Download JNTUK (Jawaharlal Nehru Technological University Kakinada) B.Tech Supplementary-Regular 2012 January R10 I Semester (1st Year 1st Sem) Mathematical Methods Question Paper.
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
Page 1 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10 Set No. 3
5.(a) Using Newton ?s interpolation formula given sin 45
0
=0.7071
sin 50
0
= 0.7660, sin 55
0
= 0.8192 and sin 60
0
= 0.8660 find sin 52
0
.
(b) Find y(-2) & y(1.5) from the following data using Lagrange ?s interpolation formula.
x -4 -1 0 2 5
f(x) 1245 33 5 9 1335
[7M+8M]
6.(a) Find First and second derivatives from the data near x = 1.5 using central forward
difference.
x 1 1.2 1.4 1.6 1.8 2
y 2.72 3.32 4.06 4.95 6.05 7.39
6
dx
(b) Using Simpson ?s rule. Evaluate
?
dividing the range into 6 equal parts.
0
1 ? x
2
[7M+8M]
7. Use Milne ?s Method to find y (0.8) from y
|
=1+y
2
, y(0) = 0, find the initial values
y(0.2) , y(0.4) and y(0.6) From Range Kutta method.
[15M]
8.(a) Fit a least square parabola to the following data
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
(b) Fit an exponential curve of the form y (x) = ae
bx
to the following data
x 1 2 3 4 5
y 2.600 3.300 4.200 5.400 6.900
[7M+8M]
Page 2 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10 Set No. 3
5.(a) Using Newton ?s interpolation formula given sin 45
0
=0.7071
sin 50
0
= 0.7660, sin 55
0
= 0.8192 and sin 60
0
= 0.8660 find sin 52
0
.
(b) Find y(-2) & y(1.5) from the following data using Lagrange ?s interpolation formula.
x -4 -1 0 2 5
f(x) 1245 33 5 9 1335
[7M+8M]
6.(a) Find First and second derivatives from the data near x = 1.5 using central forward
difference.
x 1 1.2 1.4 1.6 1.8 2
y 2.72 3.32 4.06 4.95 6.05 7.39
6
dx
(b) Using Simpson ?s rule. Evaluate
?
dividing the range into 6 equal parts.
0
1 ? x
2
[7M+8M]
7. Use Milne ?s Method to find y (0.8) from y
|
=1+y
2
, y(0) = 0, find the initial values
y(0.2) , y(0.4) and y(0.6) From Range Kutta method.
[15M]
8.(a) Fit a least square parabola to the following data
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
(b) Fit an exponential curve of the form y (x) = ae
bx
to the following data
x 1 2 3 4 5
y 2.600 3.300 4.200 5.400 6.900
[7M+8M]
Page 2 of 2
Code No: R10107 / R10
Set No. 4
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic Engineering, Civil
Engineering, Electronics & Instrumentation Engineering, Aeronautical Engineering,
Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the values of a and b for which the equations x + y + z = 3, x + 2y + 2z = 6,
x+ ay + 3z = b have
(i) no solution (ii) infinitely number of solutions (iii) unique solutions.
(b) Solve the following system of equations using Gauss ? Seidel Iteration Method
27x + 6y ? z = 85, 6x + 15y + 2z = 72, x + y + 54z = 110.
[7M+8M]
2.(a) Prove that the two Eigen vectors corresponding to the two different Eigen values are
linearly independent .
1 1 1
(b) Diagonalize the matrix A= 1 1 1 and find A
4
using the model matrix.
1
1 1
[7M+8M]
3.(a) Reduce the Quadratic form to canonical form 3x
2
+2y
2
-4xz by using orthogonal
transformation.
(b) Using Lagrange ?s Reduction Reduce the Quadratic Form
x
1
2
+ 4x
2
2
+ x
3
2
? 4x
1
x
2
+ 2x
3
x
1
? 4x
2
x
3
to canonical form. Also find the
nature, rank, index, signature.
[7M+8M]
4.(a) Using Bisection Method find the root between 2&3 of the equation x
4
-x
3
-2x
2
-6x-4=0 up
to three decimals
(b) using iteration method find an approximate root of the equation x
4
-x-13=0.
[7M+8M]
5.(a) Find log 58.75 from the following data.
x 40 45 50 55 60 65
log x 1.60206 1.65321 1.69897 1.74036 1.77815 1.81291
Using Newton ?s backward interpolation formula.
(b) Using Gauss forward interpolation formula find the value of f(25)
from the following data f(20) =24, f(24)=32, f(28)=35,f(32)=40.
[7M+8M]
Page 1 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10 Set No. 3
5.(a) Using Newton ?s interpolation formula given sin 45
0
=0.7071
sin 50
0
= 0.7660, sin 55
0
= 0.8192 and sin 60
0
= 0.8660 find sin 52
0
.
(b) Find y(-2) & y(1.5) from the following data using Lagrange ?s interpolation formula.
x -4 -1 0 2 5
f(x) 1245 33 5 9 1335
[7M+8M]
6.(a) Find First and second derivatives from the data near x = 1.5 using central forward
difference.
x 1 1.2 1.4 1.6 1.8 2
y 2.72 3.32 4.06 4.95 6.05 7.39
6
dx
(b) Using Simpson ?s rule. Evaluate
?
dividing the range into 6 equal parts.
0
1 ? x
2
[7M+8M]
7. Use Milne ?s Method to find y (0.8) from y
|
=1+y
2
, y(0) = 0, find the initial values
y(0.2) , y(0.4) and y(0.6) From Range Kutta method.
[15M]
8.(a) Fit a least square parabola to the following data
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
(b) Fit an exponential curve of the form y (x) = ae
bx
to the following data
x 1 2 3 4 5
y 2.600 3.300 4.200 5.400 6.900
[7M+8M]
Page 2 of 2
Code No: R10107 / R10
Set No. 4
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic Engineering, Civil
Engineering, Electronics & Instrumentation Engineering, Aeronautical Engineering,
Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the values of a and b for which the equations x + y + z = 3, x + 2y + 2z = 6,
x+ ay + 3z = b have
(i) no solution (ii) infinitely number of solutions (iii) unique solutions.
(b) Solve the following system of equations using Gauss ? Seidel Iteration Method
27x + 6y ? z = 85, 6x + 15y + 2z = 72, x + y + 54z = 110.
[7M+8M]
2.(a) Prove that the two Eigen vectors corresponding to the two different Eigen values are
linearly independent .
