GTU BE 1st and 2nd Semester 2110015 Vector Calculus and Linear Algebra Summer 2018 Question Paper

Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE – SEMESTER? 1
st
/ 2
nd
EXAMINATION (NEW SYLLABUS) ? SUMMER 2018
Subject Code: 2110015 Date: 17-05-2018
Subject Name: Vector Calculus and Linear Algebra
Time: 02:30 pm to 05:30 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 Objective Question (MCQ)Mark
(a) Choose the appropriate answer for the following questions. 07
1. A square matrix whose determinant is non zero is called (A) Singular (B) non-singular (C) invertible (D) both B and C
2.
If A and B are non singular matrices then
1( ) _ _ _ _ AB
?
? (A)11
AB
??
(B) AB (C)
11
BA
??
(D) none of these
3.
If
1 0 1
0 1 1
0 0 1
A
??
??
?
??
??
??
then A is in (A) Row echelon form (B) Reduced Row echelon form (C) both A
and B (D) none of these
4.
For what values of k does the system 2 , 3 3 x y x y k ? ? ? ? has
infinitely many solutions (A) K=5 (B) k=4 (C) k=6 (D) k=1
5. If in a set of vectors atleast one member can be expressed as a linear
combination of the remaining vectors then the set is (A) Linearly independent (B) Linearly dependent (C) basis (D) none
of these
6. If V is any vector space and S be a subset of V then S is called basis for
V if (A) S is Linearly independent (B) S spans V (C) both A and B (D) S is Linearly dependent
7. For what value of k the vectors u and v are orthogonal where u=(2,1,3) ,
v=(1,7, k) (A) K=-3 (B) k=1 (C) k=5 (D) k=2
(b) Choose the appropriate answer for the following questions. 07
1.
The eigen values of a matrix
10
24
A
??
?
??
??
are (A) 1,4 (B) -1,-4 (C) 1,3 (D) -1,3
2. If A is a nxn size invertible matrix then rank of A is (A) n-1 (B) n (C) 2n (D) n+1

Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE – SEMESTER? 1
st
/ 2
nd
EXAMINATION (NEW SYLLABUS) ? SUMMER 2018
Subject Code: 2110015 Date: 17-05-2018
Subject Name: Vector Calculus and Linear Algebra
Time: 02:30 pm to 05:30 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 Objective Question (MCQ)Mark
(a) Choose the appropriate answer for the following questions. 07
1. A square matrix whose determinant is non zero is called (A) Singular (B) non-singular (C) invertible (D) both B and C
2.
If A and B are non singular matrices then
1( ) _ _ _ _ AB
?
? (A)11
AB
??
(B) AB (C)
11
BA
??
(D) none of these
3.
If
1 0 1
0 1 1
0 0 1
A
??
??
?
??
??
??
then A is in (A) Row echelon form (B) Reduced Row echelon form (C) both A
and B (D) none of these
4.
For what values of k does the system 2 , 3 3 x y x y k ? ? ? ? has
infinitely many solutions (A) K=5 (B) k=4 (C) k=6 (D) k=1
5. If in a set of vectors atleast one member can be expressed as a linear
combination of the remaining vectors then the set is (A) Linearly independent (B) Linearly dependent (C) basis (D) none
of these
6. If V is any vector space and S be a subset of V then S is called basis for
V if (A) S is Linearly independent (B) S spans V (C) both A and B (D) S is Linearly dependent
7. For what value of k the vectors u and v are orthogonal where u=(2,1,3) ,
v=(1,7, k) (A) K=-3 (B) k=1 (C) k=5 (D) k=2
(b) Choose the appropriate answer for the following questions. 07
1.
The eigen values of a matrix
10
24
A
??
?
??
??
are (A) 1,4 (B) -1,-4 (C) 1,3 (D) -1,3
2. If A is a nxn size invertible matrix then rank of A is (A) n-1 (B) n (C) 2n (D) n+1

