Download GTU BE/B.Tech 2019 Winter 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 Winter 1st And 2nd Sem New And Spfu 3110014 Mathematics I Previous Question Paper

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 3110014 Date: 17/01/2020

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) Find the equations of the tenagent plane and normal line to the surface
?? 2
+ ?? 2
+ ?? 2
= 3 at the point (1,1,1)
03
(b)
Evaluate lim
?? ?0
?? ?? ?? ??????? (1+?? )
?? 2

04
(c) Using Gauss Elimination method solve the following system
-x+3y+4z =30
3x+2y-z =9
2x-y+2z =10
07

Q.2 (a) Test the convergence of the series

1
3
+ (
2
5
)
2
+ (
3
7
)
3
+??? + (
?? 2?? +1
)
?? + ???
03

(b) Discuss the Maxima and Minima of the function 3?? 2
? ?? 2
+ ?? 3
04

(c)
Find the fourier series of ?? (?? ) =
(?? ??? )
2
in the interval (0,2?)
07


OR


(c)
Change the order of integration and evaluate ? ? ?????? ?? 2
???? ???? 1
?? 1
0

07


Q.3 (a)
Find the value of ?? (
7
2
,
5
2
)
03

(b) Obtain the fourier cosine series of the function f(x) = ?? ?? in the range (0,?? ) 04

(c) Find the maximum and minimum distance fom the point (1,2,2) to the
sphere ?? 2
+ ?? 2
+ ?? 2
= 36
07


OR

Q.3 (a)
Test the convergence of the series .......
4
1
3
1
2
1
1
1
2 2 2 2
? ? ? ?
03

(b) Evaluate ?(?? 2
? ?? 2
) ???? ???? over the triangle with the vertices (0,1),
(1,1), (1,2)
04

(c) Find the volume of the solid generated by rotating the plane region
bounded by ?? =
1
?? , x=1 and x=3 about the X axis.
07
Q.4 (a)
Evaluate ? ? ?? ???? ???? sin ?? 0
?? 0

03

(b) Express f(x) = 2?? 3
+ 3?? 2
? 8?? + 7 in terms of (x-2) 04
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Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 3110014 Date: 17/01/2020

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) Find the equations of the tenagent plane and normal line to the surface
?? 2
+ ?? 2
+ ?? 2
= 3 at the point (1,1,1)
03
(b)
Evaluate lim
?? ?0
?? ?? ?? ??????? (1+?? )
?? 2

04
(c) Using Gauss Elimination method solve the following system
-x+3y+4z =30
3x+2y-z =9
2x-y+2z =10
07

Q.2 (a) Test the convergence of the series

1
3
+ (
2
5
)
2
+ (
3
7
)
3
+??? + (
?? 2?? +1
)
?? + ???
03

(b) Discuss the Maxima and Minima of the function 3?? 2
? ?? 2
+ ?? 3
04

(c)
Find the fourier series of ?? (?? ) =
(?? ??? )
2
in the interval (0,2?)
07


OR


(c)
Change the order of integration and evaluate ? ? ?????? ?? 2
???? ???? 1
?? 1
0

07


Q.3 (a)
Find the value of ?? (
7
2
,
5
2
)
03

(b) Obtain the fourier cosine series of the function f(x) = ?? ?? in the range (0,?? ) 04

(c) Find the maximum and minimum distance fom the point (1,2,2) to the
sphere ?? 2
+ ?? 2
+ ?? 2
= 36
07


OR

Q.3 (a)
Test the convergence of the series .......
4
1
3
1
2
1
1
1
2 2 2 2
? ? ? ?
03

(b) Evaluate ?(?? 2
? ?? 2
) ???? ???? over the triangle with the vertices (0,1),
(1,1), (1,2)
04

(c) Find the volume of the solid generated by rotating the plane region
bounded by ?? =
1
?? , x=1 and x=3 about the X axis.
07
Q.4 (a)
Evaluate ? ? ?? ???? ???? sin ?? 0
?? 0

03

(b) Express f(x) = 2?? 3
+ 3?? 2
? 8?? + 7 in terms of (x-2) 04

(c)
Using Gauss-Jordan method find
1 ?
A for
?
?
?
?
?
?
?
?
?
?
?
8 0 1
3 5 2
3 2 1
A
07


OR

Q.4 (a)
Using Cayley-Hamilton Theorem find
1 ?
A for
?
?
?
?
?
? ?
?
3 2
1 1
A
03

(b)
Evaluate ?
????
?? 2
+1
?
0

04

(c) Test the convergence of the series

?? 1?2
+
?? 2
3?4
+
?? 3
5?6
+
?? 4
7?8
+???
07
Q.5 (a)
Evaluate ? ? ???? ???? ???? 2
1
1
0

03
(b)
Find the eigen values and eigenvectors of the matrix [
0 1 0
0 0 1
1 ?3 3
]
04
(c)
If u = f (x-y, y-z, z-x) then show that
????
????
+
????
????
+
????
????
= 0
07
OR

Q.5 (a) Find the directional derivatives of f = ????
2
+ ???? 2
at the point (2,-1,1) ,in
the direction of i+2j+2k.
03
(b)
Test the convergence of the series ?
??? ?? 2
+1
?
?? =1

04
(c) Evaluate ? ?????? ???? ???? ???? over the positive octant of the sphere ?? 2
+
?? 2
+ ?? 2
= 4
07

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