1 1 1
(b) Diagonalize the matrix A= 1 1 1 and find A
4
using the model matrix.
1
1 1
[7M+8M]
3.(a) Reduce the Quadratic form to canonical form 3x
2
+2y
2
-4xz by using orthogonal
transformation.
(b) Using Lagrange ?s Reduction Reduce the Quadratic Form
x
1
2
+ 4x
2
2
+ x
3
2
? 4x
1
x
2
+ 2x
3
x
1
? 4x
2
x
3
to canonical form. Also find the
nature, rank, index, signature.
[7M+8M]
4.(a) Using Bisection Method find the root between 2&3 of the equation x
4
-x
3
-2x
2
-6x-4=0 up
to three decimals
(b) using iteration method find an approximate root of the equation x
4
-x-13=0.
[7M+8M]
5.(a) Find log 58.75 from the following data.
x 40 45 50 55 60 65
log x 1.60206 1.65321 1.69897 1.74036 1.77815 1.81291
Using Newton ?s backward interpolation formula.
(b) Using Gauss forward interpolation formula find the value of f(25)
from the following data f(20) =24, f(24)=32, f(28)=35,f(32)=40.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 4
6.(a) find the values of cos (1.74) from the following data.
x 1.7 1.74 1.78 1.82 1.86
Sin x 0.9857 0.9916 0.9781 0.9691 0.9584
? / 2
(b)
Evaluate
?
sin ? d ? using
0
(i) Simpson ?s 1/3 rule (ii) Simpson ?s 1/8 rule taking n = 6
[7M+8M]
7.(a) solve the differential equation
dy
=
1
, y(4) = 4and compute y(4.2) & y(4.4) using
2
dx
x + y
Taylor ?s series method.
(b) solve y
|
= y - x
2
, y(0) =1 by Picard ?s method up to the fourth approximation hence find
the value of y(0.1) , y(0.2) .
[7M+8M]
8.(a) Using least square method, fit a second degree polynomial estimate y at x=6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a least square straight line for the following data.
x 1 2 3 4 5 6
y 6 4 3 5 4 2
[7M+8M]
Page 2 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10 Set No. 3
5.(a) Using Newton ?s interpolation formula given sin 45
0
=0.7071
sin 50
0
= 0.7660, sin 55
0
= 0.8192 and sin 60
0
= 0.8660 find sin 52
0
.
(b) Find y(-2) & y(1.5) from the following data using Lagrange ?s interpolation formula.
x -4 -1 0 2 5
f(x) 1245 33 5 9 1335
[7M+8M]
6.(a) Find First and second derivatives from the data near x = 1.5 using central forward
difference.
x 1 1.2 1.4 1.6 1.8 2
y 2.72 3.32 4.06 4.95 6.05 7.39
6
dx
(b) Using Simpson ?s rule. Evaluate
?
dividing the range into 6 equal parts.
0
1 ? x
2
[7M+8M]
7. Use Milne ?s Method to find y (0.8) from y
|
=1+y
2
, y(0) = 0, find the initial values
y(0.2) , y(0.4) and y(0.6) From Range Kutta method.
[15M]
8.(a) Fit a least square parabola to the following data
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
(b) Fit an exponential curve of the form y (x) = ae
bx
to the following data
x 1 2 3 4 5
y 2.600 3.300 4.200 5.400 6.900
[7M+8M]
Page 2 of 2
Code No: R10107 / R10
Set No. 4
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic Engineering, Civil
Engineering, Electronics & Instrumentation Engineering, Aeronautical Engineering,
Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the values of a and b for which the equations x + y + z = 3, x + 2y + 2z = 6,
x+ ay + 3z = b have
(i) no solution (ii) infinitely number of solutions (iii) unique solutions.
(b) Solve the following system of equations using Gauss ? Seidel Iteration Method
27x + 6y ? z = 85, 6x + 15y + 2z = 72, x + y + 54z = 110.
[7M+8M]
2.(a) Prove that the two Eigen vectors corresponding to the two different Eigen values are
linearly independent .
1 1 1
(b) Diagonalize the matrix A= 1 1 1 and find A
4
using the model matrix.
1
1 1
[7M+8M]
3.(a) Reduce the Quadratic form to canonical form 3x
2
+2y
2
-4xz by using orthogonal
transformation.
(b) Using Lagrange ?s Reduction Reduce the Quadratic Form
x
1
2
+ 4x
2
2
+ x
3
2
? 4x
1
x
2
+ 2x
3
x
1
? 4x
2
x
3
to canonical form. Also find the
nature, rank, index, signature.
[7M+8M]
4.(a) Using Bisection Method find the root between 2&3 of the equation x
4
-x
3
-2x
2
-6x-4=0 up
to three decimals
(b) using iteration method find an approximate root of the equation x
4
-x-13=0.
[7M+8M]
5.(a) Find log 58.75 from the following data.
x 40 45 50 55 60 65
log x 1.60206 1.65321 1.69897 1.74036 1.77815 1.81291
Using Newton ?s backward interpolation formula.
(b) Using Gauss forward interpolation formula find the value of f(25)
from the following data f(20) =24, f(24)=32, f(28)=35,f(32)=40.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 4
6.(a) find the values of cos (1.74) from the following data.
x 1.7 1.74 1.78 1.82 1.86
Sin x 0.9857 0.9916 0.9781 0.9691 0.9584
? / 2
(b)
Evaluate
?
sin ? d ? using
0
(i) Simpson ?s 1/3 rule (ii) Simpson ?s 1/8 rule taking n = 6
[7M+8M]
7.(a) solve the differential equation
dy
=
1
, y(4) = 4and compute y(4.2) & y(4.4) using
2
dx
x + y
Taylor ?s series method.
(b) solve y
|
= y - x
2
, y(0) =1 by Picard ?s method up to the fourth approximation hence find
the value of y(0.1) , y(0.2) .
[7M+8M]
8.(a) Using least square method, fit a second degree polynomial estimate y at x=6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a least square straight line for the following data.
x 1 2 3 4 5 6
y 6 4 3 5 4 2
[7M+8M]
Page 2 of 2
Code No: R10106/R10
R07
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. (a) Write a note on the importance of education on environmental issues and
concerns.
(b) Describe the multidisciplinary nature of environmental studies. [7+8]
2. Discuss the importance of environmental studies with respect to the following
state-ments.
(a) We live in a world wherein natural resources are limited.
