Download GTU BE/B.Tech 2019 Summer 1st Sem And 2nd Sem Old 110009 Maths Ii Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 1st Sem And 2nd Sem Old 110009 Maths Ii Previous Question Paper

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (OLD) EXAMINATION ? SUMMER-2019
Subject Code: 110009 Date: 01/06/2019

Subject Name: Maths - II

Time: 10:30 AM TO 01:30 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) Find the value of ? so that the equations
0 3 4
0 3
0 2 2
? ? ?
? ? ?
? ? ?
z y x
z y x
z y x
?
; have a non trivial solution.
4
(ii) Verify Cauchy-Schwarz inequality for the vectors ) 0 , 1 , 3 ( ? ? u , ). 3 , 1 , 2 ( ? ? v
3
(b) Solve the following equations by Gauss elimination and back substitution,
0 5 6 3
1 3 4 2
9 2
? ? ?
? ? ?
? ? ?
z y x
z y x
z y x

7
Q-2 (a)
(i) Obtain the reduced row echelon form of the matrix
?
?
?
?
?
?
?
?
?
?
?
?
8 4 7 3
4 3 4 2
3 1 2 1
2 2 3 1
.
4

(ii) Find the rank of the matrix, if
?
?
?
?
?
?
?
?
?
?
?
? ?
2 9 3 2
2 5 1 3
0 4 2 1
A .
3
(b)
Use row operation to find
1 ?
A , if
?
?
?
?
?
?
?
?
?
?
?
8 0 1
3 5 2
3 2 1
A .
7
Q-3 (a)
(i) Find the eigen values and eigen vectors of the matrix,
?
?
?
?
?
?
?
?
?
?
?
5 0 0
6 2 0
4 1 3
A .
4

(ii) Using Cayley-Hamilton theorem, find ,
2
A if
?
?
?
?
?
?
?
3 2
4 1
A .
3

(b)
Find a matrix P that diagonalizes
?
?
?
?
?
?
?
?
2 1
0 1
A and hence find .
10
A
7

Q-4 (a)
(i) Reduce ? ? ) 0 , 2 , 0 ( ), 3 , 4 , 0 ( ), 1 , 1 , 0 ( ), 0 , 0 , 1 ( ? ? ? S to obtain a basis for .
3
R 4

(ii) Determine whether or not the vectors ? ? ) 1 , 2 , 2 ( ), 2 , 1 , 2 ( ), 2 , 2 , 1 ( in
3
R are linearly
independent.

3
(b) Let ? ? ? ? , 0 , , | , ? ? ? y R y x y x V let . , ) , ( ), , ( R V d c b a ? ? ?
Define ) , ( ) , ( ) , ( d b c a d c b a ? ? ? ? and ) , ( ) , (
?
? ? b a b a ? ? . Prove that V is a vector space.


7
Q-5 (a)
(i) Show that the transformation , :
2 2
R R T ? where ) , 2 ( ) , ( y x y x y x T ? ? ? is a linear
transformation.
4

(ii) Express the quadratic form , 6 3 2 ) , (
2 2
xy y x y x Q ? ? ? in matrix notation. 3
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Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (OLD) EXAMINATION ? SUMMER-2019
Subject Code: 110009 Date: 01/06/2019

Subject Name: Maths - II

Time: 10:30 AM TO 01:30 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) Find the value of ? so that the equations
0 3 4
0 3
0 2 2
? ? ?
? ? ?
? ? ?
z y x
z y x
z y x
?
; have a non trivial solution.
4
(ii) Verify Cauchy-Schwarz inequality for the vectors ) 0 , 1 , 3 ( ? ? u , ). 3 , 1 , 2 ( ? ? v
3
(b) Solve the following equations by Gauss elimination and back substitution,
0 5 6 3
1 3 4 2
9 2
? ? ?
? ? ?
? ? ?
z y x
z y x
z y x

7
Q-2 (a)
(i) Obtain the reduced row echelon form of the matrix
?
?
?
?
?
?
?
?
?
?
?
?
8 4 7 3
4 3 4 2
3 1 2 1
2 2 3 1
.
4

(ii) Find the rank of the matrix, if
?
?
?
?
?
?
?
?
?
?
?
? ?
2 9 3 2
2 5 1 3
0 4 2 1
A .
3
(b)
Use row operation to find
1 ?
A , if
?
?
?
?
?
?
?
?
?
?
?
8 0 1
3 5 2
3 2 1
A .
7
Q-3 (a)
(i) Find the eigen values and eigen vectors of the matrix,
?
?
?
?
?
?
?
?
?
?
?
5 0 0
6 2 0
4 1 3
A .
4

(ii) Using Cayley-Hamilton theorem, find ,
2
A if
?
?
?
?
?
?
?
3 2
4 1
A .
3

(b)
Find a matrix P that diagonalizes
?
?
?
?
?
?
?
?
2 1
0 1
A and hence find .
10
A
7

Q-4 (a)
(i) Reduce ? ? ) 0 , 2 , 0 ( ), 3 , 4 , 0 ( ), 1 , 1 , 0 ( ), 0 , 0 , 1 ( ? ? ? S to obtain a basis for .
3
R 4

(ii) Determine whether or not the vectors ? ? ) 1 , 2 , 2 ( ), 2 , 1 , 2 ( ), 2 , 2 , 1 ( in
3
R are linearly
independent.

3
(b) Let ? ? ? ? , 0 , , | , ? ? ? y R y x y x V let . , ) , ( ), , ( R V d c b a ? ? ?
Define ) , ( ) , ( ) , ( d b c a d c b a ? ? ? ? and ) , ( ) , (
?
? ? b a b a ? ? . Prove that V is a vector space.


7
Q-5 (a)
(i) Show that the transformation , :
2 2
R R T ? where ) , 2 ( ) , ( y x y x y x T ? ? ? is a linear
transformation.
4

(ii) Express the quadratic form , 6 3 2 ) , (
2 2
xy y x y x Q ? ? ? in matrix notation. 3

(b)
Find the rank and nullity of the matrix
?
?
?
?
?
?
?
?
?
?
?
?
?
0 0 0
2 0 4
1 0 2
A
7
Q-6 (a)
(i) Find a basis for the orthogonal complement of the subset of
3
R spanned by the vectors
). 2 , 6 , 7 ( ), 4 , 4 , 5 ( ), 3 , 1 , 1 (
3 2 1
? ? ? ? ? ? ? v v v
4

(ii) Determine whether the linear transformation , :
3 2
R R T ? where
) , , ( ) , ( y x y x y x T ? ? is one-one.
3
(b)
Let
3
R have the Euclidean inner product. Use the Gram-Schmidt process to transform the
basis ? ? ) 1 , 2 , 1 ( ), 0 , 1 , 1 ( ), 1 , 1 , 1 ( ? ? S into an Orthonormal basis.

7
Q-7 (a)
(i) Prove that
?
?
?
?
?
?
?
?
?
?
?
? ?
? ?
?
3 7 9 2
7 9 2 4 3
2 4 3 1
i i
i i
i i
A is a Hermitian matrix.
4

(ii) Let
2
R have the Euclidean inner product. Find the cosine of the angle ? between the
vectors ) 2 , 1 , 3 , 4 ( ? ? u and ). 3 , 2 , 1 , 2 ( ? ? v
3
(b) Find the least square solution of the linear system b AX ? and find the orthogonal
projection of b onto the column space of A , where
?
?
?
?
?
?
?
?
?
? ?
?
1 3
1 1
2 2
A ,
?
?
?
?
?
?
?
?
?
?
? ?
1
1
2
b
7


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