Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2014 June-July 10MAT11 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) 2014 June-July 10MAT11 Engineering Mathematics I Question Paper

10MAT11
PDF Eraser Free
USN
(04 Marks)
D) None of these
First Semester B.E. Degree Examination, June / July 2014
Engineering Mathematics - I
.
-2
'
Time: 3 hrs.
a.
E
.
4 Note: 1. Answer any FIVE full questions, choosing at least two from each part.
IR
2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.
3. Answer to objective type questions on sheets other than OMR will not be valued.
oD
?
-c
1
z
i g 1 a. Choose;the correct answers for the following :
O00
i) If y? (4177)
n
e
4
" cos x +n tan
-1
-
1
then y ?
? ii
00
:a" ill
-
4
Va -zr
a
i.o 4)
A) e
4
x cosx B) e
2
" sin 3x C) ex cos x
4.) 0
?B 2 X
X
5
x
7

= ? E:
ii) sin x =x? +---- ........ is,
3
3! 5! 7!
O ci
,

O 0
= .0
A) Taylor's series .-B) Exponential series C) Meclaurin's series D) None of these
2
. ti. ,
. " iii) In the Rolle's theorem if ;F'(c) = 0 then the tangent at the point x = c is,

Q 0
Er
A) parallel to y-axis B) parallel to x-axis C) parallel to both axes D) None of these
7,1 O'
' ? I; 0
00 0
? es
5' 15
A) (log x)3" B) 3(logx)n
,
C) 3" log 3' D) 3' (log
e
3)"
at
If x = sin t, y = sinpt prove that, (1? x
2
)v ?(2n +1)xy
e+
, + (p
2
?n
2
)y
n
=O. (04 Marks)
=
74 b. ,.. n+2
'
c. State and prove Cauchy's mean value theorem in [0, 16]. (06 Marks)
o
73 d. Expand 41+ sin 2x by using Meclaurin's expansion.
04
(06 Marks)
Co
IlL?01/11P
C'S
2 e
0
a. Choose the correct answers for the following :
(04 Marks)
i) The value of It 0 + xfx is, al
5
x"...
, 4a.
1
:8
A), e B) 1 C) ? D) 09
,,...
e
o
-17,.. =
tu ?
ii) The angle between two curves r = ae
e
and re
p
= b is,
... 00
? 73
2
0. .
= it
4
it
e > A) B) C) 0 D) TC
2
0

o
iii)
ls
. l
ir dx )
2
+
(dy )
2

.?
., dt dt ) dt )
Z' A) Polar form B) Parametric form C) Cartesian form D) None of these
t'
O log x
t It
0
t
x-
,
.. cot x
A) 1 B) 0 C) 2 D) ? 2
x(1+ acosx) ? bsinx
b. Find a & b, if It = 1 . (04 Marks)
' x-o
x
3
c. Find the pedal equation of the curve r
2
= a
2
cos20 (06 Marks)
d. Find the radius of curvature at any point t of the curve x = a(t +sin t) and y = a(1? cost) .
(06 Marks)
Max. Marks:100
PART ? A
iv) If y = 3' then y
n
=
1 of 4
FirstRanker.com - FirstRanker's Choice
10MAT11
PDF Eraser Free
USN
(04 Marks)
D) None of these
First Semester B.E. Degree Examination, June / July 2014
Engineering Mathematics - I
.
-2
'
Time: 3 hrs.
a.
E
.
4 Note: 1. Answer any FIVE full questions, choosing at least two from each part.
IR
2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.
3. Answer to objective type questions on sheets other than OMR will not be valued.
oD
?
-c
1
z
i g 1 a. Choose;the correct answers for the following :
O00
i) If y? (4177)
n
e
4
" cos x +n tan
-1
-
1
then y ?
? ii
00
:a" ill
-
4
Va -zr
a
i.o 4)
A) e
4
x cosx B) e
2
" sin 3x C) ex cos x
4.) 0
?B 2 X
X
5
x
7