(b) Green spaces and gardens are vital to the psychological and physical health
of city dwellers. [7+8]
3. (a) What are the di erent tropic levels of organisms in an ecosystem ?
(b) Why is a complex ecosystem more stable than one with few species? [9+6]
4. (a) Write a brief note on biodiversity and ecosystem diversity.
(b) Explain the evolution of diverse species in an ecosystem. [15]
5. (a) Oceans are ultimate sink for most of the waste we produce. Explain.
(b) List o shore sources of Marine Pollution.
(c) Explain the e ects of oil pollution on the ocean. [7+4+4]
6. Discuss brie y the provision of the following Acts:
(a) The Water ( Prevention Control of Pollution ) Act ,1974
(b) The Air (Prevention and Control of Pollution ) Act, 1981
(c) The Wildlife Protection Act 1971
(d) The Forest Conservation Act of 1980 [4+4+4+3]
7. Explain the relation between population and economic growth from the point of
view of sustainable development. [15]
8. (a) What is the methodology to be followed for study of a studying cause and
e ects of a polluted site? Write also the observations for various aspects
and data to be collected.
1 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10 Set No. 3
5.(a) Using Newton ?s interpolation formula given sin 45
0
=0.7071
sin 50
0
= 0.7660, sin 55
0
= 0.8192 and sin 60
0
= 0.8660 find sin 52
0
.
(b) Find y(-2) & y(1.5) from the following data using Lagrange ?s interpolation formula.
x -4 -1 0 2 5
f(x) 1245 33 5 9 1335
[7M+8M]
6.(a) Find First and second derivatives from the data near x = 1.5 using central forward
difference.
x 1 1.2 1.4 1.6 1.8 2
y 2.72 3.32 4.06 4.95 6.05 7.39
6
dx
(b) Using Simpson ?s rule. Evaluate
?
dividing the range into 6 equal parts.
0
1 ? x
2
[7M+8M]
7. Use Milne ?s Method to find y (0.8) from y
|
=1+y
2
, y(0) = 0, find the initial values
y(0.2) , y(0.4) and y(0.6) From Range Kutta method.
[15M]
8.(a) Fit a least square parabola to the following data
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
(b) Fit an exponential curve of the form y (x) = ae
bx
to the following data
x 1 2 3 4 5
y 2.600 3.300 4.200 5.400 6.900
[7M+8M]
Page 2 of 2
Code No: R10107 / R10
Set No. 4
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic Engineering, Civil
Engineering, Electronics & Instrumentation Engineering, Aeronautical Engineering,
Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the values of a and b for which the equations x + y + z = 3, x + 2y + 2z = 6,
x+ ay + 3z = b have
(i) no solution (ii) infinitely number of solutions (iii) unique solutions.
(b) Solve the following system of equations using Gauss ? Seidel Iteration Method
27x + 6y ? z = 85, 6x + 15y + 2z = 72, x + y + 54z = 110.
[7M+8M]
2.(a) Prove that the two Eigen vectors corresponding to the two different Eigen values are
linearly independent .
1 1 1
(b) Diagonalize the matrix A= 1 1 1 and find A
4
using the model matrix.
1
1 1
[7M+8M]
3.(a) Reduce the Quadratic form to canonical form 3x
2
+2y
2
-4xz by using orthogonal
transformation.
(b) Using Lagrange ?s Reduction Reduce the Quadratic Form
x
1
2
+ 4x
2
2
+ x
3
2
? 4x
1
x
2
+ 2x
3
x
1
? 4x
2
x
3
to canonical form. Also find the
nature, rank, index, signature.
[7M+8M]
4.(a) Using Bisection Method find the root between 2&3 of the equation x
4
-x
3
-2x
2
-6x-4=0 up
to three decimals
(b) using iteration method find an approximate root of the equation x
4
-x-13=0.
[7M+8M]
5.(a) Find log 58.75 from the following data.
x 40 45 50 55 60 65
log x 1.60206 1.65321 1.69897 1.74036 1.77815 1.81291
Using Newton ?s backward interpolation formula.
(b) Using Gauss forward interpolation formula find the value of f(25)
from the following data f(20) =24, f(24)=32, f(28)=35,f(32)=40.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 4
6.(a) find the values of cos (1.74) from the following data.
x 1.7 1.74 1.78 1.82 1.86
Sin x 0.9857 0.9916 0.9781 0.9691 0.9584
? / 2
(b)
Evaluate
?
sin ? d ? using
0
(i) Simpson ?s 1/3 rule (ii) Simpson ?s 1/8 rule taking n = 6
[7M+8M]
7.(a) solve the differential equation
dy
=
1
, y(4) = 4and compute y(4.2) & y(4.4) using
2
dx
x + y
Taylor ?s series method.
(b) solve y
|
= y - x
2
, y(0) =1 by Picard ?s method up to the fourth approximation hence find
the value of y(0.1) , y(0.2) .
[7M+8M]
8.(a) Using least square method, fit a second degree polynomial estimate y at x=6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a least square straight line for the following data.
x 1 2 3 4 5 6
y 6 4 3 5 4 2
[7M+8M]
Page 2 of 2
Code No: R10106/R10
R07
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. (a) Write a note on the importance of education on environmental issues and
concerns.
(b) Describe the multidisciplinary nature of environmental studies. [7+8]
2. Discuss the importance of environmental studies with respect to the following
state-ments.
(a) We live in a world wherein natural resources are limited.
(b) Green spaces and gardens are vital to the psychological and physical health
of city dwellers. [7+8]
3. (a) What are the di erent tropic levels of organisms in an ecosystem ?
(b) Why is a complex ecosystem more stable than one with few species? [9+6]
4. (a) Write a brief note on biodiversity and ecosystem diversity.
(b) Explain the evolution of diverse species in an ecosystem. [15]
5. (a) Oceans are ultimate sink for most of the waste we produce. Explain.
(b) List o shore sources of Marine Pollution.
(c) Explain the e ects of oil pollution on the ocean. [7+4+4]
6. Discuss brie y the provision of the following Acts:
(a) The Water ( Prevention Control of Pollution ) Act ,1974
(b) The Air (Prevention and Control of Pollution ) Act, 1981
(c) The Wildlife Protection Act 1971
(d) The Forest Conservation Act of 1980 [4+4+4+3]
7. Explain the relation between population and economic growth from the point of
view of sustainable development. [15]
8. (a) What is the methodology to be followed for study of a studying cause and
e ects of a polluted site? Write also the observations for various aspects
and data to be collected.
