Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 2017 Jan 2016 Dec 15MAT11 Engineering Mathematics I Question Paper
2a
= 1+ cos B.
r
c. Find the radius of curvature of the curve r. = an Ake.
/trks)
Marks)
(05 Marks)
OR
2 a. if x = tan(log y), find the value of (1+x
2
)y,-,4-1 + (2nx-1)3rn + (11)(n-1)Yn-i-
Module-2
w
)
3 a. Explain log(cos x) about the point x = upto 3
rd
degree terms using Taylor's series.
_ 3
valuate&l:
2
. State Euler's theorem and use it to find x ?
)(
+ y when u = tan-I
+
au au
ax ay x + y
(06 Marks)
(05 Marks)
(05 Marks)
15MAT 11
USN
4.)
First Semester B.E. Degree Examination, Dec.2016/Jan.2017
Engineering Mathematics - I
Time: 3 hrs. Max. Marks: 80
Note: Answer FIVE full questions, choosing one full question from each module.
Module- 1
I a. If y ? e
x
cos
3
x, find y
n
. (06 Marks)
b. Find the angle between the curves
a b
r = and r ? . (05 Marks)
1 + cos() 1 ? cOSO
c. Find the radius of curvature of the curve x
4
+ y
4
= 2 at the point (1, 1). (05 Marks)
OR
4 a.
b.
C.
e x
Expand Maclaurin's including 3r
d
degree terms.
u , v,
(06 Marks)
(05 Marks)
(05 Marks)
using series upto and
1 +
ex
Find
u
when u =
x3y2 + x2
y
3
with x = at
2
, y = 2at. Use Partial derivatives.
dt
if ?
X2X
' ?
X
i
X
3
X
1
X
2
find the Jacobian J
, v
, w
, value of
x
i
x
2
x
3
x,,x
2
,x
1
Module-3
5 a. A particle moves on the curve x = 2t
1
7/
1-7
4 , 4t, z = 3t ? 5, where t is the time find the
components of velocity and acceleration at time t = 1 in the direction of i ? 3j + 2k.
(06 Marks)
b. Find the divergence and curl of the vector V = (xyz)i + (3x
2
y)j + (xz
2
? y
2
z)K at the point
(2, -1, 1). (05 Marks)
c. A vector field is given by A = (x
2
+ xy
2
) i + (y
2
+ x
2
y)j, show that the field is irrotational and
find the scalar potential.
z
1 of 2
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b. Find the Pedal equation of ?
2a
= 1+ cos B.
r
c. Find the radius of curvature of the curve r. = an Ake.
/trks)
Marks)
(05 Marks)
OR
2 a. if x = tan(log y), find the value of (1+x
2
)y,-,4-1 + (2nx-1)3rn + (11)(n-1)Yn-i-
Module-2
w
)
3 a. Explain log(cos x) about the point x = upto 3
rd
degree terms using Taylor's series.
_ 3
valuate&l:
2
. State Euler's theorem and use it to find x ?
)(
+ y when u = tan-I
+
au au
ax ay x + y
(06 Marks)
(05 Marks)
(05 Marks)
15MAT 11
USN
4.)
First Semester B.E. Degree Examination, Dec.2016/Jan.2017
Engineering Mathematics - I
Time: 3 hrs. Max. Marks: 80
Note: Answer FIVE full questions, choosing one full question from each module.
Module- 1
I a. If y ? e
x
cos
3
x, find y
n
. (06 Marks)
b. Find the angle between the curves
a b
r = and r ? . (05 Marks)
1 + cos() 1 ? cOSO
c. Find the radius of curvature of the curve x
4
+ y
4
= 2 at the point (1, 1). (05 Marks)
OR
4 a.
b.
C.
e x
Expand Maclaurin's including 3r
d
degree terms.
u , v,
(06 Marks)
(05 Marks)
(05 Marks)
using series upto and
1 +
ex
Find
u
when u =
x3y2 + x2
y
3
with x = at
2
, y = 2at. Use Partial derivatives.
dt
if ?
X2X
' ?
X
i
X
3
X
1
X
2
find the Jacobian J
, v
, w
, value of
x
i
x
2
x
3
x,,x
2
,x
1
Module-3
5 a. A particle moves on the curve x = 2t
1
7/
1-7
4 , 4t, z = 3t ? 5, where t is the time find the
components of velocity and acceleration at time t = 1 in the direction of i ? 3j + 2k.
(06 Marks)
b. Find the divergence and curl of the vector V = (xyz)i + (3x
2
y)j + (xz
2
? y
2
z)K at the point
(2, -1, 1). (05 Marks)
c. A vector field is given by A = (x
2
+ xy
2
) i + (y
2
+ x
2
y)j, show that the field is irrotational and
find the scalar potential.
z
1 of 2
15MAT11
OR
6 a. Find grad (I) when if = 3x
2
y ? y
3
z
2
at the point (1, -2, -1). (06 Marks)
b. Find a for which f? (x + 3y)i + (y - 2z)j + (x + az)k is solenoidal. (05 Marks)
c. Prove that Div(curl V) = 0.
(05 Marks)
Module-4
7 a. Obtain the reduction formula of !sin' x cos
n
x dx. (06 Marks)
2a
b. Evaluate f xV2ax x
2
dx. (05 Marks)
c. Solve (2x log x ? xy) dy + 2y dx = 0.
(05 Marks)
OR
8
a. Obtain the reduction formula off cos' x dx. (06 Marks)
b. Obtain the Orthogonal trajectory of the family of curves r" cos n 0 = a.. Hence solve it.
(05 Marks)
c. A body originally at 80
?
C cools down at 60
?
C in 20 minutes, the temperature of the air being
40
?
C. What will be the temperature of the body after 40 minutes from the original?(05 Marks)
9 a. Find the rank of the matrix
. Solve b y
clitio
by Gauss ? Jordan method the system of linear equations
2x + y + z = 10, 3x + 2y + 3z = 18 , x + 4y + 9z = 16. (05 Marks)
. Find the largest eigen value and the corresponding Eigen vector by power method given that
2 0 1
-
A ? 0 2 0 . (Use [1 0 0 ]
r
as the initial vector). (Apply 4 iterations). (05 Marks)
1 0 2
OR
10 a. Use Gauss ? Seidel method to solve the equations (06 Marks)
20x + y ? 2x = 17
3x + 20y ? z = 18
2x ? 3y + 20z = 25. Carry out 2 iterations with xo ? yo ? zo -- 0.
2 ?2
-
b.
[-1
Reduce the matrix A = 1 2 1 to the diagonal form. (05 Marks)
?1 ?1 0
c. Reduce the quadratic form 3x
2
+ 5y
2
+ 3z
2
? 2yz + 2zx ? 2xy to the canonical form.
(05 Marks)
2 3 ? 1 ?1
A=
1 ?1 ?
3 1 3 ?2
6 3 0 ? 7
(06 Marks)
2 of 2
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