Download GTU BE/B.Tech 2019 Summer 1st And 2nd Sem (New And SPFU) 3110014 Mathematics ? I Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 1st And 2nd Sem (New And SPFU) 3110014 Mathematics ? I Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (NEW) EXAMINATION ? SUMMER-2019
Subject Code: 3110014 Date: 06/06/2019

Subject Name: Mathematics ? I

Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.


Marks

Q.1 (a)
Use L?Hospital?s rule to find the limit of lim
? ?1
(
? ? ?1
?
1
? ).
03
(b)
Define Gamma function and evaluate ? ? ? 2
?
?
0
.
04
(c)
Evaluate ? ? ? ? 2
? 1
? ? 3
3
0
.
07



Q.2 (a)
Define the convergence of a sequence (? ? ) and verify whether
the sequence whose ? ? ?
term is ? ? = (
? +1
? ?1
)
? converges or not.
03
(b)
Sketch the region of integration and evaluate the integral
?
(? ? 2? 2
)?
? where ? is the region inside the square |? | +
|? | = 1.
04
(c)
(i) Find the sum of the series ?
1
4
? ? ?2
and ?
4
(4? ?3)(4? +1)
? ?1
.
(ii) Use Taylor?s series to estimate ? 38?.
07

OR

(c)
Evaluate the integrals ? ?
1
(1+? 2
+? 2
)
2
? ?
0
?
0
and
? ? ?
? ? 2? ? ? 2
? 2
? ? 3
0
1
? 3 .
1
0

07
Q.3 (a)
If an electrostatic field ? acts on a liquid or a gaseous polar
dielectric, the net dipode moment ? per unit volume is ? (? ) =
? ? +? ? ? ? ? ? ?
1
? . Show that lim
? ?0
+
? (? ) = 0.
03
(b)
For what values of the constant ? does the second derivative
test guarantee that ? (? , ? ) = ? 2
+ ? + ? 2
will have a
saddle point at (0,0)? A local minimum at (0,0)?
04
(c)
Find the series radius and interval of convergence for
?
(3? ?2)
? ? ?
? =0
. For what values of ? does the series converge
absolutely?
07

OR

Q.3 (a)
Determine whether the integral ?
?
? ?1
3
0
converges or diverges.
03
(b)
Find the volume of the solid generated by revolving the region
bounded by ? =
?
? and the lines ? = 1, ? = 4 about the line
? = 1.
04
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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (NEW) EXAMINATION ? SUMMER-2019
Subject Code: 3110014 Date: 06/06/2019

Subject Name: Mathematics ? I

Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.


Marks

Q.1 (a)
Use L?Hospital?s rule to find the limit of lim
? ?1
(
? ? ?1
?
1
? ).
03
(b)
Define Gamma function and evaluate ? ? ? 2
?
?
0
.
04
(c)
Evaluate ? ? ? ? 2
? 1
? ? 3
3
0
.
07



Q.2 (a)
Define the convergence of a sequence (? ? ) and verify whether
the sequence whose ? ? ?
term is ? ? = (
? +1
? ?1
)
? converges or not.
03
(b)
Sketch the region of integration and evaluate the integral
?
(? ? 2? 2
)?
? where ? is the region inside the square |? | +
|? | = 1.
04
(c)
(i) Find the sum of the series ?
1
4
? ? ?2
and ?
4
(4? ?3)(4? +1)
? ?1
.
(ii) Use Taylor?s series to estimate ? 38?.
07

OR

(c)
Evaluate the integrals ? ?
1
(1+? 2
+? 2
)
2
? ?
0
?
0
and
? ? ?
? ? 2? ? ? 2
? 2
? ? 3
0
1
? 3 .
1
0

07
Q.3 (a)
If an electrostatic field ? acts on a liquid or a gaseous polar
dielectric, the net dipode moment ? per unit volume is ? (? ) =
? ? +? ? ? ? ? ? ?
1
? . Show that lim
? ?0
+
? (? ) = 0.
03
(b)
For what values of the constant ? does the second derivative
test guarantee that ? (? , ? ) = ? 2
+ ? + ? 2
will have a
saddle point at (0,0)? A local minimum at (0,0)?
04
(c)
Find the series radius and interval of convergence for
?
(3? ?2)
? ? ?
? =0
. For what values of ? does the series converge
absolutely?
07

OR

Q.3 (a)
Determine whether the integral ?
?
? ?1
3
0
converges or diverges.
03
(b)
Find the volume of the solid generated by revolving the region
bounded by ? =
?
? and the lines ? = 1, ? = 4 about the line
? = 1.
04
2
(c)
Check the convergence of the series
?
(? )
3
? 3
?
? =1
and ? (?1)
? (
?
? +
?
? ?
? =0
?
?
? ) .
07
Q.4 (a)
Show that the function ? (? , ? ) =
2? 2
? ? 4
+? 2
has no limit as (? , ? )
approaches to (0,0).
03
(b)
Suppose ? is a differentiable function of ? and ? and ? (? , ? ) =
? (? ? + ? , ? ? + ? ). Use the following table to calculate
? ? (0,0), ? ? (0,0), ? ? (1,2) and ? ? (1,2).
? ? ? ? ? ?
(0,0) 3 6 4 8
(1,2) 6 3 2 5

04
(c)
Find the Fourier series of 2? ?periodic function ? (? ) =
? 2
, 0 < ? < 2? and hence deduce that
? 2
6
= ?
1
? 2
?
? =0
.
07

OR

Q.4 (a)
Verify that the function ? = ? ? 2
? 2
? ? ? is a solution f the
heat conduction euation ? ? = ? 2
? ?
.
03
(b)
Find the half-range cosine series of the function
? (? ) = {
2, ?2 < ? < 0
0, 0 < ? < 2
.

04
(c)
Find the points on the sphere ? 2
+ ? 2
+ ? 2
= 4 that are
closest to and farthest from the point (3,1, ?1).
07
Q.5 (a)
Find the directional derivative ? ? ? (? , ? ) if ? (? , ? ) = ? 3
?
3? + 4? 2
and ? is the unit vector given by angle ? =
? 6
. What
is ? ? ? (1,2)?
03
(b)
Find the area of the region bounded y the curves ? = ? , ? =
? and the lines ? = 0 and ? =
? 4
.
04
(c)
Prove that ? = [
0 0 ?2
1 2 1
1 0 3
] is diagonalizable and use it to
find ? 13
.
07

OR

Q.5 (a)
Define the rank of a matrix and find the rank of the matrix ? =
[
2 ?1 0
4 5 ?3
1 ?4 7
].
03
(b)
Use Gauss-Jordan algorithm to solve the system of linear
equations 2? 1
+ 2? 2
? ? 3
+ ? 5
= 0
? 1
? ? 2
+ 2? 3
? 3? 4
+ ? 5
= 0
? 1
+ ? 2
? 2? 3
? ? 5
= 0
? 3
+ ? 4
+ ? 5
= 0
04
(c)
State Cayley-Hamilton theorem and verify if for the matrix
? = [
4 0 1
?2 1 0
?2 0 1
].
07

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