1 of 2
Code No: R10106/R10
R07
(b) Write about any polluted site you have visited and describe your ndings in
detail. [8+7]
? ? ? ? ?
2 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10 Set No. 3
5.(a) Using Newton ?s interpolation formula given sin 45
0
=0.7071
sin 50
0
= 0.7660, sin 55
0
= 0.8192 and sin 60
0
= 0.8660 find sin 52
0
.
(b) Find y(-2) & y(1.5) from the following data using Lagrange ?s interpolation formula.
x -4 -1 0 2 5
f(x) 1245 33 5 9 1335
[7M+8M]
6.(a) Find First and second derivatives from the data near x = 1.5 using central forward
difference.
x 1 1.2 1.4 1.6 1.8 2
y 2.72 3.32 4.06 4.95 6.05 7.39
6
dx
(b) Using Simpson ?s rule. Evaluate
?
dividing the range into 6 equal parts.
0
1 ? x
2
[7M+8M]
7. Use Milne ?s Method to find y (0.8) from y
|
=1+y
2
, y(0) = 0, find the initial values
y(0.2) , y(0.4) and y(0.6) From Range Kutta method.
[15M]
8.(a) Fit a least square parabola to the following data
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
(b) Fit an exponential curve of the form y (x) = ae
bx
to the following data
x 1 2 3 4 5
y 2.600 3.300 4.200 5.400 6.900
[7M+8M]
Page 2 of 2
Code No: R10107 / R10
Set No. 4
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic Engineering, Civil
Engineering, Electronics & Instrumentation Engineering, Aeronautical Engineering,
Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the values of a and b for which the equations x + y + z = 3, x + 2y + 2z = 6,
x+ ay + 3z = b have
(i) no solution (ii) infinitely number of solutions (iii) unique solutions.
(b) Solve the following system of equations using Gauss ? Seidel Iteration Method
27x + 6y ? z = 85, 6x + 15y + 2z = 72, x + y + 54z = 110.
[7M+8M]
2.(a) Prove that the two Eigen vectors corresponding to the two different Eigen values are
linearly independent .
1 1 1
(b) Diagonalize the matrix A= 1 1 1 and find A
4
using the model matrix.
1
1 1
[7M+8M]
3.(a) Reduce the Quadratic form to canonical form 3x
2
+2y
2
-4xz by using orthogonal
transformation.
(b) Using Lagrange ?s Reduction Reduce the Quadratic Form
x
1
2
+ 4x
2
2
+ x
3
2
? 4x
1
x
2
+ 2x
3
x
1
? 4x
2
x
3
to canonical form. Also find the
nature, rank, index, signature.
[7M+8M]
4.(a) Using Bisection Method find the root between 2&3 of the equation x
4
-x
3
-2x
2
-6x-4=0 up
to three decimals
(b) using iteration method find an approximate root of the equation x
4
-x-13=0.
[7M+8M]
5.(a) Find log 58.75 from the following data.
x 40 45 50 55 60 65
log x 1.60206 1.65321 1.69897 1.74036 1.77815 1.81291
Using Newton ?s backward interpolation formula.
(b) Using Gauss forward interpolation formula find the value of f(25)
from the following data f(20) =24, f(24)=32, f(28)=35,f(32)=40.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 4
6.(a) find the values of cos (1.74) from the following data.
x 1.7 1.74 1.78 1.82 1.86
Sin x 0.9857 0.9916 0.9781 0.9691 0.9584
? / 2
(b)
Evaluate
?
sin ? d ? using
0
(i) Simpson ?s 1/3 rule (ii) Simpson ?s 1/8 rule taking n = 6
[7M+8M]
7.(a) solve the differential equation
dy
=
1
, y(4) = 4and compute y(4.2) & y(4.4) using
2
dx
x + y
Taylor ?s series method.
(b) solve y
|
= y - x
2
, y(0) =1 by Picard ?s method up to the fourth approximation hence find
the value of y(0.1) , y(0.2) .
[7M+8M]
8.(a) Using least square method, fit a second degree polynomial estimate y at x=6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a least square straight line for the following data.
x 1 2 3 4 5 6
y 6 4 3 5 4 2
[7M+8M]
Page 2 of 2
Code No: R10106/R10
R07
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. (a) Write a note on the importance of education on environmental issues and
concerns.
(b) Describe the multidisciplinary nature of environmental studies. [7+8]
2. Discuss the importance of environmental studies with respect to the following
state-ments.
(a) We live in a world wherein natural resources are limited.
(b) Green spaces and gardens are vital to the psychological and physical health
of city dwellers. [7+8]
3. (a) What are the di erent tropic levels of organisms in an ecosystem ?
(b) Why is a complex ecosystem more stable than one with few species? [9+6]
4. (a) Write a brief note on biodiversity and ecosystem diversity.
(b) Explain the evolution of diverse species in an ecosystem. [15]
5. (a) Oceans are ultimate sink for most of the waste we produce. Explain.
(b) List o shore sources of Marine Pollution.
(c) Explain the e ects of oil pollution on the ocean. [7+4+4]
6. Discuss brie y the provision of the following Acts:
(a) The Water ( Prevention Control of Pollution ) Act ,1974
(b) The Air (Prevention and Control of Pollution ) Act, 1981
(c) The Wildlife Protection Act 1971
(d) The Forest Conservation Act of 1980 [4+4+4+3]
7. Explain the relation between population and economic growth from the point of
view of sustainable development. [15]
8. (a) What is the methodology to be followed for study of a studying cause and
e ects of a polluted site? Write also the observations for various aspects
and data to be collected.
1 of 2
Code No: R10106/R10
R07
(b) Write about any polluted site you have visited and describe your ndings in
detail. [8+7]
? ? ? ? ?
2 of 2
Code No: R10106/R10 Set No. 2
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. Write a detailed note on the various institutions and organizations in the eld of
Environment Education and training, their activities and focal areas. [15]
2. (a) Write a detailed note on the problems arising out of overexploitation of
forest resources.
(b) Describe how forest management is being done in India by citing any example.
[6+9]
3. (a) Brie y write about the di erent kinds of grasslands in India, also stating the
main activities in such areas.
(b) What steps can be taken to conserve grasslands and what are the common
reasons for destruction of these ecosystem? [7+8]
4. (a) What do you understand by endemic and endangered species ? How are
they categorized? Give some examples of such spcies in India.