= ? E:
ii) sin x =x? +---- ........ is,
3
3! 5! 7!
O ci
,

O 0
= .0
A) Taylor's series .-B) Exponential series C) Meclaurin's series D) None of these
2
. ti. ,
. " iii) In the Rolle's theorem if ;F'(c) = 0 then the tangent at the point x = c is,

Q 0
Er
A) parallel to y-axis B) parallel to x-axis C) parallel to both axes D) None of these
7,1 O'
' ? I; 0
00 0
? es
5' 15
A) (log x)3" B) 3(logx)n
,
C) 3" log 3' D) 3' (log
e
3)"
at
If x = sin t, y = sinpt prove that, (1? x
2
)v ?(2n +1)xy
e+
, + (p
2
?n
2
)y
n
=O. (04 Marks)
=
74 b. ,.. n+2
'
c. State and prove Cauchy's mean value theorem in [0, 16]. (06 Marks)
o
73 d. Expand 41+ sin 2x by using Meclaurin's expansion.
04
(06 Marks)
Co
IlL?01/11P
C'S
2 e
0
a. Choose the correct answers for the following :
(04 Marks)
i) The value of It 0 + xfx is, al
5
x"...
, 4a.
1
:8
A), e B) 1 C) ? D) 09
,,...
e
o
-17,.. =
tu ?
ii) The angle between two curves r = ae
e
and re
p
= b is,
... 00
? 73
2
0. .
= it
4
it
e > A) B) C) 0 D) TC
2
0

o
iii)
ls
. l
ir dx )
2
+
(dy )
2

.?
., dt dt ) dt )
Z' A) Polar form B) Parametric form C) Cartesian form D) None of these
t'
O log x
t It
0
t
x-
,
.. cot x
A) 1 B) 0 C) 2 D) ? 2
x(1+ acosx) ? bsinx
b. Find a & b, if It = 1 . (04 Marks)
' x-o
x
3
c. Find the pedal equation of the curve r
2
= a
2
cos20 (06 Marks)
d. Find the radius of curvature at any point t of the curve x = a(t +sin t) and y = a(1? cost) .
(06 Marks)
Max. Marks:100
PART ? A
iv) If y = 3' then y
n
=
1 of 4
=PDF Eraser Free
?4
094, creal
3 a. Choose the correct answer -
A) 0 B) R C) 9 D) 3
ii) Any motion in which the curl of the velocity vector is zero, then the vector v is said to
be,
= 4 verify JJ =1. (06 Marks)
- ?k ii i i
D) 1
(04 Marks)
10MAT11
(04 Marks)
au au
i) If u = (x - y)
2
+ (y - z)
2
+ (z - x)
2
then au ? + ay ? + ? is,
ax az
A) 1 B) 24 C) 2(x + y + z) D) 0
ii
2
ex cosy =
77
[1+ (x -1) -(y
71
4
) +
(x -
2
1)2
(x -1)(y - - -
1
-(y -
7
-
1
) i+ ...
4 2 4
A) (1,i) B) (0, 0) C) (1, 1) D) (4,1)
a
x
u
2 a2
u
a2
u
y

iii) At (a, b) ? = A , = B and = H
a
2
Na
and if AB - H
2
< 0 the
F
.Such a point is
called,
A) Maximum B) Minimum C) Saddle 17) Extremum
a(u, , a(x, y)
iv) If J = , J - , then DI is,
a(x, y) a(u, v)
B) 2 C)
b. then prove that x ?
au
+ y?
au
+ z
xy yz ZX
If that J u = ?, v = ?, w = ? then show
U; V
.
, W
' ' ?
z x y
koY>z
d. For the kinetic energy E = -mv
2
find' approximately the change in E as the mass m changes
(06 Marks)
4 a. Choose the correct answers for the following : (04 Marks)
i) The value of V x V9 is,
A) Constant B) Solenoidal C) Vector - ' ' " D) Irrotational
y;z) ,
iii) In orthogonal curvilinear co-ordinates the Jacobian J =
a(x,
is,
a(u, v, w)
c.
A) 0
If u=f(
1
-L,I,-)
y z x
1
ax

