Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 18 Scheme 18MAT11 Calculus and Linear Algebra Question Paper

Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 18 Scheme 18MAT11 Calculus and Linear Algebra Question Paper

USN
0.046
First Semester B.E. Degree Examination, Deg.2018/Jan.2019
Calculus and Linear Algebra
18MAT11
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
c7J
7-s
1 a.
Show that the curves r
n
= a
n
cos no and r
n
= b
11
sin nO are intersect orthogonally. (06 Marks)
b. Find the radius of curvature of the curve y = a log sece
(-
) at any point (x, y). (06 Marks)
'04
=
c. Show that the evolute of the parabola y
2
=4ax is 27ay
2
= 4(x ?2a)
3
. (08 Marks)
e
ci (=>
c 00
. N
OR
1
1
.4
+6
2 a. With usual notation, prove that tart(1) r dO ? . (06 Marks)
dr
a) '
4

?
b. Find the pedal equation of the curve r =
ae
ocota
(06 Marks)
I.
0
i
8

5 7:1
t.A =
czt et
:
7
73 s...
0
c '74 Module-2
.5 =
?,
?
?
0 -- x
2
X:
3
X
4
. -- ,..., 0
'' Td
3 a. Using Maclaurin's expansion. Prove - that -Nil + sin 2x .=:. + x +
.
1.)
(06 Marks)
ca. .., 2 6 24
cL.
0 ct
e)
r
',. ?.6
b
X X X
?: +4
a +b +c +
(07 Marks)
b. Evaluate It
u
t
4
. - 4
x ? >0
.
4

ct -.
-c,
?
?-,,,-.
c.
Find the dimensions of the rectangular bOx open at the top of maximum capacity whose
to
surface is 432 sq:c..'rn. (07 Marks)
_, .) :=4.
0. 8
a>
0 P
c.)
o ..,
>,
OR
= =

.__: r,i
Important Note
'au u
4 a. If u = f(y z, z ? x, x ? y) , show that
Ou
? + +
0,
=0. (06 Marks)
ox ay az
2
b. If u.= x
-
+ y
2
+z v = xy + yz + zx, w = x + y + z . Find Jacobian
a(u,v, w)
= .(07 Marks)
ctx, y,z)
c. Find the minimum value of x
2
+ v
2
+ z' subject to the condition x + v 7 = 3a . (07 Marks)
Module-1
c. Find the radius of curvature for the curve r = a(1 4-- COS()) . (08 Marks)
1 of 3
FirstRanker.com - FirstRanker's Choice
USN
0.046
First Semester B.E. Degree Examination, Deg.2018/Jan.2019
Calculus and Linear Algebra
18MAT11
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
c7J
7-s
1 a.
Show that the curves r
n
= a
n
cos no and r
n
= b
11
sin nO are intersect orthogonally. (06 Marks)
b. Find the radius of curvature of the curve y = a log sece
(-
) at any point (x, y). (06 Marks)
'04
=
c. Show that the evolute of the parabola y
2
=4ax is 27ay
2
= 4(x ?2a)
3
. (08 Marks)
e
ci (=>
c 00
. N
OR
1
1
.4
+6
2 a. With usual notation, prove that tart(1) r dO ? . (06 Marks)
dr
a) '
4

?
b. Find the pedal equation of the curve r =
ae
ocota
(06 Marks)
I.
0
i
8

5 7:1
t.A =
czt et
:
7
73 s...
0
c '74 Module-2
.5 =
?,
?
?
0 -- x
2
X:
3
X
4
. -- ,..., 0
'' Td
3 a. Using Maclaurin's expansion. Prove - that -Nil + sin 2x .=:. + x +
.
1.)
(06 Marks)
ca. .., 2 6 24
cL.
0 ct
e)
r
',. ?.6
b
X X X
?: +4
a +b +c +
(07 Marks)
b. Evaluate It
u
t
4
. - 4
x ? >0
.
4

ct -.
-c,
?
?-,,,-.
c.
Find the dimensions of the rectangular bOx open at the top of maximum capacity whose
to
surface is 432 sq:c..'rn. (07 Marks)
_, .) :=4.
0. 8
a>
0 P
c.)
o ..,
>,
OR
= =

.__: r,i
Important Note
'au u
4 a. If u = f(y z, z ? x, x ? y) , show that
Ou
? + +
0,
=0. (06 Marks)
ox ay az
2
b. If u.= x
-
+ y
2
+z v = xy + yz + zx, w = x + y + z . Find Jacobian
a(u,v, w)
= .(07 Marks)
ctx, y,z)
c. Find the minimum value of x
2
+ v
2
+ z' subject to the condition x + v 7 = 3a . (07 Marks)
Module-1
c. Find the radius of curvature for the curve r = a(1 4-- COS()) . (08 Marks)
1 of 3
Module-3
113
5 a. Evaluate e
-(x )
dxdy , by changing into polar coordinates.
0 0
b. Find the volume of the tetrahedron bounded by the planes :
x = 0 y = 0, z = 0, X+-+L=1.
a b c
c.
Prove that f3( ,n) =
F(rn)F(n)
F(m+n)
OR
I Vx
6
a.
Evaluate xy dy dx by change of order of integration.
0 x
z x+z
b. Evaluate
i ff f
(
x+ y + z)dy dx dz .
-10 x-z
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
c.
if
-
1
Prove that sin 0 . dOx
f
. d0 = Tr .
o
0 Vsin 0
(07 Marks)
Module-4
7 a. A body ,in air at 25?C cools from 100?C to 75?C in 1 minute, find the temperature of the
body at the end of 3 minutes. (06 Mark
Solve
dy
+
y cos x + sin y + y
=
dx sin x + x cos y + x
7
c. Solve xyp
-
- (x
-7
y
2