3.
If F is solenoidal then (A) 0 F ?? (B) 0 F ? ? ? (C) 0 F ? ? ? (D) none of these
4.
The mapping
33
: ( , , ) ( , , ) T R R d efin ed b y T x y z x y z ? ? ?
is called as (A) Contraction (B) Projection (C) Reflection (D) Rotation
5.
The linear transformation : T V W ? is one to one if and only if the
nullspace of T consists of only (A) Identity vector (B) zero vector (C) any non zero vector (D) none
of these
6.
If
11
22
A
??
?
??
??
then the rank of the matrix A is (A) 1 (B) 2 (C) 0 (D) 4
7. Let A be a skew-symmetric matrix then (A)ij ji
aa ? (B)
ij ji
aa ?? (C) 0
ii
a ? (D) both B and C

Q.2 (a)Find the unit vector normal to the surface
32
4 ( 1, 1, 2 ) xy z a t ? ? ?
03
(b)Express the matrix
3 2 6
2 7 1
5 4 0
A
?
??
??
??
??
??
??
as the sum of a symmetric and
skey-symmetric matrix.
04
(c)Investigate for what values of and ?? the equations
2 3 5 9 , 7 3 2 8 , 2 3 x y z x y z x y z ?? ? ? ? ? ? ? ? ? ? have (1) No solution (2) a unique solution (3) infinite number of
solutions
07
Q.3 (a)Find the rank of the matrix
1 2 3
456
7 8 9
??
??
??
??
??
03
(b)Find the inverse of the matrix
234
4 3 1
1 2 4
??
??
??
??
??
by Gauss Jordan Method
04
(c)For the basis ? ?
3
1 2 3
,, S v v v o f R ? where
1 2 3(1,1,1), (1,1, 0), (1, 0, 0) v v v ? ? ? Let
32
: T R R ? be the
linear transformation such that
1 2 3( ) (1, 0), ( ) (2, 1), ( ) (4, 3) T v T v T v ? ? ? ? find a formula
for
1 2 3( , , ) T x x x and then use the formula to find ( 4 , 3, 2 ) T ?

07
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE – SEMESTER? 1
st
/ 2
nd
EXAMINATION (NEW SYLLABUS) ? SUMMER 2018
Subject Code: 2110015 Date: 17-05-2018
Subject Name: Vector Calculus and Linear Algebra
Time: 02:30 pm to 05:30 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 Objective Question (MCQ)Mark
(a) Choose the appropriate answer for the following questions. 07
1. A square matrix whose determinant is non zero is called (A) Singular (B) non-singular (C) invertible (D) both B and C
2.
If A and B are non singular matrices then
1( ) _ _ _ _ AB
?
? (A)11
AB
??
(B) AB (C)
11
BA
??
(D) none of these
3.
If
1 0 1
0 1 1
0 0 1
A
??
??
?
??
??
??
then A is in (A) Row echelon form (B) Reduced Row echelon form (C) both A
and B (D) none of these
4.
For what values of k does the system 2 , 3 3 x y x y k ? ? ? ? has
infinitely many solutions (A) K=5 (B) k=4 (C) k=6 (D) k=1
5. If in a set of vectors atleast one member can be expressed as a linear
combination of the remaining vectors then the set is (A) Linearly independent (B) Linearly dependent (C) basis (D) none
of these
6. If V is any vector space and S be a subset of V then S is called basis for
V if (A) S is Linearly independent (B) S spans V (C) both A and B (D) S is Linearly dependent
7. For what value of k the vectors u and v are orthogonal where u=(2,1,3) ,
v=(1,7, k) (A) K=-3 (B) k=1 (C) k=5 (D) k=2
(b) Choose the appropriate answer for the following questions. 07
1.
The eigen values of a matrix
10
24
A
??
?
??
??
are (A) 1,4 (B) -1,-4 (C) 1,3 (D) -1,3
2. If A is a nxn size invertible matrix then rank of A is (A) n-1 (B) n (C) 2n (D) n+1

3.
If F is solenoidal then (A) 0 F ?? (B) 0 F ? ? ? (C) 0 F ? ? ? (D) none of these
4.
The mapping
33
: ( , , ) ( , , ) T R R d efin ed b y T x y z x y z ? ? ?
is called as (A) Contraction (B) Projection (C) Reflection (D) Rotation
5.
The linear transformation : T V W ? is one to one if and only if the
nullspace of T consists of only (A) Identity vector (B) zero vector (C) any non zero vector (D) none
of these
6.
If
11
22
A
??
?
??
??
then the rank of the matrix A is (A) 1 (B) 2 (C) 0 (D) 4
7. Let A be a skew-symmetric matrix then (A)ij ji
aa ? (B)
ij ji
aa ?? (C) 0
ii
a ? (D) both B and C

