GTU BE 1st and 2nd Semester 110009 MATHEMATICS-II Summer 2018 Question Paper

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER ? 1
st
/ 2
nd
(OLD) EXAMINATION ? SUMMER 2018
Subject Code: 110009 Date: 17-05-2018
Subject Name: MATHEMATICS-II
Time: 02:30 pm to 05:30 pm Total Marks: 70
Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) (i) For which values of k , u and v orthogonal?

? ? ? ? 2,1, 3 , 1, 7, u v k ??
03

(ii) Verify Cauchy-Schwarz inequality for the vectors

? ? ? ? 0, 2, 2,1 , 1, 1,1,1 uv ? ? ? ? ?
04
(b) (i) Find the rank for the matrix
1 6 8
2 5 3
7 9 4
??
??
??
??
??
.
03
(ii) Solve the following linear system by using Gauss Jordan method.
3 2 8 9
2 2 3
2 3 8
x y z
x y z
x y z
? ? ?
? ? ? ?
? ? ?
04

Q.2 (a) (i) Solve the following linear system by using Gauss Elimination method
6
2 3 1 0
2 4 1
x y z
x y z
x y z
? ? ?
? ? ?
? ? ?
03
(ii) Find the inverse of the matrix
1 0 1
1 1 1
0 1 0
??
??
?
??
??
??
.
04
(b) (i) Prove that the matrix
1 2 5 3
2 7 5
5 3 5 2
ii
A i i
ii
? ? ? ??
??
??
??
?? ??
??
is a hermitian matrix.
03
(ii) Find the eigenvalues and eigenvectors of the matrix
1 0 1
1 2 1
2 2 3
A
? ??
??
?
??
??
??
04

Q.3 (a)Show that the set of all 22 ? matrices of the form
1
1
a
b
??
??
??
with addition defined
by
1 1 1
1 1 1
a c a c
b d b d
? ? ? ? ? ? ?
??
? ? ? ? ? ?
?
? ? ? ? ? ?
and scalar multiplication defined by
11
11
a ka
k
b kb
? ? ? ?
?
? ? ? ?
? ? ? ?
is a vector space.
07 1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER ? 1
st
/ 2
nd
(OLD) EXAMINATION ? SUMMER 2018
Subject Code: 110009 Date: 17-05-2018
Subject Name: MATHEMATICS-II
Time: 02:30 pm to 05:30 pm Total Marks: 70
Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) (i) For which values of k , u and v orthogonal?

? ? ? ? 2,1, 3 , 1, 7, u v k ??
03

(ii) Verify Cauchy-Schwarz inequality for the vectors

? ? ? ? 0, 2, 2,1 , 1, 1,1,1 uv ? ? ? ? ?
04
(b) (i) Find the rank for the matrix
1 6 8
2 5 3
7 9 4
??
??
??
??
??
.
03
(ii) Solve the following linear system by using Gauss Jordan method.
3 2 8 9
2 2 3
2 3 8
x y z
x y z
x y z
? ? ?
? ? ? ?
? ? ?
04

Q.2 (a) (i) Solve the following linear system by using Gauss Elimination method
6
2 3 1 0
2 4 1
x y z
x y z
x y z
? ? ?
? ? ?
? ? ?
03
(ii) Find the inverse of the matrix
1 0 1
1 1 1
0 1 0
??
??
?
??
??
??
.
04
(b) (i) Prove that the matrix
1 2 5 3
2 7 5
5 3 5 2
ii
A i i
ii
? ? ? ??
??
??
??
?? ??
??
is a hermitian matrix.
03
(ii) Find the eigenvalues and eigenvectors of the matrix
1 0 1
1 2 1
2 2 3
A
? ??
??
?
??
??
??
04

Q.3 (a)Show that the set of all 22 ? matrices of the form
1
1
a
b
??
??
??
with addition defined
by
1 1 1
1 1 1
a c a c
b d b d
? ? ? ? ? ? ?
??
? ? ? ? ? ?
?
? ? ? ? ? ?
and scalar multiplication defined by
11
11
a ka
k
b kb
? ? ? ?
?
? ? ? ?
? ? ? ?
is a vector space.
07
2
(b) (i) Determine whether the vectors
? ? ? ? 1, 2, 3 , 3, 2,1 ? and
? ? 1, 6, 5 ?? are linearly
dependent or linearly independent.
03
(ii) Determine whether the vectors
? ? ? ? ? ? 1, 1,1 , 0,1, 2 , 3, 0, 1 ?? forms basis for
3
R
04

Q.4 (a)Extend the subset
? ? ? ? ? ?
1, 2, 5, 3 , 2, 3,1, 4 A ? ? ? ? of
4
R to the basis for vector
space
4
R .
07
(b) (i) Find two vector in
2
R with Euclidean norm whose inner product with
? ? 3,1 ? is
zero.
03
(ii) Obtain the matrix of a linear transformation
33
: T R R ? defined by
? ? ( , , ) 2 , , 3 T x y z x x y z x z ? ? ? ? with respect to the basis
? ?
1(1, 0, 0), (1,1, 0), (1,1,1) B ? and
? ?
2(1, 0, 0), (0,1, 0), (0, 0,1) B ? .
04

Q.5 (a)For the basis
? ? ,, S u v w ? of
3
R , where
? ? ? ? 1,1,1 , 1,1, 0 uv ?? and
? ? 1, 0, 0 w ? , let
33
: T R R ? be a linear transformation such that ( ) (2, 1, 4 ), ( ) (3, 0,1), ( ) ( 1, 5,1) T u T v T w ? ? ? ? ? . Find a formula for ( , , ) T x y z and
use it to find (2, 4, 1) T ? .
07
(b) State Rank-Nullity theorem.
Let
43
: T R R ? be a linear transformation defined by (1, 0, 0, 0 ) (1,1,1), (0,1, 0, 0 ) (1, 1,1), (0, 0,1, 0 ) (1, 0, 0 ), (0, 0, 0,1) (1, 0,1) T T T T ? ? ? ? ? .
Then verify the rank-nullity theorem.
07

Q.6 (a) Find the least square solution of the linear system AX b ? given by
12
12
12
7
0
27
xx
xx
xx
??
? ? ?
? ? ?
07
(b)Let
3
R have the Euclidean inner product. Use the Gram Schmidt process to
transform the basis ? ?
1 2 3
,, u u u into an orthonormal basis, where
1 2 3(1, 0 , 0 ), (3, 7 , 2 ), (0 , 4 ,1) u u u ? ? ? ?
07

Q.7 (a)Find a matrix that diagonalizes and determine
1
P AP
?
, where
2 0 2
0 3 0
0 0 3
A
? ??
??
?
??
??
??
.
07
(b) (i) Find the algebraic and geometric multiplicity of
0 1 0
0 0 1
1 3 3
A
??
??
?
??
?? ?
??
.
03
(ii)Verify Caley-Hamilton theorem for the matrix,
2 1 1
1 2 1
1 1 2
A
? ??
??
? ? ?
??
?? ?
??
.
04
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