ay
k
2
from 49 to 49.5 and the velocity `?,' Changes from 1600 to 1590.
b.
A)
h1
B)
1
h
2
h
3
h,11
2
11
3

iv) A gradient of the scalar point function 9 , V9 is,
A) Scalar function B) Vector function C) 9
Find the value of the constant a such that the vector field,
D)
D) zero
C) h,h
2
h
3

F = (axy - z
3
)i + (a - 2)x
2
j + (1- a)xz
2
k is irrotational and hence find a scalar function
iP
such that F = V9. (04 Marks)
C. Prove that eurl(curl A) = V(V.A) - V
2
A . (06 Marks)
d.
Express V
2
iv in orthogonal curvilinear co-ordinates. (06 Marks)
2 of 4
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10MAT11
PDF Eraser Free
USN
(04 Marks)
D) None of these
First Semester B.E. Degree Examination, June / July 2014
Engineering Mathematics - I
.
-2
'
Time: 3 hrs.
a.
E
.
4 Note: 1. Answer any FIVE full questions, choosing at least two from each part.
IR
2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.
3. Answer to objective type questions on sheets other than OMR will not be valued.
oD
?
-c
1
z
i g 1 a. Choose;the correct answers for the following :
O00
i) If y? (4177)
n
e
4
" cos x +n tan
-1
-
1
then y ?
? ii
00
:a" ill
-
4
Va -zr
a
i.o 4)
A) e
4
x cosx B) e
2
" sin 3x C) ex cos x
4.) 0
?B 2 X
X
5
x
7

= ? E:
ii) sin x =x? +---- ........ is,
3
3! 5! 7!
O ci
,

O 0
= .0
A) Taylor's series .-B) Exponential series C) Meclaurin's series D) None of these
2
. ti. ,
. " iii) In the Rolle's theorem if ;F'(c) = 0 then the tangent at the point x = c is,

Q 0
Er
A) parallel to y-axis B) parallel to x-axis C) parallel to both axes D) None of these
7,1 O'
' ? I; 0
00 0
? es
5' 15
A) (log x)3" B) 3(logx)n
,
C) 3" log 3' D) 3' (log
e
3)"
at
If x = sin t, y = sinpt prove that, (1? x
2
)v ?(2n +1)xy
e+
, + (p
2
?n
2
)y
n
=O. (04 Marks)
=
74 b. ,.. n+2
'
c. State and prove Cauchy's mean value theorem in [0, 16]. (06 Marks)
o
73 d. Expand 41+ sin 2x by using Meclaurin's expansion.
04
(06 Marks)
Co
IlL?01/11P
C'S
2 e
0
a. Choose the correct answers for the following :
(04 Marks)
i) The value of It 0 + xfx is, al
5
x"...
, 4a.
1
:8
A), e B) 1 C) ? D) 09
,,...
e
o
-17,.. =
tu ?
ii) The angle between two curves r = ae
e
and re
p
= b is,
... 00
? 73
2
0. .
= it
4
it
e > A) B) C) 0 D) TC
2
0

