Download GTU BE/B.Tech 2019 Winter 1st And 2nd Sem New And Spfu 2110015 Vector Calculus And Linear Algebra Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 1st And 2nd Sem New And Spfu 2110015 Vector Calculus And Linear Algebra Previous Question Paper

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 2110015 Date: 01/01/2020

Subject Name: Vector Calculus And Linear Algebra
Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:
1. Question No. 1 is compulsory. Attempt any four out of remaining Six questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q-1 (a) Objective Questions Marks
07
1.
If ? = [
1 ?5 7
0 9 7
11 9 8
] then trace of the matrix ? is

(a) 12 (b) 18 (c) 72 (d) 16
2. If ? ? = 0 then ? is said to be


(a) Rotational (b) Solenoidal (c) Compressible (d) None of these
3. If ? is 3 ? 3 invertible matrix then nullity of ? is
(a) 1 (b) 2 (c) 0 (d) 3

4.
If matrix ? = [
3 ?1 1
?1 5 ?1
1 ?1 3
] is having Eigen values 2,3,6 then Eigen
values of ? ?1
are

(a) 2,3,6
(b)
1
2
,
1
3
,
1
6
(c) 1,
2
3
,
1
3

(d) None of
these

5. Which set from ? 1
= {(? , ? , ? ) ? ? 3
/? > 0} and
? 2
= {(? , ? , ? ) ? ? 3
/ ? = ? = 0} is subspace of ? 3
.

(a) ? 1
(b) ? 2
(c) ? 1
and ? 2
(d) None.
6 If Eigen values of 3 ? 3 matrix A are 5,5,5 then Algebraic multiplicity of
matrix A is

(a) 3 (b) 1 (c) 5 (d) 0.
7 For what values of c, the vector (2, ?1, ? ) has norm 3?


(a) -3 (b) 3 (c) 0 (d) 2
(b) Objective Questions 07
1.
If ? (? , ? , ? ) = ? ? ? + ? ? ? + ? ? ?
then ? ? ?
is

(a) 1 (b) 3 (c) 2 (d) 0
2. For what value of ? the vectors ? 1
= (?1,2,4) , ? 2
= (?3,6, ? ) are Linearly
Dependent?

(a) 12 (b) 7 (c) 4 (d) 1
3.
Which one is the characteristic equation of = [
1 4
2 3
] ?

(a) ? 2
? 5? + 4 = 0 (b) ? 2
? 4? ? 5 = 0
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Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 2110015 Date: 01/01/2020

Subject Name: Vector Calculus And Linear Algebra
Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:
1. Question No. 1 is compulsory. Attempt any four out of remaining Six questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q-1 (a) Objective Questions Marks
07
1.
If ? = [
1 ?5 7
0 9 7
11 9 8
] then trace of the matrix ? is

(a) 12 (b) 18 (c) 72 (d) 16
2. If ? ? = 0 then ? is said to be


(a) Rotational (b) Solenoidal (c) Compressible (d) None of these
3. If ? is 3 ? 3 invertible matrix then nullity of ? is
(a) 1 (b) 2 (c) 0 (d) 3

4.
If matrix ? = [
3 ?1 1
?1 5 ?1
1 ?1 3
] is having Eigen values 2,3,6 then Eigen
values of ? ?1
are

(a) 2,3,6
(b)
1
2
,
1
3
,
1
6
(c) 1,
2
3
,
1
3

(d) None of
these

5. Which set from ? 1
= {(? , ? , ? ) ? ? 3
/? > 0} and
? 2
= {(? , ? , ? ) ? ? 3
/ ? = ? = 0} is subspace of ? 3
.

(a) ? 1
(b) ? 2
(c) ? 1
and ? 2
(d) None.
6 If Eigen values of 3 ? 3 matrix A are 5,5,5 then Algebraic multiplicity of
matrix A is

(a) 3 (b) 1 (c) 5 (d) 0.
7 For what values of c, the vector (2, ?1, ? ) has norm 3?


(a) -3 (b) 3 (c) 0 (d) 2
(b) Objective Questions 07
1.
If ? (? , ? , ? ) = ? ? ? + ? ? ? + ? ? ?
then ? ? ?
is

(a) 1 (b) 3 (c) 2 (d) 0
2. For what value of ? the vectors ? 1
= (?1,2,4) , ? 2
= (?3,6, ? ) are Linearly
Dependent?

(a) 12 (b) 7 (c) 4 (d) 1
3.
Which one is the characteristic equation of = [
1 4
2 3
] ?

