Download OU B-Tech First Year 2011 June 3312 Mathematics I Question Paper

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Note : Answer all questions from Part - A. Answer any five questions fr6m
Part -B.
PART A (Marks : 25)
Using the Lagrange's mean value theorem, show that I sin b - sin al
b

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2
. Find the envelope of the family ofrcurves y = 3cx- c
3
, c is a parameter. 3
3. If f(x, y, z) xy4 yz zx, x = t

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2
, y tet, = te
-
t, find
df

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4. Find the linear Taylor series polynomial approxiniation to the function
= 2x
3
+ 3y

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3
- 4x
2
y about the point (1, 2).
5. If

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-
r
-
= xi + yj + zk, show that (ii . V)i = Ci.
6. Find the directional derivative of the function x, y, =

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' z a
(1, 2, 3) in the direction of the line i = ,i = i..
7. Find the values of X such that the rank of
1 . 2 4' (
A .= 2- A 5 is 2.

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4 8 X i
8. Find tbe sum and the product of eigen values of,-)the.matrix
-9. Find the values of x fOr which the series E(4)r)" is convergent.
10. Show that the series E
sin n x

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converges absolutely.
n
2
PART B (Marks 50)
11. (a) Find the radiuS of ?curvature of the curve x= a(0 - sin 0), y a(1 - cos 0)

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at 0 = 7C. ? 5
(b) Find the evolute of the curve x .2 at, y = at
2
. 5
Y

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2
3
. 10 0 8
4 9. 6
2 7 5

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(This paper contains 2 pages) P.T.O.
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Code No. : 3312/N
FACULTY OF ENGINEERING & INFORMA.TICS
B.E. 1 Year (New) (Common to all branches) (Main) Examination, June 2011

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MATHEMATICS - I
Time : 3 Hours ] [ Max. Marks : 75
Note : Answer all questions from Part - A. Answer any five questions fr6m
Part -B.
PART A (Marks : 25)

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Using the Lagrange's mean value theorem, show that I sin b - sin al
b
2
. Find the envelope of the family ofrcurves y = 3cx- c
3

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, c is a parameter. 3
3. If f(x, y, z) xy4 yz zx, x = t
2
, y tet, = te
-

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t, find
df

4. Find the linear Taylor series polynomial approxiniation to the function
= 2x

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3
+ 3y
3
- 4x
2

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y about the point (1, 2).
5. If
-
r
-

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= xi + yj + zk, show that (ii . V)i = Ci.
6. Find the directional derivative of the function x, y, =
' z a
(1, 2, 3) in the direction of the line i = ,i = i..
7. Find the values of X such that the rank of

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1 . 2 4' (
A .= 2- A 5 is 2.
4 8 X i
8. Find tbe sum and the product of eigen values of,-)the.matrix
-9. Find the values of x fOr which the series E(4)r)" is convergent.

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10. Show that the series E
sin n x
converges absolutely.
n
2

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PART B (Marks 50)
11. (a) Find the radiuS of ?curvature of the curve x= a(0 - sin 0), y a(1 - cos 0)
at 0 = 7C. ? 5
(b) Find the evolute of the curve x .2 at, y = at
2

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. 5
Y
2
3
. 10 0 8

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4 9. 6
2 7 5
(This paper contains 2 pages) P.T.O.
Code No. : 3312/N .
6

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4
4
6
?
12. (a)

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(b)
Trace the curve r = all + cos 0).
x
2 + y2
(x, y) (0

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-
, 0)
is not continuous at Show that f(x, y) = x ? y
,
0 (x, y) . (0, 0)

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(0, 0).
?13. (a) Prove that curl (f V) = (grad f) 'x V +f curl V.
? (b) Using.Green's theorem, evaluate (x
2
+ y

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2
) dx + (y + 2x)dy, where C
is the bouridary of the region bounded by the curves y
2
= x and

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If A = (
1
1
0
? 0

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0
.1
0
'1
0'

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, Show that An . n -
2
+ A
2
- , n .3 using Cayley-

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? 15. Discuss the convergence of the series.
2! ' 3! 4!
(a) 1 +
22
+

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33

nn
+ + .. . ..
(b) 0 . x >

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n.
16. (a) If f(x, y
.
) =
2

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a
f
x+ y2
(x,
(13.1 9)

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, compute (0, b).
0 (x, y) = (0, 0)
-I
k

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(b) Find the minimum value of f(x, y, z) =x
2
+ y
2
+' z

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2
subject to the
condition xyz a
3
. ?

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00 00
17. (a) Evaluate ffe
-0(21-
Y
2

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) dx dy.
o o
(b) Test whether the vectors (1, 1, 0, 1), (1, 1, 1, 1), (4, 4, 1, 1), (1, 0, 0, 1)
5
5

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4
6
5
x
2

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y. 5
?
14. (a).
Hamilton theorem.
? ' . .( 1 2 ?

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? (b) Reduce A = . 1 2 1
I to the diagonal form. . .
--1 ?1 0
5
are linearly independent or not . 5

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2 5,700
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