(b) List some common plant and animal species of India. [9+6]
5. (a) Enumerate the diseases and other problems caused by soil pollution.
(b) How do you control soil pollution? [8+7]
6. (a) Explain the phenomenon of Global Warming and the factors contributing to
it.
(b) Explain the possible impacts of Global Warming on the food supply.
(c) What are the measures taken at Global level to control the emmission of Green
House Gases? [5+5+5]
7. (a) De ne Health Impact Assement.
(b) Outline some of the important strategies which must be taken up to minimize
environmental hazards . [4+11]
8. (a) Describe how you would methodically record the elements and resources
in an ecosystem and assess its functioning.
(b) Based on your led visits, summarize your observations and ndings of the
water resource ecosystem in your region. [8+7]
? ? ? ? ?
1 of 1
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10 Set No. 3
5.(a) Using Newton ?s interpolation formula given sin 45
0
=0.7071
sin 50
0
= 0.7660, sin 55
0
= 0.8192 and sin 60
0
= 0.8660 find sin 52
0
.
(b) Find y(-2) & y(1.5) from the following data using Lagrange ?s interpolation formula.
x -4 -1 0 2 5
f(x) 1245 33 5 9 1335
[7M+8M]
6.(a) Find First and second derivatives from the data near x = 1.5 using central forward
difference.
x 1 1.2 1.4 1.6 1.8 2
y 2.72 3.32 4.06 4.95 6.05 7.39
6
dx
(b) Using Simpson ?s rule. Evaluate
?
dividing the range into 6 equal parts.
0
1 ? x
2
[7M+8M]
7. Use Milne ?s Method to find y (0.8) from y
|
=1+y
2
, y(0) = 0, find the initial values
y(0.2) , y(0.4) and y(0.6) From Range Kutta method.
[15M]
8.(a) Fit a least square parabola to the following data
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
(b) Fit an exponential curve of the form y (x) = ae
bx
to the following data
x 1 2 3 4 5
y 2.600 3.300 4.200 5.400 6.900
[7M+8M]
Page 2 of 2
Code No: R10107 / R10
Set No. 4
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic Engineering, Civil
Engineering, Electronics & Instrumentation Engineering, Aeronautical Engineering,
Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the values of a and b for which the equations x + y + z = 3, x + 2y + 2z = 6,
x+ ay + 3z = b have
(i) no solution (ii) infinitely number of solutions (iii) unique solutions.
(b) Solve the following system of equations using Gauss ? Seidel Iteration Method
27x + 6y ? z = 85, 6x + 15y + 2z = 72, x + y + 54z = 110.
[7M+8M]
2.(a) Prove that the two Eigen vectors corresponding to the two different Eigen values are
linearly independent .
1 1 1
(b) Diagonalize the matrix A= 1 1 1 and find A
4
using the model matrix.
1
1 1
[7M+8M]
3.(a) Reduce the Quadratic form to canonical form 3x
2
+2y
2
-4xz by using orthogonal
transformation.
(b) Using Lagrange ?s Reduction Reduce the Quadratic Form
x
1
2
+ 4x
2
2
+ x
3
2
? 4x
1
x
2
+ 2x
3
x
1
? 4x
2
x
3
to canonical form. Also find the
nature, rank, index, signature.
[7M+8M]
4.(a) Using Bisection Method find the root between 2&3 of the equation x
4
-x
3
-2x
2
-6x-4=0 up
to three decimals
(b) using iteration method find an approximate root of the equation x
4
-x-13=0.
[7M+8M]
5.(a) Find log 58.75 from the following data.
x 40 45 50 55 60 65
log x 1.60206 1.65321 1.69897 1.74036 1.77815 1.81291
Using Newton ?s backward interpolation formula.
(b) Using Gauss forward interpolation formula find the value of f(25)
from the following data f(20) =24, f(24)=32, f(28)=35,f(32)=40.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 4
6.(a) find the values of cos (1.74) from the following data.
x 1.7 1.74 1.78 1.82 1.86
Sin x 0.9857 0.9916 0.9781 0.9691 0.9584
? / 2
(b)
Evaluate
?
sin ? d ? using
0
(i) Simpson ?s 1/3 rule (ii) Simpson ?s 1/8 rule taking n = 6
[7M+8M]
7.(a) solve the differential equation
dy
=
1
, y(4) = 4and compute y(4.2) & y(4.4) using
2
dx
x + y
Taylor ?s series method.
(b) solve y
|
= y - x
2
, y(0) =1 by Picard ?s method up to the fourth approximation hence find
the value of y(0.1) , y(0.2) .
[7M+8M]
8.(a) Using least square method, fit a second degree polynomial estimate y at x=6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a least square straight line for the following data.
x 1 2 3 4 5 6
y 6 4 3 5 4 2
[7M+8M]
Page 2 of 2
Code No: R10106/R10
R07
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. (a) Write a note on the importance of education on environmental issues and
concerns.
(b) Describe the multidisciplinary nature of environmental studies. [7+8]
2. Discuss the importance of environmental studies with respect to the following
state-ments.
(a) We live in a world wherein natural resources are limited.
(b) Green spaces and gardens are vital to the psychological and physical health
of city dwellers. [7+8]
3. (a) What are the di erent tropic levels of organisms in an ecosystem ?
(b) Why is a complex ecosystem more stable than one with few species? [9+6]
4. (a) Write a brief note on biodiversity and ecosystem diversity.
(b) Explain the evolution of diverse species in an ecosystem. [15]
5. (a) Oceans are ultimate sink for most of the waste we produce. Explain.
(b) List o shore sources of Marine Pollution.
(c) Explain the e ects of oil pollution on the ocean. [7+4+4]
6. Discuss brie y the provision of the following Acts:
(a) The Water ( Prevention Control of Pollution ) Act ,1974
(b) The Air (Prevention and Control of Pollution ) Act, 1981
(c) The Wildlife Protection Act 1971
(d) The Forest Conservation Act of 1980 [4+4+4+3]
7. Explain the relation between population and economic growth from the point of
view of sustainable development. [15]
8. (a) What is the methodology to be followed for study of a studying cause and
e ects of a polluted site? Write also the observations for various aspects
and data to be collected.
1 of 2
Code No: R10106/R10
R07
(b) Write about any polluted site you have visited and describe your ndings in
detail. [8+7]
? ? ? ? ?
