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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