Download OU B-Tech First Year 2015 June 9002 Mathematics I Question Paper

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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

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c is a parameter.
2
X xy + x + y
x + y
4 ; (x, y) = (2,2)

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is discontinuous at the point (2, 2).
3 Prove that f (x, y)
; (x, y) # (2,2)
+ c
2

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where
0
0
-6
A=

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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

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Note: Answer all questions from Part-A. Answer any FIVE questions from Part-B.
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
x-axis, the line y = 2x and t parabola x

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2
= 4ay.
6 If a is a constant vector,an xi+ yj + zk then prove that ?X(axi-
-
)

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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

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11 6 1
is equal to 3.
(2)
(3)
(2)

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(3)
(2)
(3)
(2)
(3)

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9 Test the convergence of the series
(2)

10 Show by an example that every convergent series need not be absolute
convergent.

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(
3
)
.. 2
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PART - A (25 Marks)
I Let f
1
= and f(0)
W

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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

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c is a parameter.
2
X xy + x + y
x + y
4 ; (x, y) = (2,2)

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is discontinuous at the point (2, 2).
3 Prove that f (x, y)
; (x, y) # (2,2)
+ c
2

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where
0
0
-6
A=

--- Content provided by⁠ FirstRanker.com ---

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

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Note: Answer all questions from Part-A. Answer any FIVE questions from Part-B.
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
x-axis, the line y = 2x and t parabola x

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2
= 4ay.
6 If a is a constant vector,an xi+ yj + zk then prove that ?X(axi-
-
)

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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

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11 6 1
is equal to 3.
(2)
(3)
(2)

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(3)
(2)
(3)
(2)
(3)

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9 Test the convergence of the series
(2)

10 Show by an example that every convergent series need not be absolute
convergent.

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(
3
)
.. 2
2yz)dz is independent of the

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Code No. 90021 M
2 -
PART - B (50 Marks)
11 a) State and prove Lagrange's mean value theorem.
(

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5
)
b) Find the evolute of x
2
= 4ay.

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(5)
12 a) Find the shortest distance between the line y = 10 - 2x and the ellipse
X L
y
2

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= 1
4 9
xy
b) Show that Elm does not exist.
(x,y )--* (0 ,0) XL y

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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

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)dy +
path of integration from (1, 2, 2) to (2,
,
,3
7

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-
4). Evaluate the integral.
14 a) Prove that eigen values of Hermitian matrix are real
ii) a skew-Hermitian matrix are zero or purely imaginary.
(

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5
)
(
5
)

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b) Examine is positive definite.
. .. x'
15 a) Discuss the convergence of the series L
1 35 (2n - 1)

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2.46..(2n) 2n
b) Testthe convergence of the series 1 + 3x + 5x
2
+ 7x
3

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+
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

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- 1 2 2
(
ar
\2 f
ar

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17 a) If x = r case, y = r sineY;.then find +
air aii
b) If u = log [x
2
+ xy + y

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2
] then find X - y .
ax ay
******
(

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5
)
(5)
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