Download OU B-Tech First Year 2013 December 6002 Mathematics I Question Paper

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

,?
,
Pirtef
Dora. 24/3
Code No. 6002

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

FACULTY OF ENGINEERING & INFORMATICS
B.E. I Year (Common to all Branches) (Suppl.) Examination, December 2013
Subject: Mathematics ? I
Time: 3 Hours Max.Marks: 75
Note: Answer all questions from Part A. Answer any five questions from Part B.

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

PART ? A (25 Marks)
1. Find the Taylor's series expansion of f(x) = 2
x
about x=0. (2)
2. Find the radius of curvature of the curve r = a sine + b cos@ at e. zr/2 .

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

(
3
)
2
X -

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

3. Show that lim does not exist. (2)
(x.Y)-)(
00
) X
2

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

y
az
4. If z=y+f(u), u = ?
x
, show that u ?+

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

az

? = 1.
(
3

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

)
ax ay
1 2
5. Evaluate ex
2

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

dx dy by changing the order of integration. (2)
0 2y
6. Find a vector that gives the direction of maximum rate of increase for f(x,y,z)=6xyz
at (-1,2,1).
(

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

3
)
7. Find the values of A and p such that the system of equations x+y+z = 6,
x+2y+3z = 10, x + 2y + Az = p has an infinite number of solutions. (2)
B. Show that the vectors (2,2,0), (3,0,2), (2,-2,2) are linearly independent.

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

(
3
)
1 ,
9. Discuss the convergence of the series Z(1 +?)n p > 0. (2)

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

n'
10. Test whether the series
(-1)n
converges absolutely or not.
(

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

3
)
nVn
PART ? B (50 Marks)
11.(a) State and prove Rolle's theorem.

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

(
6
)
(b) Find the envelope of the family of curves x tan a + y sec a = 5, a is a parameter. (4)
12.(a) Trace the curve y = x

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

3
? 12 x 16.
(
6
)

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

(b) Examine f(x,y) = x
4
+ 2x
2
y ? x

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

2
+ 3y
2
for maximum and minimum values. (4)
13.(a) Show that V = 12xi 15y

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

2
j + k is irrotational and find a scalar function f(x,y,z) such
that V = grad f.
(
5

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

)
(b) Use the divergence theorem to evaluate JJ F,n ds, where F = 4xi ? 2y
2
j + z
2

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

k and
S is the surface bounding the region x
2
+y
2

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

= 4, z=0 and z=3.
(
5
)
...2.

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

FirstRanker.com - FirstRanker's Choice
.
r^;
11.
1

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

3
,?
,
Pirtef
Dora. 24/3

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

Code No. 6002
FACULTY OF ENGINEERING & INFORMATICS
B.E. I Year (Common to all Branches) (Suppl.) Examination, December 2013
Subject: Mathematics ? I
Time: 3 Hours Max.Marks: 75

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

Note: Answer all questions from Part A. Answer any five questions from Part B.
PART ? A (25 Marks)
1. Find the Taylor's series expansion of f(x) = 2
x
about x=0. (2)

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

2. Find the radius of curvature of the curve r = a sine + b cos@ at e. zr/2 .
(
3
)
2

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

X -
3. Show that lim does not exist. (2)
(x.Y)-)(
00
) X

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

2
y
az
4. If z=y+f(u), u = ?
x

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

, show that u ?+
az

? = 1.
(

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

3
)
ax ay
1 2
5. Evaluate ex

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

2
dx dy by changing the order of integration. (2)
0 2y
6. Find a vector that gives the direction of maximum rate of increase for f(x,y,z)=6xyz
at (-1,2,1).

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

(
3
)
7. Find the values of A and p such that the system of equations x+y+z = 6,
x+2y+3z = 10, x + 2y + Az = p has an infinite number of solutions. (2)

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

B. Show that the vectors (2,2,0), (3,0,2), (2,-2,2) are linearly independent.
(
3
)
1 ,

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

9. Discuss the convergence of the series Z(1 +?)n p > 0. (2)
n'
10. Test whether the series
(-1)n
converges absolutely or not.

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

(
3
)
nVn
PART ? B (50 Marks)

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

11.(a) State and prove Rolle's theorem.
(
6
)
(b) Find the envelope of the family of curves x tan a + y sec a = 5, a is a parameter. (4)

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

12.(a) Trace the curve y = x
3
? 12 x 16.
(
6

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

)
(b) Examine f(x,y) = x
4
+ 2x
2

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

y ? x
2
+ 3y
2
for maximum and minimum values. (4)

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

13.(a) Show that V = 12xi 15y
2
j + k is irrotational and find a scalar function f(x,y,z) such
that V = grad f.
(

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

5
)
(b) Use the divergence theorem to evaluate JJ F,n ds, where F = 4xi ? 2y
2
j + z

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

2
k and
S is the surface bounding the region x
2
+y

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

2
= 4, z=0 and z=3.
(
5
)

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

...2.
(4)
(
6
)

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

(5)
Code No. 6002
-2-
(1 2 3 4
2 1 4 3

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

14.(a) If -4, 10, -./2 are the three eigen values of A =
3 4 2 1
3 1 2
)

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

find the eigen values of
(b) Find the canonical form, nature, index and signature of the quadratic form
Q = 8 x
i
2

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

+ 7 x
2
, + 3 x
3
2

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

- 12x1 x2 ? 8x2x3 + 4x3xi.
15. Test the convergence of the series
a)
b)
1 2

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

+ +
3
+
1.3.5 3.5.7
(r

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

11
)
2
x
2n

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

5.7.9
(2n)i
16.(a) Find the evolute of the curve y
2
=4ax.

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

1
x2y(x-y)
, (x y) #(0,0)
(b) For the function f(x,y) = x
2

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

+ y
2

0 , (x, y) = (0,0),
a

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

2f 5 2f
show that at (0,0).
ax ay ay ax
17.(a) Show that Vx(V x V) = V (V. V) -V
2

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

V
( 2 3 1 0 4
.\

3 1 2 -1 1

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

4 -1 3 -2 -2
5 4 3 -1 5 ,
(b) Find the rank of the matrix A =
FirstRanker.com - FirstRanker's Choice