(a) ? 2
? 5? + 4 = 0 (b) ? 2
? 4? ? 5 = 0
(c) ? 2
+ 4? + 5 = 0 (d) ? 2
+ 5? + 4 = 0
4. Which matrix represents one to one transformation
(a) [
1 1
0 0
] (b) [
2 7
4 14
] (c) [
2 7
1 14
] (d) [
2 1
6 3
]

5.
Matrix ? = [
? 2 + 3? 2 ? 3? 0
] is

(a) Symmetric (b) Skew-symmetric (c) Hermitian (d) None of these
6. If ? and ? are vectors in an Inner product space then
(a) |? , ? ?| = ? ? ? (b) |? , ? ?| ? ? ? ?
(c) |? , ? ?| ? ? ? ? (d) None of these.
7. Each vector in ? 2
can be rotated in counter clockwise direction with 90
?
is
followed by the matrix,

(a) [
1 0
0 ?1
] (b) [
0 1
?1 0
] (c) [
0 ?1
1 0
] (d) [
1 1
0 0
]



Q-2 (a)
Find the Rank of a Matrix ? = [
1 ?1 2 ?1
2 1 ?2 ?2
?1 2 ?4 1
3 0 0 ?3
]
03
(b) Determine whether the given vectors ? 1
= (2, ?1,3); ? 2
= (4,1,2); ? 3
=
(8, ?1,8) Span ? 3
.
04
(c) For which values of ?a? will the following system have no solutions?
Exactly one solution? Infinitely many solutions?
? + 2? ? 3? = 4
3? ? ? + 5? = 2
4? + ? + (? 2
? 14)? = ? + 2
07

Q-3 (a) Define Singular Matrix. Find the inverse of the matrix A using Gauss Jordan
Method if it is invertible ? = [
1 0 1
?1 1 ?1
0 1 0
]
03
(b)
Express the matrix ? = [
1 5 7
?1 ?2 ?4
8 2 13
] as the sum of a symmetric and a
skew symmetric matrix
04
(c) Fine a basis for the nullspace, row space and column space of the matrix
? = [
?1 2 ?1 5 6
4 ?4 ?4 ?12 ?8
2 0 ?6 ?2 4
?3 1 7 ?2 12
]
Also determine rank and nullity of the matrix.
07

Q-4 (a) Let ? 1
: ? 2
? ? 2
, ? 2
: ? 2
? ? 3
be transformation given by
? 1
(? , ? ) = (? + ? , ? ) and ? 2
( ? , ? ) = (2? , ? , ? + ? ).
Show that ? 1
is linear transformation and also find formula for ? 2
? ? 1
.
03
(b) Let ? : ? 2
? ? 2
be the linear operator defined by 04
FirstRanker.com - FirstRanker's Choice
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (NEW) EXAMINATION ? WINTER 2019
Subject Code: 2110015 Date: 01/01/2020

Subject Name: Vector Calculus And Linear Algebra
Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:
1. Question No. 1 is compulsory. Attempt any four out of remaining Six questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q-1 (a) Objective Questions Marks
07
1.
If ? = [
1 ?5 7
0 9 7
11 9 8
] then trace of the matrix ? is

(a) 12 (b) 18 (c) 72 (d) 16
2. If ? ? = 0 then ? is said to be


(a) Rotational (b) Solenoidal (c) Compressible (d) None of these
3. If ? is 3 ? 3 invertible matrix then nullity of ? is
(a) 1 (b) 2 (c) 0 (d) 3

4.
If matrix ? = [
3 ?1 1
?1 5 ?1
1 ?1 3
] is having Eigen values 2,3,6 then Eigen
values of ? ?1
are

(a) 2,3,6
(b)
1
2
,
1
3
,
1
6
(c) 1,
2
3
,
1
3

(d) None of
these

5. Which set from ? 1
= {(? , ? , ? ) ? ? 3
/? > 0} and
? 2
= {(? , ? , ? ) ? ? 3
/ ? = ? = 0} is subspace of ? 3
.

(a) ? 1
(b) ? 2
(c) ? 1
and ? 2
(d) None.
6 If Eigen values of 3 ? 3 matrix A are 5,5,5 then Algebraic multiplicity of
matrix A is

(a) 3 (b) 1 (c) 5 (d) 0.
7 For what values of c, the vector (2, ?1, ? ) has norm 3?


(a) -3 (b) 3 (c) 0 (d) 2
(b) Objective Questions 07
1.
If ? (? , ? , ? ) = ? ? ? + ? ? ? + ? ? ?
then ? ? ?
is

(a) 1 (b) 3 (c) 2 (d) 0
2. For what value of ? the vectors ? 1
= (?1,2,4) , ? 2
= (?3,6, ? ) are Linearly
Dependent?

(a) 12 (b) 7 (c) 4 (d) 1
3.
Which one is the characteristic equation of = [
1 4
2 3
] ?