o
iii)
ls
. l
ir dx )
2
+
(dy )
2

.?
., dt dt ) dt )
Z' A) Polar form B) Parametric form C) Cartesian form D) None of these
t'
O log x
t It
0
t
x-
,
.. cot x
A) 1 B) 0 C) 2 D) ? 2
x(1+ acosx) ? bsinx
b. Find a & b, if It = 1 . (04 Marks)
' x-o
x
3
c. Find the pedal equation of the curve r
2
= a
2
cos20 (06 Marks)
d. Find the radius of curvature at any point t of the curve x = a(t +sin t) and y = a(1? cost) .
(06 Marks)
Max. Marks:100
PART ? A
iv) If y = 3' then y
n
=
1 of 4
=PDF Eraser Free
?4
094, creal
3 a. Choose the correct answer -
A) 0 B) R C) 9 D) 3
ii) Any motion in which the curl of the velocity vector is zero, then the vector v is said to
be,
= 4 verify JJ =1. (06 Marks)
- ?k ii i i
D) 1
(04 Marks)
10MAT11
(04 Marks)
au au
i) If u = (x - y)
2
+ (y - z)
2
+ (z - x)
2
then au ? + ay ? + ? is,
ax az
A) 1 B) 24 C) 2(x + y + z) D) 0
ii
2
ex cosy =
77
[1+ (x -1) -(y
71
4
) +
(x -
2
1)2
(x -1)(y - - -
1
-(y -
7
-
1
) i+ ...
4 2 4
A) (1,i) B) (0, 0) C) (1, 1) D) (4,1)
a
x
u
2 a2
u
a2
u
y

iii) At (a, b) ? = A , = B and = H
a
2
Na
and if AB - H
2
< 0 the
F
.Such a point is
called,
A) Maximum B) Minimum C) Saddle 17) Extremum
a(u, , a(x, y)
iv) If J = , J - , then DI is,
a(x, y) a(u, v)
B) 2 C)
b. then prove that x ?
au
+ y?
au
+ z
xy yz ZX
If that J u = ?, v = ?, w = ? then show
U; V
.
, W
' ' ?
z x y
koY>z
d. For the kinetic energy E = -mv
2
find' approximately the change in E as the mass m changes
(06 Marks)
4 a. Choose the correct answers for the following : (04 Marks)
i) The value of V x V9 is,
A) Constant B) Solenoidal C) Vector - ' ' " D) Irrotational
y;z) ,
iii) In orthogonal curvilinear co-ordinates the Jacobian J =
a(x,
is,
a(u, v, w)
c.
A) 0
If u=f(
1
-L,I,-)
y z x
1
ax

ay
k
2
from 49 to 49.5 and the velocity `?,' Changes from 1600 to 1590.
b.
A)
h1
B)
1
h
2
h
3
h,11
2
11
3

iv) A gradient of the scalar point function 9 , V9 is,
A) Scalar function B) Vector function C) 9
Find the value of the constant a such that the vector field,
D)
D) zero
C) h,h
2
h
3

F = (axy - z
3
)i + (a - 2)x
2
j + (1- a)xz
2
k is irrotational and hence find a scalar function
iP
such that F = V9. (04 Marks)
C. Prove that eurl(curl A) = V(V.A) - V
2
A . (06 Marks)
d.
Express V
2
iv in orthogonal curvilinear co-ordinates. (06 Marks)
2 of 4
PDF Eraser Free
10MAT11
5 a. Choose the correct answers for the following : (04 Marks)
i) The value of icos
3
(4x)dx is,
0
A) B)
C)
D)
3 6 2
ii) If the equation of the curve remains unchange after changing 0 to ? 0 the curve
r = f(0) is symmetrical about,
A) A line perpendicular to initial line through pole
B) Radially symmetric about the point pole.
C) Symmetry does not exist
D) Initial line
volume of the curve r = a(1+ cos0) about the initial line is,
4na
3
2na
3
8na
3

B)
C)
3 3
iv) The assymptote for the curve x
3
+ y
3
= 3axy is equal to,
A) x+y+i=
.
0 B) x?y?a=0 C) No Assymptote
'
b. Evaluate s
log(1 + cos x)
dx

