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

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

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (OLD) EXAMINATION ? WINTER 2019
Subject Code: 110008 Date: 17/01/2020

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) State Sandwich theorem, using it find ) ( lim
0
x g
x ?
if x x g x sec 3 ) ( 3
2
? ? ? for all . x
4
(ii) Can Rolle?s theorem for ? ? 1 , 1 , ) ( ? ? ? x x x f applied?
3

(b)
If ) , , ( x z z y y x f u ? ? ? ? , Prove that 0 ?
?
?
?
?
?
?
?
?
z
u
y
u
x
u

7
Q-2 (a)
(i) Use Taylor?s series to find the expansion of x
e
log in powers of ). 1 ( ? x
4

(ii) Use L?Hospital rule, Evaluate
x
x
x cot
log
lim
0 ?
3
(b) Trace the curve ? 2 cos
2 2
a r ? . 7
Q-3 (a)
(i) Test the convergence of
?
?
?
?
0
3
1 2
n
n
n
.
4

(ii) Does the sequence
?
?
?
?
?
?
? 3
3
n
monotone?
3

(b)
If
?
?
?
?
?
?
?
?
?
?
?
?
y x
y x
u
3 3
1
tan Prove that . sin 3 cos 2 2
2
2
2
2
2
2
2
u u
y
u
y
y x
u
xy
x
u
x ?
?
?
?
? ?
?
?
?
?

7
Q-4 (a)
(i) Test the convergence of
?
?
?1
! 3
n
n
n
n
n
, by Ratio Test.
4
(ii) Discuss the convergence if the series . ...... .......... 5 9 4 5 9 4 5 9 4 ? ? ? ? ? ? ? ? ? 3
(b)
Find the extremum values for xy y x y x f 3 ) , (
3 3
? ? ? . 7

Q-5 (a)
(i) Expand 2 3
2
? ? y y x in the neighbourhood of the point ) 2 , 1 ( ? . 4
(ii) Find the equation for tangent plane and normal line at the point ) 1 , 1 , 1 ( on the surface
3
2 2 2
? ? ? z y x .
3

(b) Find the Volume of sphere
2 2 2 2
a z y x ? ? ? . 7

Q-6 (a)
(i) Evaluate
? ?
? ?
?
0 x
y
dA
y
e
, by changing the order of integration.
4
(ii) Find the value of m if k x z j z my i y x F ) 6 5 ( ) 4 ( ) 2 ( ? ? ? ? ? ? is solenoidal.
3
(b)
Evaluate
??
?
R
dydx y x ) ( , where R is the region bounded by . 2 , , 2 , 0 ? ? ? ? ? x y x y x x
7
Q-7 (a) (i) Evaluate
? ? ?
?
1
0
2
0
2 2 2
0
) cos (
?
? ? drdz rd z r
z
4
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Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (OLD) EXAMINATION ? WINTER 2019
Subject Code: 110008 Date: 17/01/2020

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) State Sandwich theorem, using it find ) ( lim
0
x g
x ?
if x x g x sec 3 ) ( 3
2
? ? ? for all . x
4
(ii) Can Rolle?s theorem for ? ? 1 , 1 , ) ( ? ? ? x x x f applied?
3

(b)
If ) , , ( x z z y y x f u ? ? ? ? , Prove that 0 ?
?
?
?
?
?
?
?
?
z
u
y
u
x
u

7
Q-2 (a)
(i) Use Taylor?s series to find the expansion of x
e
log in powers of ). 1 ( ? x
4

(ii) Use L?Hospital rule, Evaluate
x
x
x cot
log
lim
0 ?
3
(b) Trace the curve ? 2 cos
2 2
a r ? . 7
Q-3 (a)
(i) Test the convergence of
?
?
?
?
0
3
1 2
n
n
n
.
4

(ii) Does the sequence
?
?
?
?
?
?
? 3
3
n
monotone?
3

(b)
If
?
?
?
?
?
?
?
?
?
?
?
?
y x
y x
u
3 3
1
tan Prove that . sin 3 cos 2 2
2
2
2
2
2
2
2
u u
y
u
y
y x
u
xy
x
u
x ?
?
?
?
? ?
?
?
?
?

7
Q-4 (a)
(i) Test the convergence of
?
?
?1
! 3
n
n
n
n
n
, by Ratio Test.
4
(ii) Discuss the convergence if the series . ...... .......... 5 9 4 5 9 4 5 9 4 ? ? ? ? ? ? ? ? ? 3
(b)
Find the extremum values for xy y x y x f 3 ) , (
3 3
? ? ? . 7

Q-5 (a)
(i) Expand 2 3
2
? ? y y x in the neighbourhood of the point ) 2 , 1 ( ? . 4
(ii) Find the equation for tangent plane and normal line at the point ) 1 , 1 , 1 ( on the surface
3
2 2 2
? ? ? z y x .
3

(b) Find the Volume of sphere
2 2 2 2
a z y x ? ? ? . 7

Q-6 (a)
(i) Evaluate
? ?
? ?
?
0 x
y
dA
y
e
, by changing the order of integration.
4
(ii) Find the value of m if k x z j z my i y x F ) 6 5 ( ) 4 ( ) 2 ( ? ? ? ? ? ? is solenoidal.
3
(b)
Evaluate
??
?
R
dydx y x ) ( , where R is the region bounded by . 2 , , 2 , 0 ? ? ? ? ? x y x y x x
7
Q-7 (a) (i) Evaluate
? ? ?
?
1
0
2
0
2 2 2
0
) cos (
?
? ? drdz rd z r
z
4

(ii) Using Green?s theorem to evaluate the integral ? ?
?
?
C
dy x dx y
2 2
, where : C The
triangle bounded by 0 , 1 , 0 ? ? ? ? y y x x .
3
(b)
Use divergence theorem to evaluate ? ?
??
? ?
S
zdxdz x ydzdx x dydz x
2 2 3
where S is the
closed surface consisting of the cylinder
2 2 2
a y x ? ? and the circular discs 0 ? z and
b z ? .
7

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