Download Visvesvaraya Technological University (VTU) BE/B.Tech First And Second Semester (1st sem and 2nd sem) 2019-2020 Jan ( Bachelor of Engineering) 18 Scheme 8MAT11 Calculus and Linear Algebra Previous Question Paper
c
\?, H1KOD ?
USN
8MAT11
Module-2
3 a. Find the Macluarin's series for tanx upto the term x
4
.
I
b. Evaluate lim
3
X
a' +bx +c'
(06 Marks)
(07 Marks)
First Semester B.E. Degree Examination, Dec.201klae.2020
Calculus and Linear Algebra
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
de'
1
a.
With usual notations prove that tan c = r ?
dr
h. Find the angle between the curves r = sine + cos() and r = 2 sin?
c. Show that the radius of curvature for the catenary of uniform strength
x
y = a log sec ? is a sec (x/a).
a
(06 Marks)
(06 Marks)
(08 Marks)
OR
2 a.
Show that the pairs of curves r = a(1 + cost)) and
Orthogonally.
b. Find the pedal equation of the curve r
n
= a" cos ne.
c. Show that the evolute of y
2
= 4ax is 27ay
2
= 4(x + a)
3
.
r = b(1-cos0) intersect each other
(06 Marks)
(06 Marks)
(08 Marks)
c. If
i7 u
= f(x-y, y-z, z-x), prove that - - +
au
+
au
= 0
OR
4 a. Expand log (sec x) upto the term containing x
4
using Maclaurin's series.
b. Find the extreme values of the function f(x, y) = x
3
+ y
3
- 3x - 12y + 20.
Find
i)(11
'
v
'
w)
where u = x
2
+ y
2
+ z
2
, v = xy + yz + zx, w = x + y + z.
0(x? y, z)
Module-3
N1?x 1/
1?
x
2
5
a.
Evaluate
J
J
f xyz dzdydx
0 0 0
2 \'4 x`
b. Evaluate f f (2 - x)dydx by changing the order of integration.
-2 0
c. Prove that 13(
(m) .
(n)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
1 of 2
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LIBRARY
c
\?, H1KOD ?
USN
8MAT11
Module-2
3 a. Find the Macluarin's series for tanx upto the term x
4
.
I
b. Evaluate lim
3
X
a' +bx +c'
(06 Marks)
(07 Marks)
First Semester B.E. Degree Examination, Dec.201klae.2020
Calculus and Linear Algebra
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
de'
1
a.
With usual notations prove that tan c = r ?
dr
h. Find the angle between the curves r = sine + cos() and r = 2 sin?
c. Show that the radius of curvature for the catenary of uniform strength
x
y = a log sec ? is a sec (x/a).
a
(06 Marks)
(06 Marks)
(08 Marks)
OR
2 a.
Show that the pairs of curves r = a(1 + cost)) and
Orthogonally.
b. Find the pedal equation of the curve r
n
= a" cos ne.
c. Show that the evolute of y
2
= 4ax is 27ay
2
= 4(x + a)
3
.
r = b(1-cos0) intersect each other
(06 Marks)
(06 Marks)
(08 Marks)
c. If
i7 u
= f(x-y, y-z, z-x), prove that - - +
au
+
au
= 0
OR
4 a. Expand log (sec x) upto the term containing x
4
using Maclaurin's series.
b. Find the extreme values of the function f(x, y) = x
3
+ y
3
- 3x - 12y + 20.
Find
i)(11
'
v
'
w)
where u = x
2
+ y
2
+ z
2
, v = xy + yz + zx, w = x + y + z.
0(x? y, z)
Module-3
N1?x 1/
1?
x
2
5
a.
Evaluate
J
J
f xyz dzdydx
0 0 0
2 \'4 x`
b. Evaluate f f (2 - x)dydx by changing the order of integration.
-2 0
c. Prove that 13(
(m) .
(n)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
1 of 2
18MAT11
OR
6 a. Evaluate f f ydx dy over the region bounded by the first quadrant of the ellipse x
'
y
- + =1.
a
-
b
-
(06 Marks)
b. Find by double integration the area enclosed by the curve r = a (1 + CosO) between 0 = 0
and 0 = 1t. (07 Marks)
n/2
"
in
R
,. ,
2
c. Show that f , x f A/Sin? dO = it .
AiSin0 ?
(07 Marks)
Module-4
dy y cos x + sin y + y
7 a. Solve + = 0 (06 Marks)
dx sin x + x cosy + x
dr
b. Solve rSinO - Cos() = r' (07 Marks)
dO
C. A series circuit with resistance R, inductance L and electromotive force E is governed by the
Ri differential equation L di ? + RI = E , where L and R are constants and initially the current i is
dt
zero. Find the current at any time t.
OR
(07 Marks)
8 a.
b.
Solve (4xy + 3y
2
? x)dx + x (x + 2y)dy = 0.
1
Find the orthogonal trajectories of the family of parabolas y
-
= 4ax.
(06 Marks)
(07 Marks)
c. Solve p"
1
+ 2py cotx = y
-
i
. (07 Marks)
[1 2 3 2
9
a. Find the rank of 2 3 5 1 by elementary row transformations. (06 Marks)
1 3 4 5
b. Apply Gauss-Jordan method to solve the system of equations
2x
1
+ x
2
+ 3x3 = 1,
4x, + 4x
2
+ 7X3 = 1,
2x] + 5x2 + 9X3 = 3. (07 Marks)
c.
Find the largest Eigen value and the corresponding Eigen vector of the matrix
2 0 1
A = 0 2 0 by power method. Using initial vector (100)
T
. (07 Marks)
1 0 2
R
10 a. Solve by Gauss elimination method
x 2y + 3z --- 2,
3x y + 4z --- - 4,
2x
-
F y ? 2z = 5
b. Solve the system of equations by Gauss-Seidal method
20x + y ? 2z 17,
3x + 20y ? z -18,
2x 3y + 20z 25
(06 Marks)
(07 Marks)
?1 3
4
c.
Reduce the matrix A = to the diagonal form. (07 Marks)
2 of 2
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This post was last modified on 02 March 2020