Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2015 Jan 2014 Dec 14MAT11 Engineering Mathematics I Question Paper

Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 2015 Jan 2014 Dec 14MAT11 Engineering Mathematics I Question Paper

(06 Marks)
c.
Examine the function f(x, y) = 1+sin(x
2
+y
2
) for extremum. (07 Marks)
Evaluate .
(sin x); .
b.
X 0 X
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14MAT11

First Semester B.E. Degree Examination, Dec.2014/Jan.2015
Engineering Mathematics - I
6
t
u?
Time: 3 hrs.
,,,,
,..
a.
71 Note: Answer any FIVE full questions, selecting ONE full question from each part,
E
co
iz
PART ? 1
E
I
a. If Y = cos(m log x) , prove that x
2
y,
i+2
+ (2n +1)xy
ri+
,
+
(m
2
+n
2
)yn
_0.
(07 Marks)
j 0
a
4 a"
zt = b. Find the angle of intersection between the curves r = a log 0 and r = (06 Marks)
o?.?
E'1'
c5
c. Derive an expression to find radius of curvature in Cartesian form. (07 Marks)
II
bo
to
a
.
it3
2 a.
-4- If sin - ' y = 2 1og(x +1) prove that (x2
+1)Yn+2
? (2n +1)(x +1)y,
i
+ (n
2
+ 4)y
ri
= 0 .

9
4)
(07 Marks)
b. Find the pedal equation r
n
= sec hne . (06 Marks)
.V.:
3 . -3
=

C.
07 Marks)
o
2 2
8112
? .-
_, r
i

8 =
U o
?
cr

? it:)
ozi
z
3 a. Find the first four non zero terms in the expansion of f )=
e
3

:_
i
. (07 Marks)
o
cis x + y au
0 4
5 'rt b. If cosu =
i
_ show that x au
cotu
? + y? = -
?
(06 Marks)
-*/ x + .4y
ax ay 2
Max. Marks:100
loge -
-100s,
PART -2
Important Note
w) a(u, v,
C. Find where u = x
2
+ y
2
+ z
2
, v = xy + yz + zx and w=x+y+z. Hence interpret
a(x, y, z)
the result.
(07 Marks)
4 a. If w = f(x, y) , x = rcos0 , y = r sin 0 show that
rat)
2
+
1 at1
2
-
(aw)
2
1 ra-wy
ax) a
y
) ar 1
7
ao
(07 Marks)
PART - 3
5 a.
A particle moves along the curve x = 2t
2
, y = t
2
-4t , z = 3t -5 . Find the components of
velocity and acceleration at t = 1 in the direction i -23+2k . (07 Marks)
b. Using differentiation under integral sign, evaluate
e
-ax
sin x
dx . (07 Marks)
0
c. Use general rules to trace the curve y
2
(a- x) = x
3
, a > 0 (06 Marks)
1 oft
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(06 Marks)
c.
Examine the function f(x, y) = 1+sin(x
2
+y
2
) for extremum. (07 Marks)
Evaluate .
(sin x); .
b.
X 0 X
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USN

14MAT11

First Semester B.E. Degree Examination, Dec.2014/Jan.2015
Engineering Mathematics - I
6
t
u?
Time: 3 hrs.
,,,,
,..
a.
71 Note: Answer any FIVE full questions, selecting ONE full question from each part,
E
co
iz
PART ? 1
E
I
a. If Y = cos(m log x) , prove that x
2
y,
i+2
+ (2n +1)xy
ri+
,
+
(m
2
+n
2
)yn
_0.
(07 Marks)
j 0
a
4 a"
zt = b. Find the angle of intersection between the curves r = a log 0 and r = (06 Marks)
o?.?
E'1'
c5
c. Derive an expression to find radius of curvature in Cartesian form. (07 Marks)
II
bo
to
a
.
it3
2 a.
-4- If sin - ' y = 2 1og(x +1) prove that (x2
+1)Yn+2
? (2n +1)(x +1)y,
i
+ (n
2
+ 4)y
ri
= 0 .

9
4)
(07 Marks)
b. Find the pedal equation r
n
= sec hne . (06 Marks)
.V.:
3 . -3
=

C.
07 Marks)
o
2 2
8112
? .-
_, r
i

8 =
U o
?
cr

? it:)
ozi
z
3 a. Find the first four non zero terms in the expansion of f )=
e
3

:_
i
. (07 Marks)
o
cis x + y au
0 4
5 'rt b. If cosu =
i
_ show that x au
cotu
? + y? = -
?
(06 Marks)
-*/ x + .4y
ax ay 2
Max. Marks:100
loge -
-100s,
PART -2
Important Note
w) a(u, v,
C. Find where u = x
2
+ y
2
+ z
2
, v = xy + yz + zx and w=x+y+z. Hence interpret
a(x, y, z)
the result.
(07 Marks)
4 a. If w = f(x, y) , x = rcos0 , y = r sin 0 show that
rat)
2
+
1 at1
2
-
(aw)
2
1 ra-wy
ax) a
y
) ar 1
7
ao
(07 Marks)
PART - 3
5 a.
A particle moves along the curve x = 2t
2
, y = t
2
-4t , z = 3t -5 . Find the components of
velocity and acceleration at t = 1 in the direction i -23+2k . (07 Marks)
b. Using differentiation under integral sign, evaluate
e
-ax
sin x
dx . (07 Marks)
0
c. Use general rules to trace the curve y
2
(a- x) = x
3
, a > 0 (06 Marks)
1 oft
alize the matrix, A =
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14MAT11
6
a-
b.
c.
If v = wx r , prove that
Show that div(curlA)=
If r + yj + zk and
o
r
curly = 2w where w is
.
= r . Find grad div[?
r

a constant vector. (07 Marks)
(06 Marks)
. (07 Marks)
PART 4
2
Obtain the reduction formula for 'cos' xdx . (07 Marks)
b
Solve (xy
3
+ y)dx + 2(x
2
y
2
+ x + y
4
)dy = 0 . (06 Marks)
c.
Show that the orthogonal trajectories of the family of cardioids r = a cos
2
(?
(31
) is another
2
(0
family of cardio ids r= b sin
2
?
2
(07 Marks)
,.
(
8 a. Evaluate 1 xsin
2
x cos
*
xdx . (07 Marks)
o
b. Solve ? y tan x = y
2
sec x .
0
(06 Marks) ( ?
dy

dx
c. If the temperature , of the air is 30?C and t subs e cools from 100?C to 70?C in 15
minutes, find when the temperature will be 40? .
PART? 5
9 a. Solve ? y + 2z =12 x +2,y +3z =11, 2x ? 2y z = 2 by Gauss elimination method.
(07 Marks)
(06 Marks)
?1 1 2
0 ?2 ?1 (07 Marks)
0 0 ?3
c.

Determine the largest eigen value and the corresponding eigen vector of
1 3 ?1
A= 3 2 4 .
?1 4 10
Starting with [o, 0, 1]
T
as the initial eigenvector. Perform 5 iterations. (07 Marks)
10 a. Show that the transformation y, = x, + 2x, + 5x
3
, y, = 2x, + 4x, +11x, , y
3
= ?x, + 2x
3
is
regular and find the inverse transformation. (06 Marks)
b. Solve by LU decomposition method 2x + y + 4z =12 , 8x-3y + 2z = 20 , 4x +11y z =33.
(07 Marks)
c. Reduce the quadratic form 2x
2
+ 2y
2
? 2xy ?2yz-2zx into canonical form. Hence indicate
its nature, rank, index and signature. (07 Marks)
2 of 2
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