2 of 2
Code No: R10106/R10 Set No. 2
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. Write a detailed note on the various institutions and organizations in the eld of
Environment Education and training, their activities and focal areas. [15]
2. (a) Write a detailed note on the problems arising out of overexploitation of
forest resources.
(b) Describe how forest management is being done in India by citing any example.
[6+9]
3. (a) Brie y write about the di erent kinds of grasslands in India, also stating the
main activities in such areas.
(b) What steps can be taken to conserve grasslands and what are the common
reasons for destruction of these ecosystem? [7+8]
4. (a) What do you understand by endemic and endangered species ? How are
they categorized? Give some examples of such spcies in India.
(b) List some common plant and animal species of India. [9+6]
5. (a) Enumerate the diseases and other problems caused by soil pollution.
(b) How do you control soil pollution? [8+7]
6. (a) Explain the phenomenon of Global Warming and the factors contributing to
it.
(b) Explain the possible impacts of Global Warming on the food supply.
(c) What are the measures taken at Global level to control the emmission of Green
House Gases? [5+5+5]
7. (a) De ne Health Impact Assement.
(b) Outline some of the important strategies which must be taken up to minimize
environmental hazards . [4+11]
8. (a) Describe how you would methodically record the elements and resources
in an ecosystem and assess its functioning.
(b) Based on your led visits, summarize your observations and ndings of the
water resource ecosystem in your region. [8+7]
? ? ? ? ?
1 of 1
Code No: R10106/R10 Set No. 3
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. Explain how the ideas and activities of some internationally known environmental
thinkers has in uenced environment policy. [15]
2. (a) Why is it important to conserve forest ecosystems?
(b) What are the ways in which forest resources are misused and what is the
outcome? [8+7]
3. (a) Explain the term `energy cycle' and how the organisms in the ecosystem
are dependent on it.
(b) What is ecological succession? What are the di erent stages of development
of an ecosystem? [8+7]
4. (a) Explain the concept of ex-situ conservation and illustrate your answer with
examples.
(b) What is an Integrated Protected Areas and how does it help in conservation
of biological diversity. [9+6]
5. (a) List the wastes that are prohibited from processing along with municipal
solid waste. Discuss.
(b) Brie y describe the methods of heating and disposal of solid waste. [8+7]
6. (a) What are the major issues associated with resettlement and rehabilitation?
(b) Bring out the main elements of water conservation. [8+7]
7. Explain with examples the links between the activities of man which are hazardous
to human health and environment. [15]
8. List and write brie y the main characteristics of any ve plant and ve animal
species which belong to your region or any area which you have studied. [8+7]
? ? ? ? ?
1 of 1
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
8 1 3 6
1.(a) Reduce the matrix 0 3 2 2 in to its normal form and hence find its Rank.
?
8
?
1
?
3
4
(b) Solve the following system of equations using gauss elimination method
2x
1
+x
2
+2x
3
+x
4
=6, 6x
1
-x
2
+6x
3
+12x
4
=36
4x
1
+3x
2
+3x
3
-3x
4
=1,2x
1
+2x
2
-x
3
+x
4
=10.
[7M+8M]
2.(a) Prove that the sum of the Eigen values of a square matrix is equal to its trace of the
matrix and Product of the Eigen values is equal to its determinant
(b) Verify cayley ?Hamilton theorem and hence find its inverse of the matrix
1 0 1
A= 2 1 ? 1 .
?
1
1
1
[7M+8M]
3. Reduce the quadratic from x
2
+3y
2
+3z
2
+4t
2
+4xy- 2xz+6xt+4yt+2yz the canonical
from and hence find the nature, index, rank , and signature of the quadratic from.
[15M]
4.(a) Find a root of the equation x
3
- x - 4 = 0 using regula false method.
(b) Find a real root of the equation xe
x
- cos x = 0 using Newton-Raphson method.
? n ? 1 ? 1
5.(a) Evaluate (i) tan
1
= tan
1
n
2 2n
[7M+8M]
2
n
ax + b
sin( px + q)(iii) e (ii)
(b) Appling Newton ?s forward interpolation formula , compute the value of 5.5 , given that 5
=2.236, 6 = 2.449, 7 = 2.646 , 8 = 2.828
[7M+8M]
Page 1 of 2
Set No. 1
Code No: R10107 / R10
6.(a) Find the first derivative of the function tabulated below at the point x=1.5.
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 3.375 7.0 13.625 24 38.87 59
1
(b)
Evaluate
?
e
?
x
2
dx using
0
(i) Simpson ?s 1/3 rule taking h=0.2 (ii) Trapezoidal rule.
[7M+8M]
7. (a) Find y(0.2) using modified Euler ?s method given that
dy
= x ? y, y(0) = 1, with h = 0.1
dx
(b) Find y (0.1) and y (0.2) using Runge -Kutta method fourth order given that
y' = xy + y
2
, y (0) =1.
[7M+8M]
8.(a) Fit a power function to the following data and estimate y at x=12.
Price 20 16 10 11 14
Demand 22 14 120 89 56
(b) Fit a least square parabola to the following data.
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
[7M+8M]
Page 2 of 2
Set No. 2
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Reduce the matrix to Echelon form and hence find its Rank
2 ? 4 3 ?1 0
? 2 ? 1 ?4
A =
1 2
?
0 1 1 3 1
4
? 7 4 ?4 5
(b) Solve the equations
10x
1
+ x
2
+ x
3
= 12, x
1
+ 10x
2
? x
3
= 10 and x
1
? 2x
2
+ 10x
3
= 9 by Gauss Joldan method.
[7M+8M]
2.(a) Find the Eigen Values and Eigen vectors of A
-1
. Where
?2 2 ?3
A = 2 1 ?6
? ?
2 0
1
(b) State and Prove Cayley ? Hamilton theorem.
[7M+8M]
3. Reduce the Quadratic form 3x
2
+ 3 y
2
+ 3z
2
+ 2 xy + 2xz ? 2 yz into sum of squares
form by an orthogonal transformation and hence find nature, rank, index and signature.
[15M]
4.(a) Find a real root of xe
x
= 2 using Regula ?Falsi method.
(b) Find real root of the equation 1 + tan
-1
x ? x = 0 near x = 1 correct up to 4 decimal
places using iteration method.
[7M+8M]
5.(a) Find f (1.28). If f (1.15) = 1.0723, f (1.20) = 1.0954, f (1.25) = 1.1180,
and f (1.30) = 1.1401.
