Download OU (Osmania University) B.Tech (Bachelor of Technology) First Year (1st Year) 2015 June 9002 Mathematics I Question Paper
I Let f
1
= and f(0)
W
I. Find an interval in which f(1) lies.
3  x
2
2 Find the equation of the envelope of the family of straight linOs
c is a parameter.
2
X xy + x + y
x + y
4 ; (x, y) = (2,2)
is discontinuous at the point (2, 2).
3 Prove that f (x, y)
; (x, y) # (2,2)
+ c
2
where
0
0
6
A=
Code No. 9002 / M
FACULTY OF ENGINEERING AND INFORMATICS
B.E. I Year (Main) Examination, May / June 2016
Subject : Mathematics  I
Time : 3 hours Max. Marks : 75
Note: Answer all questions from PartA. Answer any FIVE questions from PartB.
4 If f(x, y) = tan (x y), find an approxima pvaluef f(I .1, 0.8) using the Taylor
series linear approximation.
5 Evaluate the double integral .?ty dx dy, , where IT is the region bounded by the
xaxis, the line y = 2x and t parabola x
2
= 4ay.
6 If a is a constant vector,an xi+ yj + zk then prove that ?X(axi

)
7 Test whether the vectors (1,0,0), (0,2,0), (0,0,3) are linearly independent or not.
8 Find all values of A for which rank of the matrix.
1 0 0
2  1 0
0 2 I
11 6 1
is equal to 3.
(2)
(3)
(2)
(3)
(2)
(3)
(2)
(3)
9 Test the convergence of the series
(2)
10 Show by an example that every convergent series need not be absolute
convergent.
(
3
)
.. 2
FirstRanker.com  FirstRanker's Choice
PART  A (25 Marks)
I Let f
1
= and f(0)
W
I. Find an interval in which f(1) lies.
3  x
2
2 Find the equation of the envelope of the family of straight linOs
c is a parameter.
2
X xy + x + y
x + y
4 ; (x, y) = (2,2)
is discontinuous at the point (2, 2).
3 Prove that f (x, y)
; (x, y) # (2,2)
+ c
2
where
0
0
6
A=
Code No. 9002 / M
FACULTY OF ENGINEERING AND INFORMATICS
B.E. I Year (Main) Examination, May / June 2016
Subject : Mathematics  I
Time : 3 hours Max. Marks : 75
Note: Answer all questions from PartA. Answer any FIVE questions from PartB.
4 If f(x, y) = tan (x y), find an approxima pvaluef f(I .1, 0.8) using the Taylor
series linear approximation.
5 Evaluate the double integral .?ty dx dy, , where IT is the region bounded by the
xaxis, the line y = 2x and t parabola x
2
= 4ay.
6 If a is a constant vector,an xi+ yj + zk then prove that ?X(axi

)
7 Test whether the vectors (1,0,0), (0,2,0), (0,0,3) are linearly independent or not.
8 Find all values of A for which rank of the matrix.
1 0 0
2  1 0
0 2 I
11 6 1
is equal to 3.
(2)
(3)
(2)
(3)
(2)
(3)
(2)
(3)
9 Test the convergence of the series
(2)
10 Show by an example that every convergent series need not be absolute
convergent.
(
3
)
.. 2
2yz)dz is independent of the
Code No. 90021 M
2 
PART  B (50 Marks)
11 a) State and prove Lagrange's mean value theorem.
(
5
)
b) Find the evolute of x
2
= 4ay.
(5)
12 a) Find the shortest distance between the line y = 10  2x and the ellipse
X L
y
2
= 1
4 9
xy
b) Show that Elm does not exist.
(x,y )* (0 ,0) XL y
13 a) Show that the vector field defined by e vector function
xyz(yzi + xzj + xyk) is
conservative.
b) Show that J(yz Odx + (z + xz + z
2
)dy +
path of integration from (1, 2, 2) to (2,
,
,3
7

4). Evaluate the integral.
14 a) Prove that eigen values of Hermitian matrix are real
ii) a skewHermitian matrix are zero or purely imaginary.
(
5
)
(
5
)
b) Examine is positive definite.
. .. x'
15 a) Discuss the convergence of the series L
1 35 (2n  1)
2.46..(2n) 2n
b) Testthe convergence of the series 1 + 3x + 5x
2
+ 7x
3
+
16 a) State and prove Cayley Hamilton theorem.
b) Find the eigen values and the corresponding eigen vectors.
1 2 2
A= 0 2 1
 1 2 2
(
ar
\2 f
ar
17 a) If x = r case, y = r sineY;.then find +
air aii
b) If u = log [x
2
+ xy + y
2
] then find X  y .
ax ay
******
(
5
)
(5)
FirstRanker.com  FirstRanker's Choice
This post was last modified on 20 November 2019