0
cos x
2a
c. Evaluate V2ax ? x
2
dx .
(06 Marks)
d. Find the area of surface of revolution abOut x-axis of the astroid x + y
3
= a
33
. (06 Marks)
6 a. Choose the correct answers for:the following : (04 Marks)
i) In the homogeneous differential equation,
elY
=
(xY)
the degree of the function,
dx f
2
(xy)
f
i
(xy) and f
2
(xy) are,
A) Different B) Relatively prime C) Same D) None of these
dy
ii) The integrating factor of the differential equation, ? + cot xy = cos x is,
dx
AXtilisx B) sin x C) ? sin x D) cot x
iii)
dy
Replacing ? by ? ?
dy
in the differential equation f x, y, dy = 0 we get the
dx - dx dx
differential equation of,
A) Polar trajectory B) Orthogonal trajectory
C) Parametric trajectory D) Parallel trajectory.
iv) Two families of curves are said to be orthogonal if every member of either family cuts
each member of the other family at,
D) 27c
A) Zero angle B) Right angle C) a L,, ?
6 3
b. Solve (1+ eVY)dx + eYY1? dy = 0 . ( (04 Marks)
Y
c. Solve
i ci
+ x sin 2y = x
3
cos
2
y. (06 Marks)
dx
d. Find the orthogonal trajectories of r
2
= a
2
cos
2
0 . (06 Marks)
3 of 4
Tca
3

D)
3
D)x+y?a=0
(rivsei
(04 Marks)
FirstRanker.com - FirstRanker's Choice
10MAT11
PDF Eraser Free
USN
(04 Marks)
D) None of these
First Semester B.E. Degree Examination, June / July 2014
Engineering Mathematics - I
.
-2
'
Time: 3 hrs.
a.
E
.
4 Note: 1. Answer any FIVE full questions, choosing at least two from each part.
IR
2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.
3. Answer to objective type questions on sheets other than OMR will not be valued.
oD
?
-c
1
z
i g 1 a. Choose;the correct answers for the following :
O00
i) If y? (4177)
n
e
4
" cos x +n tan
-1
-
1
then y ?
? ii
00
:a" ill
-
4
Va -zr
a
i.o 4)
A) e
4
x cosx B) e
2
" sin 3x C) ex cos x
4.) 0
?B 2 X
X
5
x
7

= ? E:
ii) sin x =x? +---- ........ is,
3
3! 5! 7!
O ci
,

O 0
= .0
A) Taylor's series .-B) Exponential series C) Meclaurin's series D) None of these
2
. ti. ,
. " iii) In the Rolle's theorem if ;F'(c) = 0 then the tangent at the point x = c is,

Q 0
Er
A) parallel to y-axis B) parallel to x-axis C) parallel to both axes D) None of these
7,1 O'
' ? I; 0
00 0
? es
5' 15
A) (log x)3" B) 3(logx)n
,
C) 3" log 3' D) 3' (log
e
3)"
at
If x = sin t, y = sinpt prove that, (1? x
2
)v ?(2n +1)xy
e+
, + (p
2
?n
2
)y
n
=O. (04 Marks)
=
74 b. ,.. n+2
'
c. State and prove Cauchy's mean value theorem in [0, 16]. (06 Marks)
o
73 d. Expand 41+ sin 2x by using Meclaurin's expansion.
04
(06 Marks)
Co
IlL?01/11P
C'S
2 e
0
a. Choose the correct answers for the following :
(04 Marks)
i) The value of It 0 + xfx is, al
5
x"...
, 4a.
1
:8
A), e B) 1 C) ? D) 09
,,...
e
o
-17,.. =
tu ?
ii) The angle between two curves r = ae
e
and re
p
= b is,
... 00
? 73
2
0. .
= it
4
it
e > A) B) C) 0 D) TC
2
0

