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