Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 18 Scheme 18INIAT11 Calculus and Linear Algebra Question Paper
18INIAT11
Al
of EMI
S
IL"
First Semester B.F. Degree Examination?June/Jul:s 2019
Calculus and Linear Algebra
Time: 3 hrs.
Max. Marks: 100
Note: Answer an;' FIVE full questions, choosing ONE full question from each module.
Module-1
de'
1 a. With usual notation, prove that tan fi = r (06 Marks)
dr
b. Find the radius of curvature of a
2
y = ? a
3
at the point where the curve cuts the x-axis.
(06 Marks)
C. Show that the evolute of the parabola y
2
= 4ax is 27ay
2
= 4(x ? 20
3
. (08 Marks)
(06 Marks)
(06 Marks)
(08 Marks)
(06 Nlarks)
(07 Marks)
(07 Marks)
Module-2
3 a. Expand log( I + cosx) by Maclaurin's series up to the term containing x
4
.
lira +b' +c' \
b. Evaluate
c.
OR
x y z
then
"eu
4 a.
If u = f tnen prove that x ? ? + y + z ? -- =0
y z x
j
OR
2
a. Prove that the pedal equation of the curve r" = a"cosnO is a
n
.p = r"
+1
.
b.
Show that for the curve r( ? cos0) = 2a, p
2
varies as C.
C.
Find the angle between the polar curves r = a( cos()) and r = + cos0).
x -0l
3
Find the extreme values of the function tax, y) = x
3
+ y
3
? 3x ? 12y + 20.
(06 Marks)
b. If u x + 3y
-
?
v 4x
-
yz, w = 2z
-
? xy. Evaluate
r(u,v,w)
at the point (1, -I, 0).
i)(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
ax
x dy?dx a > 0
,
h. Find the area bounded between the circle x
-
7
+ y
-
= a
-
and the line x + y = a.
m. n
c.
Prove that 13( m,
n
(06 Marks)
(07 Marks)
(07 Marks)
1 of 2
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USN
18INIAT11
Al
of EMI
S
IL"
First Semester B.F. Degree Examination?June/Jul:s 2019
Calculus and Linear Algebra
Time: 3 hrs.
Max. Marks: 100
Note: Answer an;' FIVE full questions, choosing ONE full question from each module.
Module-1
de'
1 a. With usual notation, prove that tan fi = r (06 Marks)
dr
b. Find the radius of curvature of a
2
y = ? a
3
at the point where the curve cuts the x-axis.
(06 Marks)
C. Show that the evolute of the parabola y
2
= 4ax is 27ay
2
= 4(x ? 20
3
. (08 Marks)
(06 Marks)
(06 Marks)
(08 Marks)
(06 Nlarks)
(07 Marks)
(07 Marks)
Module-2
3 a. Expand log( I + cosx) by Maclaurin's series up to the term containing x
4
.
lira +b' +c' \
b. Evaluate
c.
OR
x y z
then
"eu
4 a.
If u = f tnen prove that x ? ? + y + z ? -- =0
y z x
j
OR
2
a. Prove that the pedal equation of the curve r" = a"cosnO is a
n
.p = r"
+1
.
b.
Show that for the curve r( ? cos0) = 2a, p
2
varies as C.
C.
Find the angle between the polar curves r = a( cos()) and r = + cos0).
x -0l
3
Find the extreme values of the function tax, y) = x
3
+ y
3
? 3x ? 12y + 20.
(06 Marks)
b. If u x + 3y
-
?
v 4x
-
yz, w = 2z
-
? xy. Evaluate
r(u,v,w)
at the point (1, -I, 0).
i)(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
ax
x dy?dx a > 0
,
h. Find the area bounded between the circle x
-
7
+ y
-
= a
-
and the line x + y = a.
m. n
c.
Prove that 13( m,
n
(06 Marks)
(07 Marks)
(07 Marks)
1 of 2
" 1
2 - 2
3
A
5 -4 6
-1 -3 2 -2
2 4 -1 6
by applying elementary Row transformations. (06 Marks)
18MAT11
OR
c h
6
a. Evaluate
.
1 f f(x
2
+ + ) dz.dy.dx
(06 Marks)
?b -a
b. Find Find the area bounded by the ellipse ?_-
-1 by double integration. (07 Marks)
a
-
dO x
7:
c. Show that 1-N/sin 0.d0 (07 Marks)
Vs
in 0
Module-4
7 a. Solve (1 + e' )dx + e '
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
b. Find the orthogonal trajectory of the family of the curves r
n
?eosn0 = 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 = (07 Marks)
Module-5
9 a_ Find the rank of the matrix
b. Solve the following system of equations by Gauss-Jordan method:
x + y + z 9, 2x + y - z = 0, 2x + 5y + 7z = 52 (07 Marks)
c.
Using Rayleigh's power method find the largest eigen value and corresponding eigen vecto
0 1
of the matrix A = 0 2 0 with X
(
''' = (1, 0, 0)
1
' as the initial eigen vector carry out
0 2
5 iterations.
(07 Marks)
10 a.
OR
For what values of r. and? the system of equations.
x + y + z = 6, x+ 2y + 3z --- - 10, x + 2y + Xy. p may have
i) Unique solution ii) Infinite number of solutions n ) No solution. (06 Marks)
? 1
h. Reduce the matrix A = into diagonal form. (07 Marks)
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 oft
-
P
a
s
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? F
.?
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This post was last modified on 01 January 2020