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Name of the Examinati on.. B . Tech. End Semester Examination February 2023
(Delayed Autumn Semester)
Branch
: CSE, ECE, EE, ME
Semester
: 1st
Title of the Course
: Advanced Calculus I
Course Code
: MALB 101 I
Engineering Mathematics I
MALB 103
Time: Three Hours
Maximum Marks: 50
Section A (10 Marks)
Section A contains Question number 1 to 10 of 01 Mark each.
Multiple options may be correct in MCQ's.
Q.l. Which of the following series are divergent
00
1
00 1 + 3n2
(A). ~
~
+ 1)
2n(2n
(B). ~
1 + n2
~
00
n
(C). ~
~
y4n2
3n + 1
D).
(
L 3n 2
+1
oo
2-
n=l
00
Q.2. Radius of convergence of series
L tlnnsx"
is
n=l
(A). 2
(B). 0
(C). 1
(D). Infinite
Q.3. The norm of the partition P = {O,0.2, 0.5, 0.7, 0.9, 1.4, 1.7, 2} is
(A). 1.4
(B). 0.2
(C). 0.5
(D). 0.9
[J:- r:-
QA. The value of the integral
X-Y
X
dzdydx
is
(D). 114
(C). 5/6
(B). 7/6
(A). 7/4
Q.5. Which of the following fields are conservative
-+
F
(A).
---+
F
(C).
-+
-+
-+-
~
= vzi + xz.j + xyk
= xi + yj + zk
-+
-+
Q.6. The inflection points
Q.7. Given f(x, y)
-+
-+
-+
= ysinzi + xsinzj + xycoszk
(D). F = (y + z)i + (x + z)j + (x + y)k
4
for the function y = 3'x - tanx are ---(B). F
---+
-+
= x2 + kxy + y2.
-+
-+
-+
For what values ofk the second derivative test
locating extreme values is inconclusive?
Q.8. The double integral
J3 J 1
o {ill
as
_
ey3
dydx by changing the order of integration is written
.
) - x Y + y z + zx at the point
Q.9. The directional derivative of the _:unct~onf~,:,
Z PO - 1 2) in the direction of 11= 3i
Q.I0.' The, curl of the vector ----+
F (x, y, z)
+ 6j -
2k
IS
3]7
= 2x [2y + z
l
~
2 2.....
+ 2x2y j + 3x z k IS __
Section B (20 Marks)
. num b er 11 t 0 15) of 04 Marks
Section B contains 5 questions (Question
2 each. 6
..
X
Q.ll. Find the absolute maxima and rmruma
0 f the functi10nf(x ,-y) - x2 + y + xy on the rectangular plate 0
< x < 5, - 3 < Y < 3.
~
1+2+3+
+ 22 + 32 +
Q.12. Discuss the convergence of the series ~
+n
+ n2 x
12
n
.
n=1
Q.13. Find the volume of the solid generated by revolving the region bounded by curves
y = e-x, y = 0, x = 0, x = 1 about the x-axis.
Q.14. Find the volume of the region bounded above by the paraboloid Z = x2
+ y2
and
= x, x = 0, x + y = 2 in the xy-plane.
Evaluate the surface integral of the vector field F = z27 + xJ - 3zk in the
below by the triangle enclosed by the lines y
Q.15.
outward normal direction away from z-axis through the cone Z =
V
x2
+ y2,
0
< Z < 1.
Section C (20 Marks)
Section C contains 4 questions (Question number 16 to 19) of 05 Marks each.
Q.16. Define ratio and root test for convergence of series. Discuss the convergence of the
n!n!
L 2(2n)!
.
00
series
n
n=1
Q.17. Convert the integral
II IHl
x
-1 0
(x2
+ y2)dzdxdy
into an equivalent integral in
0
cylindrical coordinates and hence evaluate the integral.
Q.18. Using Green's theorem in a plane calculate the counterclockwise circulation and
-+
3 2.....
1
....
outward flux for the force field F = x Y i + - x4y j on the region bounded by curves
y
= x, y = x2 -
2
X.
-....
Q.19. Calculate the flux of the curl of the field F
S : r(r, e) = rcose] + rSinOI
the outward unit normal.
+ (4 -
r2)k, 0
= 2zi-. + 3xj..... + 5yk..... across the surface
< r < 2,0 < 0 < 21l in the direction of