Q.2 (a)Find the unit vector normal to the surface
32
4 ( 1, 1, 2 ) xy z a t ? ? ?
03
(b)Express the matrix
3 2 6
2 7 1
5 4 0
A
?
??
??
??
??
??
??
as the sum of a symmetric and
skey-symmetric matrix.
04
(c)Investigate for what values of and ?? the equations
2 3 5 9 , 7 3 2 8 , 2 3 x y z x y z x y z ?? ? ? ? ? ? ? ? ? ? have (1) No solution (2) a unique solution (3) infinite number of
solutions
07
Q.3 (a)Find the rank of the matrix
1 2 3
456
7 8 9
??
??
??
??
??
03
(b)Find the inverse of the matrix
234
4 3 1
1 2 4
??
??
??
??
??
by Gauss Jordan Method
04
(c)For the basis ? ?
3
1 2 3
,, S v v v o f R ? where
1 2 3(1,1,1), (1,1, 0), (1, 0, 0) v v v ? ? ? Let
32
: T R R ? be the
linear transformation such that
1 2 3( ) (1, 0), ( ) (2, 1), ( ) (4, 3) T v T v T v ? ? ? ? find a formula
for
1 2 3( , , ) T x x x and then use the formula to find ( 4 , 3, 2 ) T ?

07

Q.4 (a)Determine whether the vector ( 5 ,1 1, 7 ) v ? ? ? is a linear combination
of the vectors
1 2 3(1, 2, 2), (0, 5, 5), (2, 0, 8) v v v ? ? ? ?
03
(b)Solve the linear system
4 , 2 , 2 2 2 x y z x y z x y z ? ? ? ? ? ? ? ? ? ? ? by gauss
elimination method.
04
(c)
Let
3
R have the Euclidean inner product. Use the gram schmidt process
to transform the basis
1 2 3( , , ) u u u in to Orthonormal basis where
1 2 3(1, 0, 0), (3, 7, 2), (0, 4,1) u u u ? ? ? ?
07
Q.5 (a) Find the eigen values and corresponding eigen vectors of
2 1 2
15
A
?
??
?
??
?
??
03
(b)Let
2 3 1
1 2 1
1 1 2
A a n d b
?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
then find the least squares
solutions to AX=b
04
(c)
Let
33
: T R R ? be a linear operator and
1 2 3( , , ) B v v v ? a basis for
3
R . Suppose that
1 2 3( ) (1,1, 0), ( ) (1, 0, 1), ( ) (2,1, 1) T v T v T v ? ? ? ? ? then (1) Is (1,2,1) in R(T) ? (2) Find a basis for R(T).
07
Q.6 (a)Find the work done by the force
2(3 3 ) 3 F x x i zj k a lo n g ? ? ? ?
the straight line
, 0 1 . ti tj tk t ? ? ? ? .
03
(b)Check whether the vectors (2,-3,1), (4,1,1),(0,-7,1) is a basis for
3
R
04
(c) Verify Green?s Theorem for
22( ) 1 F x y i xj a n d C is x y ? ? ? ? ?
07
Q.7 (a)Find the directional derivative of
22
4 (1, 2, 1) xz x yz a t ? ? ? in the
direction of 22 i j k ??
03
(b)Show that ( co s ) ( sin ) ( )
xx
F e y yz i xz e y j xy z k ? ? ? ? ? ?
is conservative and find the potential function.
04
(c)Let ? ?
1 2 1, 2( , ) / , ( , ), ( ) V a b a b R a n d let v v v w w w ? ? ? ?
then define
1 2 1 2 1 1 2 2( , ) ( , ) ( 1, 1) v v w w v w v w ? ? ? ? ? ? and
1 2 1 2( , ) ( 1, 1) c v v cv c cv c ? ? ? ? ? then verify that V is a vector
space.
07
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