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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