Normal Form

(b) Solve by Gauss seidal method x+4y+15z=24, x+12y+z=26, 10x+y-2z=10

[7+8]

2. (a) Find Eigen Vectors of

5 4

1 2

(b) If is an Eigen value of A then prove that

jAj

is an Eigen value of Adj. A

[7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Using Newton- Raphson's method compute

p

41 correct to four decimal places.

(b) Find a real root of the equatione

x

=x+2in the interval [1, 1.4] using bisection

method. [8+7]

5. (a) Apply Gauss backward interpolation formula to nd y when x = 26 form the

following table:

x 20 24 28 32

Y 2854 3162 3544 3992

(b) Using Lagrange's interpolation formula, nd the value of y when x = 2 from

the following data:

x 1 3 4 6

y 4 40 85 259

[8+7]

6. (a) Find the value of f

0

(x) at x=0.01 from the following table using Bessel's

formula.

x 0.01 0.02 0.03 0.04 0.05 0.06

f(x) 0.1023 0.1047 0.1071 0.1096 0.1122 0.1148

(b) Find the area bounded by the curve y = e

x

2

2

, x - axis between x = 0 and x

= 3 by using Simpson's 3/8 rule. [8+7]

1 of 2

Code No: R10107/R10 Set No. 2

7. (a) Solve y

1

=x-y, y(0)=1 by modied Euler's method and nd y(0.1), y(0.2)

(b) Apply third order R-K method to nd y(0.25) where y

1

=1+xy, y(0)=1 [8+7]

8. (a) Fit a power curve y=ax

b

to the following data

x 5 6 7 8 9 10

y 133 55 23 7 2 2

(b) Fit a curve of the type y= a+bx+cx

2

to the following data

x 0 1 2 3 4 5 6

y 14 18 23 29 36 40 46

[7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 3

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank using Normal Form A =

2

6

6

4

1

2

3

6

2

4

2

8

3

3

1

7

0

2

3

5

3

7

7

5

(b) Solve Homogeneous equations x

1

+2x

2

+3x

3

=0 , 2x

1

+3x

2

+x

3

=0,

4x

1

+5x

2

+4x

3

=0 , X

1

+x

2

-2x

3

=0 [7+8]

2. (a)Find Eigen values and Eigen vectors of

8 4

2 2

(b) If is an Eigen value of A then prove that

1

is an Eigen value of A

1

if it

exists [7+8]

3. Find the rank, signature and index of the quadratic form 2x

2

1

+x

2

2

3x

2

3

+ 12x

1

x

2

4x

1

x

3

8x

2

x

3

by reducing it to normal form .Also write the linear transformation

which brings about the normal reduction [15]

4. (a) Find out square root of 25 given x

0

=2, x

1=

7 using Bisection method

(b) Solve the equation x

3

+ 2x

2

+ 10x = 20by iteration method [8+7]

5. (a) Use gauss forward interpolation formula to estimate f(32), given f(25) =

0.2707, f(30) = 0.3027, f(35) = 0.3386, f(40) = 0.3794.

(b) Find the interpolating polynomial f(x) from the table given below.

x 0 1 4 5

f(x) 4 3 24 39

[8+7]

6. (a) Using the table below, nd f

0

(0)

x 0 2 3 4 7 9

f(x) 4 26 58 110 460 920

(b) Evaluate

R

1

0

p

1 +x

3

dx taking h = 0.1 using Simpson's 3/8

th

rule. [8+7]

7. (a) Solve y

1

=x+y subject to the condition y(0)=1 by Taylor series method and

hence nd y(0.2), y(0.4)

(b) Solve y

1

=x-y, y(0)=1 by Picard's method and hence nd y at x=0.2 [8+7]

1 of 2

Code No: R10107/R10 Set No. 3

8. (a) Fit a curve of the type y= a+bx+cx

2

to the following data

x 10 15 20 25 30 35

y 35.3 32.4 29.2 26.1 23.2 20.5

(b) Fit a curve of the type y=ab

x

to the following data by the method of least

squares

x 1 2 5 10 20 30 40 50

Y 98.2 91.7 81.3 64 36.4 32.6 7.1 11.3 [7+8]

?????

2 of 2

Code No: R10107/R10 Set No. 4

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Find rank of matrix using Normal form A =

2

4

1 2 3

2 2 1

3 0 4

2

3

1

3

5

(b) Solve system of equations, if consistent 2x-y-z=2 , x+2y+z=2, 4x-7y-5z=2

[7+8]

2. Verify Cayley - Hamilton theorem and nd A

1

if A =

2

4

2 1 2

1 2 1

1 1 2

3

5

[15]

3. Reduce the quadratic form to canonical from by an orthogonal reduction and state

the nature of the quadratic form 5x

2

+ 26y

2

+ 6xy + 4yz + 14zx. Also nd its rank

signature and index. [15]

4. (a) Using Newton-Raphson's method nd the square root of a number and hence

nd the square root of 24.

(b) Find a real root of the equation x=e

x

, using Bisection method [8+7]

5. (a) Apply Gauss's forward formula to nd f(x) at x = 3.5 from the table below.

X 2 3 4 5

F(x) 2.626 3.454 4.784 6.986

(b) Find sin 45

0

using Gauss's backward interpolation formula given that sin 20

0

= 0.342, sin 30

0

= 0.502, sin 40

0

=0.642, sin 50

0

= 0.766, sin 60

0

=0.866, sin

70

0

= 0.939, sin 80

0

= 0.984. [8+7]

6. (a) Given the following table. Find f

0

(1) and f

00

(3)

x 0 2 4 6 8

f(x) 7 13 43 145 367

(b) Find approximate value of

R

1:04

1

f(x)dxusing the following table.

x 1 1.01 1.02 1.03 1.04

f(x) 3.953 4.066 4.182 4.300 4.421

[8+7]

7. (a) Given that

dy

dx

=

(1+x

2

)y

2

2

, y(0)=1, y(0.1)=1.06, y(0.2)=1.12, y(0.3)=1.21 then

evaluate y(0.4) by Milne`s predictor corrector method

1 of 2

Code No: R10107/R10 Set No. 4

(b) Solve

dy

dx

=

yx

y+x

, y(0)= 1 estimate y(0.1) and y(0.2) using Euler's method in 5

steps [8+7]

8. (a) Fit a least square parabola y= a+bx+cx

2

to the following data

x 1 2 3 4 5

y 5 12 25 44 69

(b) Fit a straight line of the form y= a+bx to the following data

x 1 2 3 4 5

y 5 12 26 60 90

[8+7]

?????

2 of 2

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