Download OU B-Tech First Year 2012 January 5002 Mathematics I Question Paper

Download OU (Osmania University) B.Tech (Bachelor of Technology) First Year (1st Year) 2012 January 5002 Mathematics I Question Paper

1111111111111111111311 Code No. 5002/N
FACULTY OF ENGINEERING AND INFORMATICS
S.E. 1 Year (New) (Common to All Branches) (Suppl.) Examination,
January 2012
MATHEMATICS ? I
Time: 3 Hours] [Max. Marks: 75
Note: Answer all questions from Part A Answer any five questions from Part B.,
PART A (25 Marks)
1. Are these vectors linearly dependent ? Verify.
(4, 2, 1) (2, 3, 2) (1, 1, 4). 3
2. Find the sum of the Eigen values of A 2
2 1 3 2
5 6 3 2
A= 3 4 1 2
1 0 0 2
3. Test for convergence 1(
?

1

2n+1 ?
n+1
4. Discuss the convergence
2n+5 ?
5. Expand f(x) = cot x about x =
7
-
1
4

6. Find the radius of curvature at origin of the curve y x = x
2
2xy + y
2
.
urn
xy
7.
Determine (x,y)-*(0,0) x
2
+y
2

2
3
2
3
(This paper contains 3 pages) P.T.O.
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1111111111111111111311 Code No. 5002/N
FACULTY OF ENGINEERING AND INFORMATICS
S.E. 1 Year (New) (Common to All Branches) (Suppl.) Examination,
January 2012
MATHEMATICS ? I
Time: 3 Hours] [Max. Marks: 75
Note: Answer all questions from Part A Answer any five questions from Part B.,
PART A (25 Marks)
1. Are these vectors linearly dependent ? Verify.
(4, 2, 1) (2, 3, 2) (1, 1, 4). 3
2. Find the sum of the Eigen values of A 2
2 1 3 2
5 6 3 2
A= 3 4 1 2
1 0 0 2
3. Test for convergence 1(
?

1

2n+1 ?
n+1
4. Discuss the convergence
2n+5 ?
5. Expand f(x) = cot x about x =
7
-
1
4

6. Find the radius of curvature at origin of the curve y x = x
2
2xy + y
2
.
urn
xy
7.
Determine (x,y)-*(0,0) x
2
+y
2

2
3
2
3
(This paper contains 3 pages) P.T.O.
1101111111111111111111 Code No. 50021N
8. If u = sin -1
x
+ tan-1 YX
(
, show that x ?
au
+ y ?
au
= 0
y
ax ay
2
9. Evaluate : fx
2
y dxdy over the first quadrant of x
2
+ y
2
= 1. 3
10. Find the directional derivative of f(x, y) = x
3
y
2
at (1, 1) in the direction
of 2i +3j .
2
PART - B (5x10=50 Marks)
11. a) Using Cayley-Hamilton theorem, find the inverse 5
b) Reduce the quadratic forms to Canonical forms x1 + 3x + 3x -2x
2
x
3
. 5
n! 2n
12. a) Discuss the convergence of L,
n=1 n
n
5
b) Test the series E
1

for convergence. 5
13. a) Verify Lagranges Mean Value theorem for f(x) = x
3
- 3x- 1 in (- 11/7, 13t7). 5
y
b) Find the envelope of the family of straight lines x -
a
+-
b
= 1 where a + b = c, c is
constant. 5
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1111111111111111111311 Code No. 5002/N
FACULTY OF ENGINEERING AND INFORMATICS
S.E. 1 Year (New) (Common to All Branches) (Suppl.) Examination,
January 2012
MATHEMATICS ? I
Time: 3 Hours] [Max. Marks: 75
Note: Answer all questions from Part A Answer any five questions from Part B.,
PART A (25 Marks)
1. Are these vectors linearly dependent ? Verify.
(4, 2, 1) (2, 3, 2) (1, 1, 4). 3
2. Find the sum of the Eigen values of A 2
2 1 3 2
5 6 3 2
A= 3 4 1 2
1 0 0 2
3. Test for convergence 1(
?

1

2n+1 ?
n+1
4. Discuss the convergence
2n+5 ?
5. Expand f(x) = cot x about x =
7
-
1
4

6. Find the radius of curvature at origin of the curve y x = x
2
2xy + y
2
.
urn
xy
7.
Determine (x,y)-*(0,0) x
2
+y
2

2
3
2
3
(This paper contains 3 pages) P.T.O.
1101111111111111111111 Code No. 50021N
8. If u = sin -1
x
+ tan-1 YX
(
, show that x ?
au
+ y ?
au
= 0
y
ax ay
2
9. Evaluate : fx
2
y dxdy over the first quadrant of x
2
+ y
2
= 1. 3
10. Find the directional derivative of f(x, y) = x
3
y
2
at (1, 1) in the direction
of 2i +3j .
2
PART - B (5x10=50 Marks)
11. a) Using Cayley-Hamilton theorem, find the inverse 5
b) Reduce the quadratic forms to Canonical forms x1 + 3x + 3x -2x
2
x
3
. 5
n! 2n
12. a) Discuss the convergence of L,
n=1 n
n
5
b) Test the series E
1

for convergence. 5
13. a) Verify Lagranges Mean Value theorem for f(x) = x
3
- 3x- 1 in (- 11/7, 13t7). 5
y
b) Find the envelope of the family of straight lines x -
a
+-
b
= 1 where a + b = c, c is
constant. 5
111111110111 M11111EN Code No. 50021N
14. a) Find the radius of curvature y
2
= 4ax at (ate , 2at). 5
b) Find local maxima and minima of f(x) = 3x
4
- 2x
3
6x
2
+ 6x + 1. 5
X
2
y
2
15. a) Find the area of the ellipse

+
?
2
-
= 1 using Green's theorem.
a b
b) If A is a constant vector and R = xi + show that
v x 4A ?
A) = A x
16. a) Reduce the matrix to normal form and find rank. 5
- 3 - 6 1 2
--

1 2 -3 3
1 2 1 1
5
5
b) Discuss the convergence of the series
n+1)3
x
n
, x > 0.
n
n+1
5
17. a) If v = (x
2
+ y
2
+ z
2
)
-112
5
a
2
u a
2
u a
2
u

find 2 +
2 + 2
aX ay aZ
b) Evaluate if E -rldS where F = 6zi 4x1 + yk wherp S is the portion of the
plane 2x + 3y + 6z = 12. 5
3 1600
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