o
iii)
ls
. l
ir dx )
2
+
(dy )
2

.?
., dt dt ) dt )
Z' A) Polar form B) Parametric form C) Cartesian form D) None of these
t'
O log x
t It
0
t
x-
,
.. cot x
A) 1 B) 0 C) 2 D) ? 2
x(1+ acosx) ? bsinx
b. Find a & b, if It = 1 . (04 Marks)
' x-o
x
3
c. Find the pedal equation of the curve r
2
= a
2
cos20 (06 Marks)
d. Find the radius of curvature at any point t of the curve x = a(t +sin t) and y = a(1? cost) .
(06 Marks)
Max. Marks:100
PART ? A
iv) If y = 3' then y
n
=
1 of 4
=PDF Eraser Free
?4
094, creal
3 a. Choose the correct answer -
A) 0 B) R C) 9 D) 3
ii) Any motion in which the curl of the velocity vector is zero, then the vector v is said to
be,
= 4 verify JJ =1. (06 Marks)
- ?k ii i i
D) 1
(04 Marks)
10MAT11
(04 Marks)
au au
i) If u = (x - y)
2
+ (y - z)
2
+ (z - x)
2
then au ? + ay ? + ? is,
ax az
A) 1 B) 24 C) 2(x + y + z) D) 0
ii
2
ex cosy =
77
[1+ (x -1) -(y
71
4
) +
(x -
2
1)2
(x -1)(y - - -
1
-(y -
7
-
1
) i+ ...
4 2 4
A) (1,i) B) (0, 0) C) (1, 1) D) (4,1)
a
x
u
2 a2
u
a2
u
y

iii) At (a, b) ? = A , = B and = H
a
2
Na
and if AB - H
2
< 0 the
F
.Such a point is
called,
A) Maximum B) Minimum C) Saddle 17) Extremum
a(u, , a(x, y)
iv) If J = , J - , then DI is,
a(x, y) a(u, v)
B) 2 C)
b. then prove that x ?
au
+ y?
au
+ z
xy yz ZX
If that J u = ?, v = ?, w = ? then show
U; V
.
, W
' ' ?
z x y
koY>z
d. For the kinetic energy E = -mv
2
find' approximately the change in E as the mass m changes
(06 Marks)
4 a. Choose the correct answers for the following : (04 Marks)
i) The value of V x V9 is,
A) Constant B) Solenoidal C) Vector - ' ' " D) Irrotational
y;z) ,
iii) In orthogonal curvilinear co-ordinates the Jacobian J =
a(x,
is,
a(u, v, w)
c.
A) 0
If u=f(
1
-L,I,-)
y z x
1
ax

ay
k
2
from 49 to 49.5 and the velocity `?,' Changes from 1600 to 1590.
b.
A)
h1
B)
1
h
2
h
3
h,11
2
11
3

iv) A gradient of the scalar point function 9 , V9 is,
A) Scalar function B) Vector function C) 9
Find the value of the constant a such that the vector field,
D)
D) zero
C) h,h
2
h
3

F = (axy - z
3
)i + (a - 2)x
2
j + (1- a)xz
2
k is irrotational and hence find a scalar function
iP
such that F = V9. (04 Marks)
C. Prove that eurl(curl A) = V(V.A) - V
2
A . (06 Marks)
d.
Express V
2
iv in orthogonal curvilinear co-ordinates. (06 Marks)
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10MAT11
5 a. Choose the correct answers for the following : (04 Marks)
i) The value of icos
3
(4x)dx is,
0
A) B)
C)
D)
3 6 2
ii) If the equation of the curve remains unchange after changing 0 to ? 0 the curve
r = f(0) is symmetrical about,
A) A line perpendicular to initial line through pole
B) Radially symmetric about the point pole.
C) Symmetry does not exist
D) Initial line
volume of the curve r = a(1+ cos0) about the initial line is,
4na
3
2na
3
8na
3

B)
C)
3 3
iv) The assymptote for the curve x
3
+ y
3
= 3axy is equal to,
A) x+y+i=
.
0 B) x?y?a=0 C) No Assymptote
'
b. Evaluate s
log(1 + cos x)
dx