)p + xy = 0 .
(07 Marks)
(07 Marks)
OR
, dv
8 a.
Solve + y tan x = y see x.
dx
(06 Marks)
b. Show that the family of parabolas y
7
= 4a(x + a) is self orthogonal. (07 Marks)
c. Find the general solution of the equation (px y)(py + x) = 0 by reducing into Clairaut's
from, taking the substitution X = x
2
, Y = y
2
. (07 Marks)
2 of3
FirstRanker.com - FirstRanker's Choice
USN
0.046
First Semester B.E. Degree Examination, Deg.2018/Jan.2019
Calculus and Linear Algebra
18MAT11
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
c7J
7-s
1 a.
Show that the curves r
n
= a
n
cos no and r
n
= b
11
sin nO are intersect orthogonally. (06 Marks)
b. Find the radius of curvature of the curve y = a log sece
(-
) at any point (x, y). (06 Marks)
'04
=
c. Show that the evolute of the parabola y
2
=4ax is 27ay
2
= 4(x ?2a)
3
. (08 Marks)
e
ci (=>
c 00
. N
OR
1
1
.4
+6
2 a. With usual notation, prove that tart(1) r dO ? . (06 Marks)
dr
a) '
4

?
b. Find the pedal equation of the curve r =
ae
ocota
(06 Marks)
I.
0
i
8

5 7:1
t.A =
czt et
:
7
73 s...
0
c '74 Module-2
.5 =
?,
?
?
0 -- x
2
X:
3
X
4
. -- ,..., 0
'' Td
3 a. Using Maclaurin's expansion. Prove - that -Nil + sin 2x .=:. + x +
.
1.)
(06 Marks)
ca. .., 2 6 24
cL.
0 ct
e)
r
',. ?.6
b
X X X
?: +4
a +b +c +
(07 Marks)
b. Evaluate It
u
t
4
. - 4
x ? >0
.
4

ct -.
-c,
?
?-,,,-.
c.
Find the dimensions of the rectangular bOx open at the top of maximum capacity whose
to
surface is 432 sq:c..'rn. (07 Marks)
_, .) :=4.
0. 8
a>
0 P
c.)
o ..,
>,
OR
= =

.__: r,i
Important Note
'au u
4 a. If u = f(y z, z ? x, x ? y) , show that
Ou
? + +
0,
=0. (06 Marks)
ox ay az
2
b. If u.= x
-
+ y
2
+z v = xy + yz + zx, w = x + y + z . Find Jacobian
a(u,v, w)
= .(07 Marks)
ctx, y,z)
c. Find the minimum value of x
2
+ v
2
+ z' subject to the condition x + v 7 = 3a . (07 Marks)
Module-1
c. Find the radius of curvature for the curve r = a(1 4-- COS()) . (08 Marks)
1 of 3
Module-3
113
5 a. Evaluate e
-(x )
dxdy , by changing into polar coordinates.
0 0
b. Find the volume of the tetrahedron bounded by the planes :
x = 0 y = 0, z = 0, X+-+L=1.
a b c
c.
Prove that f3( ,n) =
F(rn)F(n)
F(m+n)
OR
I Vx
6
a.
Evaluate xy dy dx by change of order of integration.
0 x
z x+z
b. Evaluate
i ff f
(
x+ y + z)dy dx dz .
-10 x-z
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
c.
if
-
1
Prove that sin 0 . dOx
f
. d0 = Tr .
o
0 Vsin 0
(07 Marks)
Module-4
7 a. A body ,in air at 25?C cools from 100?C to 75?C in 1 minute, find the temperature of the
body at the end of 3 minutes. (06 Mark
Solve
dy
+
y cos x + sin y + y
=
dx sin x + x cos y + x
7
c. Solve xyp
-
- (x
-7
y
2

)p + xy = 0 .
(07 Marks)
(07 Marks)
OR
, dv
8 a.
Solve + y tan x = y see x.
dx
(06 Marks)
b. Show that the family of parabolas y
7
= 4a(x + a) is self orthogonal. (07 Marks)
c. Find the general solution of the equation (px y)(py + x) = 0 by reducing into Clairaut's
from, taking the substitution X = x
2
, Y = y
2
. (07 Marks)
2 of3
as the initial eigen vector (carry out 6 iterations).
(07 Marks)
SOC4eiya.
CHMOD
4

Irk
LIBRARY
(06 Marks)
18MAT1 1
Module-5
9 a. Find the rank of the matrix :
1 2 ?2 3
2 5 ?4 6
A = (07 Marks)
?1 ?3 2 ?2
2 4 ?1
6
b. Solve the system of equations
12x+ y+ z=31
2x + 8y z = 24
3x + 4y + 10z = 58
By Gauss ?Siedal method.
c. Diagonalize the matrix :
?1 3
?2 4
A =
(07 Marks)
(06 Marks)
OR
10 a.
For what values of X, and M the system of equations :
x + 2y + 3z = 6
x + 3y + 5z = 9
2x + 5y + =M
has i) no solution ii) a unique solution iii) infinite number of solution. (07 Marks)
b.
Find the largest eigen value and the corresponding eigen vector of :
6 ?2
A= ? 2 3 -
2 ?1 3
by Rayleigh's power method, use [1 1
Solve the system of`equations :
x+ y+ z=9
2x + y? z = 0
2x + 5y + 7z = 52
By Gauss elimination method.
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