(b) Find the cubic polynomial which takes the values
x 0 1 2 5
f(x) 2 3 12 147
using Lagranges interpolation formula.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 2
6.(a) Find the values of f
|
(1) using the data.
x 1.0 1.5 2.0 2.5 3.0
f(x) 27 106.75 324 783.75 1621
? / 2
?
e
sin
x
.dx taking h= ? /6
(b) Evaluate
0
using
(i) Trapezoidal rule.
(ii)Simpson ?s 1/3rule.
[7M+8M]
7. Find the solution of
dy
= x ? y , y(0) = 1. at x = 0.4 and h = 0.1using Miline ?s method.
dx
Use Euler ?s modified method to evaluate y(0.1), y(0.2) and y(0.3).
[15M]
8.(a) Using least square method fit a second degree polynomial estimate y at x = 6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a power curve of the form y(x) = ax
b
to the data.
x 1 2 3 4 5 6
y 4.0 5.7 6.9 8.0 8.9 9.8
[7M+8M]
Page 2 of 2
Set No. 3
Code No: R10107 / R10
I B.Tech I Semester Regular/Supplementary Examinations January
2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic
Engineering, Civil Engineering, Electronics & Instrumentation Engineering,
Aeronautical Engineering, Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the non ?singular matrices P&Q such that P A Q is in the normal from where
1 3 6 ? 1
A = 1 4 5 1
5 4
1
3
(b) Solve x + 2y + z = 3, 2x + 3y + 2z = 5, 3x - 5y + 5z = 2, 3x + 9y ? z = 4.
[7M+8M]
2.(a) Find the Eigen Values and the corresponding Eigen vectors of the matrix
? 22 ? 3
2 1 ? 6
?
1
?
20
(b) State Cayley ? Hamilton theorem. Find the characteristic Equation of the matrix
2 1 1
A = 0 1 0 and hence find the matrix represented by
1
1
2
A
8
? 5 A
7
+ 7 A
6
? 3A
5
+ A
4
? 5 A
3
+ 8 A
2
? 2 A + I .
[7M+8M]
3.(a) Reduce the following Quadratic from to canonical form by diagonalization
6x
2
+ 3 y
2
+ 3z
2
? 4 yz ? 4zx ? 2 xy
(b) Using Lagrange ?s reduction, transform
x
1
2
? 4x
2
2
+ 5x
3
2
+ 2x
1
x
2
? 4x
1
x
3
+ 2x
4
2
? 6x
3
x
4
to canonical form and
hence find rank, nature, index and signature.
[7M+8M]
4.(a) Using Bisection method find a square root of 26 correct up to three decimal places.
(b) Using Newton Raphson method compute 41 correct to Four decimal places.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10 Set No. 3
5.(a) Using Newton ?s interpolation formula given sin 45
0
=0.7071
sin 50
0
= 0.7660, sin 55
0
= 0.8192 and sin 60
0
= 0.8660 find sin 52
0
.
(b) Find y(-2) & y(1.5) from the following data using Lagrange ?s interpolation formula.
x -4 -1 0 2 5
f(x) 1245 33 5 9 1335
[7M+8M]
6.(a) Find First and second derivatives from the data near x = 1.5 using central forward
difference.
x 1 1.2 1.4 1.6 1.8 2
y 2.72 3.32 4.06 4.95 6.05 7.39
6
dx
(b) Using Simpson ?s rule. Evaluate
?
dividing the range into 6 equal parts.
0
1 ? x
2
[7M+8M]
7. Use Milne ?s Method to find y (0.8) from y
|
=1+y
2
, y(0) = 0, find the initial values
y(0.2) , y(0.4) and y(0.6) From Range Kutta method.
[15M]
8.(a) Fit a least square parabola to the following data
x 0 0.2 0.4 0.7 0.9 1.0
y 1.016 0.768 0.648 0.401 0.272 0.193
(b) Fit an exponential curve of the form y (x) = ae
bx
to the following data
x 1 2 3 4 5
y 2.600 3.300 4.200 5.400 6.900
[7M+8M]
Page 2 of 2
Code No: R10107 / R10
Set No. 4
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICAL METHODS
(Common to Computer Science Engineering, Electrical & Electronic Engineering, Civil
Engineering, Electronics & Instrumentation Engineering, Aeronautical Engineering,
Bio-Technology & Automobile Engineering.)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the values of a and b for which the equations x + y + z = 3, x + 2y + 2z = 6,
x+ ay + 3z = b have
(i) no solution (ii) infinitely number of solutions (iii) unique solutions.
(b) Solve the following system of equations using Gauss ? Seidel Iteration Method
27x + 6y ? z = 85, 6x + 15y + 2z = 72, x + y + 54z = 110.
[7M+8M]
2.(a) Prove that the two Eigen vectors corresponding to the two different Eigen values are
linearly independent .
1 1 1
(b) Diagonalize the matrix A= 1 1 1 and find A
4
using the model matrix.
1
1 1
[7M+8M]
3.(a) Reduce the Quadratic form to canonical form 3x
2
+2y
2
-4xz by using orthogonal
transformation.
(b) Using Lagrange ?s Reduction Reduce the Quadratic Form
x
1
2
+ 4x
2
2
+ x
3
2
? 4x
1
x
2
+ 2x
3
x
1
? 4x
2
x
3
to canonical form. Also find the
nature, rank, index, signature.
[7M+8M]
4.(a) Using Bisection Method find the root between 2&3 of the equation x
4
-x
3
-2x
2
-6x-4=0 up
to three decimals
(b) using iteration method find an approximate root of the equation x
4
-x-13=0.
[7M+8M]
5.(a) Find log 58.75 from the following data.
x 40 45 50 55 60 65
log x 1.60206 1.65321 1.69897 1.74036 1.77815 1.81291
Using Newton ?s backward interpolation formula.
(b) Using Gauss forward interpolation formula find the value of f(25)
from the following data f(20) =24, f(24)=32, f(28)=35,f(32)=40.
[7M+8M]
Page 1 of 2
Code No: R10107 / R10
Set No. 4
6.(a) find the values of cos (1.74) from the following data.
x 1.7 1.74 1.78 1.82 1.86
Sin x 0.9857 0.9916 0.9781 0.9691 0.9584
? / 2
(b)
Evaluate
?
sin ? d ? using
0
(i) Simpson ?s 1/3 rule (ii) Simpson ?s 1/8 rule taking n = 6
[7M+8M]
7.(a) solve the differential equation
dy
=
1
, y(4) = 4and compute y(4.2) & y(4.4) using
2
dx
x + y
Taylor ?s series method.
