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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