(a) ? 2
? 5? + 4 = 0 (b) ? 2
? 4? ? 5 = 0
(c) ? 2
+ 4? + 5 = 0 (d) ? 2
+ 5? + 4 = 0
4. Which matrix represents one to one transformation
(a) [
1 1
0 0
] (b) [
2 7
4 14
] (c) [
2 7
1 14
] (d) [
2 1
6 3
]

5.
Matrix ? = [
? 2 + 3? 2 ? 3? 0
] is

(a) Symmetric (b) Skew-symmetric (c) Hermitian (d) None of these
6. If ? and ? are vectors in an Inner product space then
(a) |? , ? ?| = ? ? ? (b) |? , ? ?| ? ? ? ?
(c) |? , ? ?| ? ? ? ? (d) None of these.
7. Each vector in ? 2
can be rotated in counter clockwise direction with 90
?
is
followed by the matrix,

(a) [
1 0
0 ?1
] (b) [
0 1
?1 0
] (c) [
0 ?1
1 0
] (d) [
1 1
0 0
]



Q-2 (a)
Find the Rank of a Matrix ? = [
1 ?1 2 ?1
2 1 ?2 ?2
?1 2 ?4 1
3 0 0 ?3
]
03
(b) Determine whether the given vectors ? 1
= (2, ?1,3); ? 2
= (4,1,2); ? 3
=
(8, ?1,8) Span ? 3
.
04
(c) For which values of ?a? will the following system have no solutions?
Exactly one solution? Infinitely many solutions?
? + 2? ? 3? = 4
3? ? ? + 5? = 2
4? + ? + (? 2
? 14)? = ? + 2
07

Q-3 (a) Define Singular Matrix. Find the inverse of the matrix A using Gauss Jordan
Method if it is invertible ? = [
1 0 1
?1 1 ?1
0 1 0
]
03
(b)
Express the matrix ? = [
1 5 7
?1 ?2 ?4
8 2 13
] as the sum of a symmetric and a
skew symmetric matrix
04
(c) Fine a basis for the nullspace, row space and column space of the matrix
? = [
?1 2 ?1 5 6
4 ?4 ?4 ?12 ?8
2 0 ?6 ?2 4
?3 1 7 ?2 12
]
Also determine rank and nullity of the matrix.
07

Q-4 (a) Let ? 1
: ? 2
? ? 2
, ? 2
: ? 2
? ? 3
be transformation given by
? 1
(? , ? ) = (? + ? , ? ) and ? 2
( ? , ? ) = (2? , ? , ? + ? ).
Show that ? 1
is linear transformation and also find formula for ? 2
? ? 1
.
03
(b) Let ? : ? 2
? ? 2
be the linear operator defined by 04

**********
? (? , ? ) = (2? ? ? , ?8? + 4? ) . Find a basis for ker(? ) and basis for ? (? ).
(c)
Find a matrix P that diagonalize ? , where ? = [
1 ?6 ?4
0 4 2
0 ?6 ?3
]
And determine ? ?1
? .
07

Q-5 (a) Find constants ? , ? , ? so that
? = (? + 2? + ? )? ? + (? ? 3? ? ? )? ? + (4? + ? + 2? )? ?
is irrotational.

03
(b)
Let the vector space ? 2
have the inner product ? , ? ? = ? ? (? )? (? ) ? 1
?1

(i) Find ? ? for ? = ? 2
.
(ii) Find ? (? , ? ) of ? = 1 and ? = ? .
04
(c) Using Gram-Schmidt process orthonormalize the set of linearly independent
vectors ? 1
= (1, 0, 1, 1), ? 2
= (?1, 0, ?1, 1) and ? 3
= (0, ?1, 1, 1)
of ? 4
with standard inner product.
07

Q-6 (a) The temperature at any point in space is given by ? = ? + ? + ? .
Determine the derivative of ? in the direction of the vector 3? ? ? 4? ?
at the
point (1,1,1).
03
(b) Find the orthogonal projection of ? = (2,1,3) on the subspace of ? 3

spanned by the vectors ? 1
= (1, 1, 0), ? 2
= (1, 2, 1).
04

(c) Verify Green?s theorem for the field ? = (? ? ? )? ? + ? ? ? and the region ?
bounded by the unit circle ? : ? (? ) = (cos ? )? ? + (? )? ? ; 0 ? ? ? 2?
07


Q-7 (a) Find the co ordinate vector of ? = 2 ? ? + ? 2
relative to the basis
? = {? 1,
? 2
, ? 3
} where ? 1
= 1 + ? , ? 2
= 1 + ? 2
, ? 3
= ? + ? 2

03
(b) Let ? : ? 3
? ? 3
be multiplication by ? determine whether ? has inverse. If
so find ? ?1
(? 1
, ? 2
, ? 3
), where ? = [
1 4 ?1
1 2 1
?1 1 0
]
04
(c) Determine whether ? +
of all positive real numbers with operators
? + ? = ? and ? = ? ? as a Vector Space.
07
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