(b) solve y
|
= y - x
2
, y(0) =1 by Picard ?s method up to the fourth approximation hence find
the value of y(0.1) , y(0.2) .
[7M+8M]
8.(a) Using least square method, fit a second degree polynomial estimate y at x=6.5
x 0 1 2 3 4 5 6 7 8
y 12 10.5 10 8 7 8 7.5 8.5 9
(b) Fit a least square straight line for the following data.
x 1 2 3 4 5 6
y 6 4 3 5 4 2
[7M+8M]
Page 2 of 2
Code No: R10106/R10
R07
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. (a) Write a note on the importance of education on environmental issues and
concerns.
(b) Describe the multidisciplinary nature of environmental studies. [7+8]
2. Discuss the importance of environmental studies with respect to the following
state-ments.
(a) We live in a world wherein natural resources are limited.
(b) Green spaces and gardens are vital to the psychological and physical health
of city dwellers. [7+8]
3. (a) What are the di erent tropic levels of organisms in an ecosystem ?
(b) Why is a complex ecosystem more stable than one with few species? [9+6]
4. (a) Write a brief note on biodiversity and ecosystem diversity.
(b) Explain the evolution of diverse species in an ecosystem. [15]
5. (a) Oceans are ultimate sink for most of the waste we produce. Explain.
(b) List o shore sources of Marine Pollution.
(c) Explain the e ects of oil pollution on the ocean. [7+4+4]
6. Discuss brie y the provision of the following Acts:
(a) The Water ( Prevention Control of Pollution ) Act ,1974
(b) The Air (Prevention and Control of Pollution ) Act, 1981
(c) The Wildlife Protection Act 1971
(d) The Forest Conservation Act of 1980 [4+4+4+3]
7. Explain the relation between population and economic growth from the point of
view of sustainable development. [15]
8. (a) What is the methodology to be followed for study of a studying cause and
e ects of a polluted site? Write also the observations for various aspects
and data to be collected.
1 of 2
Code No: R10106/R10
R07
(b) Write about any polluted site you have visited and describe your ndings in
detail. [8+7]
? ? ? ? ?
2 of 2
Code No: R10106/R10 Set No. 2
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. Write a detailed note on the various institutions and organizations in the eld of
Environment Education and training, their activities and focal areas. [15]
2. (a) Write a detailed note on the problems arising out of overexploitation of
forest resources.
(b) Describe how forest management is being done in India by citing any example.
[6+9]
3. (a) Brie y write about the di erent kinds of grasslands in India, also stating the
main activities in such areas.
(b) What steps can be taken to conserve grasslands and what are the common
reasons for destruction of these ecosystem? [7+8]
4. (a) What do you understand by endemic and endangered species ? How are
they categorized? Give some examples of such spcies in India.
(b) List some common plant and animal species of India. [9+6]
5. (a) Enumerate the diseases and other problems caused by soil pollution.
(b) How do you control soil pollution? [8+7]
6. (a) Explain the phenomenon of Global Warming and the factors contributing to
it.
(b) Explain the possible impacts of Global Warming on the food supply.
(c) What are the measures taken at Global level to control the emmission of Green
House Gases? [5+5+5]
7. (a) De ne Health Impact Assement.
(b) Outline some of the important strategies which must be taken up to minimize
environmental hazards . [4+11]
8. (a) Describe how you would methodically record the elements and resources
in an ecosystem and assess its functioning.
(b) Based on your led visits, summarize your observations and ndings of the
water resource ecosystem in your region. [8+7]
? ? ? ? ?
1 of 1
Code No: R10106/R10 Set No. 3
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. Explain how the ideas and activities of some internationally known environmental
thinkers has in uenced environment policy. [15]
2. (a) Why is it important to conserve forest ecosystems?
(b) What are the ways in which forest resources are misused and what is the
outcome? [8+7]
3. (a) Explain the term `energy cycle' and how the organisms in the ecosystem
are dependent on it.
(b) What is ecological succession? What are the di erent stages of development
of an ecosystem? [8+7]
4. (a) Explain the concept of ex-situ conservation and illustrate your answer with
examples.
(b) What is an Integrated Protected Areas and how does it help in conservation
of biological diversity. [9+6]
5. (a) List the wastes that are prohibited from processing along with municipal
solid waste. Discuss.
(b) Brie y describe the methods of heating and disposal of solid waste. [8+7]
6. (a) What are the major issues associated with resettlement and rehabilitation?
(b) Bring out the main elements of water conservation. [8+7]
7. Explain with examples the links between the activities of man which are hazardous
to human health and environment. [15]
8. List and write brie y the main characteristics of any ve plant and ve animal
species which belong to your region or any area which you have studied. [8+7]
? ? ? ? ?
1 of 1
Code No: R10106/R10 Set No. 4
I B.Tech I Semester Regular/Supplementary Examinations, Jan 2012
ENVIRONMENTAL STUDIES
( Common to Mechanical Engineering, Electronics & Communication Engineering,
Chemical Engineering, Bio-Medical Engineering, Information Technology,
Electronics & Control Engineering, Production Technology and Mining)
Time: 3 hours Max Marks: 75 Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. Mention brie y the contributions made by the following:
(a) BNHS
(b) Indira Gandhi
(c) Botanical Survey of India
(d) Madhav Gadgil [4+4+4+3]
2. (a) Why is it important to conserve forest ecosystems?
(b) What are the ways in which forest resources are misused and what is the
outcome? [8+7]
3. How do di erent development activities, including construction of dams, a ect the
various aquatic ecosystems and what actions need to be taken to conserve them?
[8+7]
4. (a) Explain the concept of in-situ conservation of biodiversity. Illustrate your
answer with examples.
(b) What is an Integrated Protected Area System? How do these contribute to
preservation of biodiversity? [9+6]
5. (a) What is signi cance of the term inversion in the dissipation of pollutants in
the atmosphere ?
(b) List the meteorological parameters in uencing the disposal of air pollutants
in the atmosphere. [15]
6. (a) What are the ways in which individuals can help us in environmental man-
agement.
(b) Describe Narmada Bachao Andolan. [7+8]
7. (a) Explain the importance of value education in the context of the environment.
(b) Write a note on environmental values. [7+8]
8. Explain the causes and e ects of air pollution by describing any urban or industrial
area that you have studied. [15]
? ? ? ? ?
1 of 1
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