Download GTU BE/B.Tech 2019 Summer 1st Sem And 2nd Sem Old 110008 Maths I Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 1st Sem And 2nd Sem Old 110008 Maths I Previous Question Paper

Page 1 of 2

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (OLD) EXAMINATION ? SUMMER-2019
Subject Code: 110008 Date: 06/06/2019

Subject Name: Maths - I

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

Instructions:

1. Attempt any five questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q.1 (a)
(i) Evaluate : lim
?? ?1
(1 ? ?? )tan (
????
2
)
(ii) Evaluate : lim
?? ?0
(
1
?? )
1?cos ??
(iii) Find Jacobian
?? (?? ,?? )
?? (?? ,?? )
for functions ?? = ?????????? , ?? = ?????????? .
02

02

03

(b)
(i) Sketch the region and find the area bounded by ellipse
?? 2
9
+
?? 2
4
= 1.
(ii) Using Lagrange?s mean value theorem, prove that

?? ??? 1+?? 2
< ?????? ?1
?? ? ?????? ?1
?? <
?? ??? 1+?? 2

03

04

Q.2 (a)
(i) Verify Rolle?s theorem for f(x) = x(x + 3)?? ?
?? 2
in?3 ? ?? ? 0.
(ii) Find two non-negative numbers whose sum is 9 such that the product of one
number and the square of the other is maximum.

04

03
(b)
(i) Prove that tan
?1
(
?1+?? 2
?1
?? ) =
1
2
(?? ?
?? 3
3
+
?? 5
5
? ? ).
(ii) Find the absolute maximum and minimum values of ?? (?? ) =
?? 3
?? +2
in
interval [?1, 1].
04

03

Q.3 (a)
(i) Test the convergence of the series:?
?? ?? ?? ?? (?? +1)
?? , ?? > 0.
?
?? =1

(ii) Test the convergence of the series: ?
??? ?? 2
+1

?
?? =1


04

03
(b)
(i) Test the convergence of the series:
1
2
?
2
5
+
3
10
?
4
17
+ ?
(ii) Test the convergence of the series: ?
?? !
?? ??
?
?? =1


04

03

Q.4 (a) (i) Find extreme values of f(x, y) = x
3
+ 3xy
2
? 3x
2
? 3y
2
+ 7
(ii) Find all first and first and second order partial derivatives for
f(x, y) = x
2
sin y + y
2
cos x. Hence, verify mixed derivative Theorem.

04

03
(b)
(i) If ?? = tan
?1
(
?? 3
+?? 3
?? ??? ); show that ?? 2
?? 2
?? ?? ?? 2
+ 2????
?? 2
?? ????????
+ ?? 2
?? 2
?? ?? ?? 2
= 2?????????????? 3?? .
(ii) If ?? = ?? (?? ? ?? , ?? ? ?? , ?? ? ?? ) then prove that
????
????
+
????
????
+
????
????
= 0.





04

03



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Page 1 of 2

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (OLD) EXAMINATION ? SUMMER-2019
Subject Code: 110008 Date: 06/06/2019

Subject Name: Maths - I

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

Instructions:

1. Attempt any five questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q.1 (a)
(i) Evaluate : lim
?? ?1
(1 ? ?? )tan (
????
2
)
(ii) Evaluate : lim
?? ?0
(
1
?? )
1?cos ??
(iii) Find Jacobian
?? (?? ,?? )
?? (?? ,?? )
for functions ?? = ?????????? , ?? = ?????????? .
02

02

03

(b)
(i) Sketch the region and find the area bounded by ellipse
?? 2
9
+
?? 2
4
= 1.
(ii) Using Lagrange?s mean value theorem, prove that

?? ??? 1+?? 2
< ?????? ?1
?? ? ?????? ?1
?? <
?? ??? 1+?? 2

03

04

Q.2 (a)
(i) Verify Rolle?s theorem for f(x) = x(x + 3)?? ?
?? 2
in?3 ? ?? ? 0.
(ii) Find two non-negative numbers whose sum is 9 such that the product of one
number and the square of the other is maximum.

04

03
(b)
(i) Prove that tan
?1
(
?1+?? 2
?1
?? ) =
1
2
(?? ?
?? 3
3
+
?? 5
5
? ? ).
(ii) Find the absolute maximum and minimum values of ?? (?? ) =
?? 3
?? +2
in
interval [?1, 1].
04

03

Q.3 (a)
(i) Test the convergence of the series:?
?? ?? ?? ?? (?? +1)
?? , ?? > 0.
?
?? =1

(ii) Test the convergence of the series: ?
??? ?? 2
+1

?
?? =1


04

03
(b)
(i) Test the convergence of the series:
1
2
?
2
5
+
3
10
?
4
17
+ ?
(ii) Test the convergence of the series: ?
?? !
?? ??
?
?? =1


04

03

Q.4 (a) (i) Find extreme values of f(x, y) = x
3
+ 3xy
2
? 3x
2
? 3y
2
+ 7
(ii) Find all first and first and second order partial derivatives for
f(x, y) = x
2
sin y + y
2
cos x. Hence, verify mixed derivative Theorem.

04

03
(b)
(i) If ?? = tan
?1
(
?? 3
+?? 3
?? ??? ); show that ?? 2
?? 2
?? ?? ?? 2
+ 2????
?? 2
?? ????????
+ ?? 2
?? 2
?? ?? ?? 2
= 2?????????????? 3?? .
(ii) If ?? = ?? (?? ? ?? , ?? ? ?? , ?? ? ?? ) then prove that
????
????
+
????
????
+
????
????
= 0.





04

03



Page 2 of 2






























Q.5 (a) Sketch the region of integration and evaluate by reversing the order of
Integration for integral ? ? ???????? 2?????
?? 2
4?? 4?? 0


07
(b)
(i) Evaluate the integral ? ? ?? ?(?? 2
+?? 2
)
???????? ?
0
?
0
by changing into polar
coordinates.
(ii) Evaluate ? ? ? ?? ?? ???????????? ?? +?? 0
1??? 0
1
0
.
04


03

Q.6 (a) (i) Find directional derivative of ? = ?? ?? 2
+ ?? ?? 2
at point (2, ?1, 1) in the
direction of vector ?? ? + 2?? ? + 2?? ?
.
(ii) If ?? ? = ?? 3
?? ? + (2?? 3
?
1
5?? 2
) ?? ? , then show that ?? ? ?
?? ?? ?
????
= ?? ?
.

04

03
(b) Evaluate ? ?? ?
?

?? ?? ?? ? , where, ?? ?
= (?? 2
+ ?? 2
)?? ? ? 2???? ?? ? . Where, C is the rectangle
in XY-plane bounded by ?? = 0, ?? = ?? , ?? = ?? , ?? = 0.
07

Q.7 (a) Verify Green?s theorem for ? [(?? 2
? 2???? )???? + (?? 2
?? + 3)???? ]

?? . Where, C is the
boundary of the region bounded by ?? = ?? 2
and the line ?? = ?? .

07
(b) (i) Show that
?? ?
= (?? 2
? ?? 2
+ 3???? ? 2?? )?? ? + (3???? + 2???? )?? ? + (3???? ? 2???? + 2?? )?? ?
is both
solenoidal and irrotational.
(ii) Evaluate ? [?? ?? ???? + 2?????? ? ???? ]

?? by Stoke?s theorem. where, C is the curve
?? 2
+ ?? 2
= 4, ?? = 2.

04


03


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