Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 18 Scheme 18IVIAT11 Calculus and Linear Algebra Question Paper
U
a
CO
CO
1
a.
b.
c.
USN
18IVIAT11
of Ems!
First Semester B.F. Degree Examination, June/July 2019
Calculus and Linear Algebra
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE MI questions, choosing ONE full question from each module.
2 a.
b.
c.
4 a.
With usual notation,
Find the radius of
Show that the evolute
Prove that the pedal
Show that for the
Find the angle between
Expand log(1+ cosx)
lim jax
Evaluate
x 01
Find the extreme
(x y z
If u f
y z x
Module-I
(06 Marks)
x-axis.
(06 Marks)
(08 Marks)
(06 M a rks)
(06 Marks)
(08 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
prove that tan (1)? r
(10
(
dr
curvature of a
2
y = x
3
? a
3
at
of the parabola y
2
= 4ax
OR
equation of the curve r
n
curve r(1 ? cos0) = 2a, p
2
varies
the polar curves r = a(1
Module-2
.
the point where the curve cuts the
is 27ay
2
= 4(x ? 2a)'.
= a"cosnO is a
n
.p = r"'
I
.
as r'.
? cos()) and r = b(1 + cos0).
by Maclaurin's series up
+b`
to the term containing x
4
.
= x
3
+ y
3
? 3x ? 12y + 20.
?u
au
+ z ? ? =0
cz
3
values of the function tax, y)
OR
, au
then prove that x ? ? + y
3 a.
b.
c.
b. If u = x + 3y
2
? z
3
, v = 4x
-
yz, w = 2z
-
? xy. Evaluate a(u, v,
w)
at the point (1 -1, 0).
a( x, y,z)
(07 Marks)
c. A rectangular box, open at the top, is to have a volume of 32 cubic feet. Find the dimensions
of the box, if the total surface area is minimum. (07 Marks)
Module-3
5 a. Evaluate by changing the order of integration
f
fx
2
?dy?dx , a> 0
h. Find the area bounded between the circle x
2
+ y
2
= a
2
and the line x + y = a.
F
-
1 ? i
-
1
in + n
c.
Prove that I3(m,
(06 Marks)
(07 Marks)
(07 Marks)
1 of 2
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C.)
U
a
CO
CO
1
a.
b.
c.
USN
18IVIAT11
of Ems!
First Semester B.F. Degree Examination, June/July 2019
Calculus and Linear Algebra
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE MI questions, choosing ONE full question from each module.
2 a.
b.
c.
4 a.
With usual notation,
Find the radius of
Show that the evolute
Prove that the pedal
Show that for the
Find the angle between
Expand log(1+ cosx)
lim jax
Evaluate
x 01
Find the extreme
(x y z
If u f
y z x
Module-I
(06 Marks)
x-axis.
(06 Marks)
(08 Marks)
(06 M a rks)
(06 Marks)
(08 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
prove that tan (1)? r
(10
(
dr
curvature of a
2
y = x
3
? a
3
at
of the parabola y
2
= 4ax
OR
equation of the curve r
n
curve r(1 ? cos0) = 2a, p
2
varies
the polar curves r = a(1
Module-2
.
the point where the curve cuts the
is 27ay
2
= 4(x ? 2a)'.
= a"cosnO is a
n
.p = r"'
I
.
as r'.
? cos()) and r = b(1 + cos0).
by Maclaurin's series up
+b`
to the term containing x
4
.
= x
3
+ y
3
? 3x ? 12y + 20.
?u
au
+ z ? ? =0
cz
3
values of the function tax, y)
OR
, au
then prove that x ? ? + y
3 a.
b.
c.
b. If u = x + 3y
2
? z
3
, v = 4x
-
yz, w = 2z
-
? xy. Evaluate a(u, v,
w)
at the point (1 -1, 0).
a( x, y,z)
(07 Marks)
c. A rectangular box, open at the top, is to have a volume of 32 cubic feet. Find the dimensions
of the box, if the total surface area is minimum. (07 Marks)
Module-3
5 a. Evaluate by changing the order of integration
f
fx
2
?dy?dx , a> 0
h. Find the area bounded between the circle x
2
+ y
2
= a
2
and the line x + y = a.
F
-
1 ? i
-
1
in + n
c.
Prove that I3(m,
(06 Marks)
(07 Marks)
(07 Marks)
1 of 2
18MAT11
OR
c h a
6 a.
Evaluate
J J
(x
-
+ y
-
+ z
-
) dz.dy.dx
-c -h
y
2
b. Find the area bounded by the ellipse
x
+ =1 by double integration.
a
-
b
-
dO
c. Show that x J,/sin 0.d0 =
0 Vsin 0
0
Module-4
(06 Marks)
(07 Marks)
(07 Marks)
7 a. Solve (1 + e' )dx + e' ' 1--
x
dy = 0 (06 Marks)
Y
b. If the air is maintained at 30?C and the temperature of the body cools from 80?C to 60?C in
12 minutes. Find the temperature of the body after 24 minutes. (07 Marks)
c. Solve yp
2
+ (x ? y) p ? x = 0.
(07 Marks)
OR
8 a. Solve ?
dy
+ y ? tan x = y ? sec x (06 Marks)
dx
h. Find the orthogonal trajectory of the family of the curves rn?cosnO = a
n
, where a is a
parameter. (07 Marks)
c. Solve the equation (px y) ? (py + x) = 2p by reducing into Clairaut's form taking the
substitution X = x
-
, Y = y
2
. (07 Marks)
9 a.
b.
c. Using Rayleigh's power method find the largest eigen value and corresponding eigen vecto
of the matrix A =
'2 0 1N
0 2 0
?
I 0 2
/
with X'
?
= (I, 0, 0)
1
as the initial eigen vector carry out
5 iterations. (07 Marks)
OR
10 a. For what values of and t the system of equations.
x + y + z = 6, x+ 2y + 3z = 10, x + 2y + X.z = j_t may have
i) Unique solution ii) Infinite number of solutions iii) No solution. (06 Marks)
b. Reduce the matrix A =
(-1 3
into
)
diagonal lbrm. (07 (harks)
2
4
c.
Solve the following system of equations by Gauss-Seidel method
20x + y ? 2z = 17, 3x + 20y ? z = ?18, 2x ? 3y + 20z = 25. Carry out 3 iterations. (07 Marks)
2 of 2
S
o
ciety
--
;;"-\
\
a nouxnru Nst-
Find the rank of the matrix
(
l 2 ?2 3
/ 5
?4 6
Module-5
A = by applying elementary Row transformations. (06 Marks)
?1 ?3 2 ?2
2 4 ?1 6
Solve the following syste
-
n of equations by Gauss-Jordan method:
x + y + z = 9, 2x + y ? z = 0, 2x + 5y + 7z = 52 (07 Nlarks)
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This post was last modified on 01 January 2020