Download OU B-Tech First Year 2014 June 6002 Mathematics I Question Paper

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(
5
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Code No. 6002 / M
FACULTY OF ENGINEERING and INFORMATICS
B.E, I - Year (Main) Examination, June 2014
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.
PART - A (25 Marks)
1 Define rank of a matrix. (2)
2 Show that the vectors (1, 2, 3), (2, 3, 4) and (3, 4, 5) are linearly dependent.
(

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3
)
3 State the necessary condition for a positive series a? to be convergent. (2)
4 Discuss the convergence of
f

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.
(
3
)
5 Using the Lagrange mean value theorem, show that sin b - sin a! 0

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-
) - (2)
6 Find the radius of curvature for the curve y = x
2
- 6x + 10 at (3, 1).

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(
3
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x
3

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y
7 Show that li m ---- does not exist. (2)
v
,,.),(0.0)
x

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6
?y
2

8 Expand f(x,

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y) = x2 + 3
y
2
9x - 9y + 26 in Taylor series of maximum order about
(2, 2).

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3
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9 Find of , if f(x, y, z) = lo e
(

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x
2 + y2 + z2)
(2)
10 Show that the vector (x + yz) i + (4y - z
2

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x)j + (2xz 4z)k is solenoidal
(
3
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PART - B (50 Marks)

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11 a) Test for consistency and solve 2x 3y + 7z = 5 , 3x + y 3z = 13,
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5
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2x + 19y 47z = 32.

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b) Verify Cayley - Hamilton theorem for the matrix A =
3
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5
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b)
13 a)
b)
14 a)
b)

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15 a)
b)
Test the series
1
V 11

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4
+ 1 11 ?
1 -
1
.

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1 for convergence.
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5
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Verify Rolle's theorem for the function f(x) = (x + 2) (x - 3) in the interval [-2, 3]. (5)

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Find the evolute of the curve x
2
= 4ay.
(
5

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)
Find all asymptotes of the curve 1
,
= x + -
I

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--
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5
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Discuss the maxima and minima of f(x, y) = 4x

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2
+ 2y
2
+ 4xy - 10x - 2y - 3.
(

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5
)
Show that V
2
7.." = ti(ri + i?"

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--2
, where r = i , = xi + zk
(
5
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If S is any closed surface enclosing a volume V and F= ox 4
3
j - prove
that T ds +1.) C) V.
(

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5
)
?
16 a) Find the eigen values and the corresponding eigen vectors of A = 0 4 l . (5)
0) 0 6

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J

b) Discuss convergence of I
I
, +

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1
2
-
1
, { ........................................................

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2 3
-

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4
2
5
17 Verify Green's theorem for f Kly -8
.

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1
,2
)(ix + (4 , -6.1.)(11.
,
where C is the boundary of

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the region bounded by x = 0, y = 0 and x + y , 1. (10)
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