0
cos x
2a
c. Evaluate V2ax ? x
2
dx .
(06 Marks)
d. Find the area of surface of revolution abOut x-axis of the astroid x + y
3
= a
33
. (06 Marks)
6 a. Choose the correct answers for:the following : (04 Marks)
i) In the homogeneous differential equation,
elY
=
(xY)
the degree of the function,
dx f
2
(xy)
f
i
(xy) and f
2
(xy) are,
A) Different B) Relatively prime C) Same D) None of these
dy
ii) The integrating factor of the differential equation, ? + cot xy = cos x is,
dx
AXtilisx B) sin x C) ? sin x D) cot x
iii)
dy
Replacing ? by ? ?
dy
in the differential equation f x, y, dy = 0 we get the
dx - dx dx
differential equation of,
A) Polar trajectory B) Orthogonal trajectory
C) Parametric trajectory D) Parallel trajectory.
iv) Two families of curves are said to be orthogonal if every member of either family cuts
each member of the other family at,
D) 27c
A) Zero angle B) Right angle C) a L,, ?
6 3
b. Solve (1+ eVY)dx + eYY1? dy = 0 . ( (04 Marks)
Y
c. Solve
i ci
+ x sin 2y = x
3
cos
2
y. (06 Marks)
dx
d. Find the orthogonal trajectories of r
2
= a
2
cos
2
0 . (06 Marks)
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Tca
3

D)
3
D)x+y?a=0
(rivsei
(04 Marks)
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10MATli
4
.-h, t
7 a. Choose the correct answer (04 Marks)
A = 0
[7
0
0
7
0
0
0
7
is called,
A) Scalar matrix B) Diagonal matrix C) Identity matrix D) None of these
ii) If r = n and x = y = z = 0. The equations have only solution.
A) Non trivial B) Trivial C) Unique D) Infinite
iii) In Gauss Jordan method, the coefficient matrix can be reduced to,
A) Echelon form B) Unit matrix C) Triangular form D) Diagonal matrix
iv) The inverse square matrix A is given by,
adjA
:adjA
1
b. Find the Rank of the matrix, 2
1
c. Investigate the values of 2 and .i such that the system of equations, x + y + z = 6 ,
x + 2y + 3z =10 , x +2y + Xz = g may be i) Unique solution ii) Infinite solution iii) No
solution. (06 Marks)
d.
Using Gauss elimination method solve, ?
2x
1
? X
2
+3)(
3
=1, ? 3X
1
-= 4X
2
5X
3
= 0, X
2
?
.0
5X
3
(05 Marks)
8 a. Choose the correct answers for the following : (04 Marks)
i) A square matrix A of order 3 has 3 linearly independent eigen vectors then a matrix P
can be found such that P
-1
AP is a,
A) Diagonal matrix B) Unit matrix
C) Singular matrix D) Symmetric matrix
ii) The eigen values of matrix [
2
,-- are,
42 2
A) 2 ? B) 2 ? C) D) None of these
iii) Solving the equations x + 2y + 3z = 0 , 3x + 4y + 4z = 0, 7x. +10y +12z = 0 . x, y and
z values are,
A) x=y=z=0 B) x=y=z= 1 C) x y z D) None of these
iv) The index and significance of the quadratic form, x
2
, + 2x
2
2
? 3x; are respectively
and
A) Index = 1, Signature = 1 B) Index = 1, Signature =
C) Index = 2, Signature = 1 D) None of these.
Find all the eigen values and the corresponding eigen vectors of the matrix,
b.
1)
A) Al l B)
I
A
I
C) adjA
2 3 2
3 5 1 . (05 Marks)
3 4 5
8 ?6 2
A = ?6 7 ?4
(04 Marks)
2 ?4 3
L
c.
Reduce the matrix A =
11 ?4 ?7
7 ? 2 ? 5 into a diagonal matrix.
10 ? 4 ? 6
(06 Marks)

d.
Reduce the quadratic form 3x2
5y2 3z2
2yz + 2zx ? 2xy to the canonical form.
(06 Marks)
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