Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER–I & II (NEW) EXAMINATION – WINTER 2020
Subject Code:3110014 Date:16/03/2021
Subject Name:Mathematics – I
Time:10:30 AM TO 12:30 PM Total Marks:47
Instructions:
1. Attempt any THREE questions from Q1 to Q6.
2. Q7 is compulsory.
3. Make suitable assumptions wherever necessary.
4. Figures to the right indicate full marks.
Marks
Q.1 (a) Expand sin x in powers of ( x / 2) . 03
(b) lim tan 2 x x 2 04
Evaluate .
x 0 x 2 tan 2 x
(c) 3x 5 03
(i) Check the convergence of 4 x 4 7 dx.
(ii) The region between the curve y x , 0 x 4 and the line x = 4 is 04
revolved about the x – axis to generate a solid. Find its volume.
Q.2 (a) x y u u 03
If u cos ec 1 2 , show that x
2
y tan u.
x y x y
(b)
2n 5 04
Check the convergence of the series n .
n 1 3
(c) 03
(i) Test the convergence of the series n 1 n .
n 1
n 04
2
(ii) Test the convergence of n
n 1
3
1
.
Q.3 (a) Solve the following equations by Gauss’ elimination method: 03
x y z 6, x 2 y 3 z 14,2 x 4 y 7 z 30.
(b) u u u 04
If u f ( x y, y z, z x) , prove that 0.
x y z
(c) (i) Find the equation of the tangent plane and normal line to the surface 03
x 2 2 y 2 3z 2 12 at (1, 2, -1).
(ii) For f ( x, y) x 3 y 3 3xy , find the maximum and minimum values. 04
Q.4 (a) 8 0 0 16 03
Find the rank of the matrix 0 0 0 6 .
0 9 9 9
(b) 2u 2u 04
If u f ( x at ) g ( x at ) , prove that 2 a 2 2 .
t x
1
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER–I & II (NEW) EXAMINATION – WINTER 2020
Subject Code:3110014 Date:16/03/2021
Subject Name:Mathematics – I
Time:10:30 AM TO 12:30 PM Total Marks:47
Instructions:
1. Attempt any THREE questions from Q1 to Q6.
2. Q7 is compulsory.
3. Make suitable assumptions wherever necessary.
4. Figures to the right indicate full marks.
Marks
Q.1 (a) Expand sin x in powers of ( x / 2) . 03
(b) lim tan 2 x x 2 04
Evaluate .
x 0 x 2 tan 2 x
(c) 3x 5 03
(i) Check the convergence of 4 x 4 7 dx.
(ii) The region between the curve y x , 0 x 4 and the line x = 4 is 04
revolved about the x – axis to generate a solid. Find its volume.
Q.2 (a) x y u u 03
If u cos ec 1 2 , show that x
2
y tan u.
x y x y
(b)
2n 5 04
Check the convergence of the series n .
n 1 3
(c) 03
(i) Test the convergence of the series n 1 n .
n 1
n 04
2
(ii) Test the convergence of n
n 1
3
1
.
Q.3 (a) Solve the following equations by Gauss’ elimination method: 03
x y z 6, x 2 y 3 z 14,2 x 4 y 7 z 30.
(b) u u u 04
If u f ( x y, y z, z x) , prove that 0.
x y z
(c) (i) Find the equation of the tangent plane and normal line to the surface 03
x 2 2 y 2 3z 2 12 at (1, 2, -1).
(ii) For f ( x, y) x 3 y 3 3xy , find the maximum and minimum values. 04
Q.4 (a) 8 0 0 16 03
Find the rank of the matrix 0 0 0 6 .
0 9 9 9
(b) 2u 2u 04
If u f ( x at ) g ( x at ) , prove that 2 a 2 2 .
t x
1
