GTU BE 1st and 2nd Semester 110015 Vector Calculus and Linear Algebra Summer 2018 Question Paper

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER? 2
nd
? EXAMINATION ? SUMMER 2018
Subject Code:110015 Date: 17-05-2018
Subject Name: Vector Calculus and Linear Algebra
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) 1
If ?? = [
1 0 0
5 2 0
12 15 3
] then find the eigen values of ?? and hence find eigen values
of ?? 5
and ?? -1
.
4
2
Prove that ?? = [
1 3 + 4?? -2?? 3 – 4?? 2 9 – 7?? 2?? 9 + 7?? 3
] is a Hermitian matrix.
3 (b) 1 Solve the following system of equations
?? + 5?? = 2
11?? + ?? + 2?? = 3
?? + 5?? + 2?? = 1
Using Gauss elimination method
4
2 Determine whether or not vectors (1, -2, 1), (2, 1, -1), (7, -4, 1) in ?? 3
are
linearly independent.
3
Q-2 (a) 1 Investigate for what values of ? and ? the equations
?? + ?? + ?? = 6
?? + 2?? + 3?? = 10
x + 2y + ? z = ?
have (i) a unique solution (ii) no solution
4 1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER? 2
nd
? EXAMINATION ? SUMMER 2018
Subject Code:110015 Date: 17-05-2018
Subject Name: Vector Calculus and Linear Algebra
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) 1
If ?? = [
1 0 0
5 2 0
12 15 3
] then find the eigen values of ?? and hence find eigen values
of ?? 5
and ?? -1
.
4
2
Prove that ?? = [
1 3 + 4?? -2?? 3 – 4?? 2 9 – 7?? 2?? 9 + 7?? 3
] is a Hermitian matrix.
3 (b) 1 Solve the following system of equations
?? + 5?? = 2
11?? + ?? + 2?? = 3
?? + 5?? + 2?? = 1
Using Gauss elimination method
4
2 Determine whether or not vectors (1, -2, 1), (2, 1, -1), (7, -4, 1) in ?? 3
are
linearly independent.
3
Q-2 (a) 1 Investigate for what values of ? and ? the equations
?? + ?? + ?? = 6
?? + 2?? + 3?? = 10
x + 2y + ? z = ?
have (i) a unique solution (ii) no solution
4
2

2
Obtain the reduced row echelon form of the matrix ?? = [
1 3 2 2
1 2 1 3
2 4 3 4
3 7 4 8
]
3
(b)
Prove that the set of all 2×2 matrices of the form [
?? 1
1 ?? ] with the operations
defined as
[
?? 1
1 ?? ] + [
?? 1
1 ?? ] = [
?? + ?? 1
1 ?? + ?? ] & k [
?? 1
1 ?? ] = [
???? 1
1 ????
] is a vector space.
7
Q-3 (a) For the basis ?? = {?? 1
, ?? 2
, ?? 3
} of ?? 3
where ?? 1
= (1, 1, 1), ?? 2
= (1, 1, 0) and ?? 3
=(1, 0, 0). Let ?? : ?? 3
? ?? 3
be a linear transformation such that (?? 1
) = (2, -1, 4),
?? (?? 2
) = (3, 0, 1), ?? (?? 3
) = (-1, 5, 1). Find a formula for ?? (?? 1
, ?? 2
, ?? 3
)and use it to find ?? (2, 4, -1).
7 (b) 1 (i) Find the Euclidean inner product u.v where
) 3 , 4 , 2 , 2 ( ), 5 , 4 , 1 , 3 ( ? ? ? ? ? v u (ii) For which values of k are ?? = (2, 1, 3) and ?? = (1, 7, ?? )
orthogonal?
4

2 For ?? = (2, 1, 3), ?? = (2, -1, 3) verify Cauchy-Schwarz inequality holds. 3
Q-4 (a) 1
Find eigen values and eigenvectors of the matrix ?? = [
3 1 4
0 2 6
0 0 5
]
4
2 Define Symmetric matrix and Skew-symmetric matrix by giving example. 3 (b) 1 Translate and rotate the coordinate axes, if necessary, to put the conic 9?? 2
–
4???? + 6?? 2
– 10?? – 20?? = 5 in standard position. Find the equation of the conic
in the final coordinate system.
4
2
Use Cayley-Hamilton theorem to find ?? -1
fro ?? = [
1 3 7
4 2 3
1 2 1
]
3
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER? 2
nd
? EXAMINATION ? SUMMER 2018
Subject Code:110015 Date: 17-05-2018
Subject Name: Vector Calculus and Linear Algebra
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) 1
If ?? = [
1 0 0
5 2 0
12 15 3
] then find the eigen values of ?? and hence find eigen values
of ?? 5
and ?? -1
.
4
2
Prove that ?? = [
1 3 + 4?? -2?? 3 – 4?? 2 9 – 7?? 2?? 9 + 7?? 3
] is a Hermitian matrix.
3 (b) 1 Solve the following system of equations
?? + 5?? = 2
11?? + ?? + 2?? = 3
?? + 5?? + 2?? = 1
Using Gauss elimination method
4
2 Determine whether or not vectors (1, -2, 1), (2, 1, -1), (7, -4, 1) in ?? 3
are
linearly independent.
3
Q-2 (a) 1 Investigate for what values of ? and ? the equations
?? + ?? + ?? = 6
?? + 2?? + 3?? = 10
x + 2y + ? z = ?
have (i) a unique solution (ii) no solution
4
2

2
Obtain the reduced row echelon form of the matrix ?? = [
1 3 2 2
1 2 1 3
2 4 3 4
3 7 4 8
]
3
(b)
Prove that the set of all 2×2 matrices of the form [
?? 1
1 ?? ] with the operations
defined as
[
?? 1
1 ?? ] + [
?? 1
1 ?? ] = [
?? + ?? 1
1 ?? + ?? ] & k [
?? 1
1 ?? ] = [
???? 1
1 ????
] is a vector space.
7
Q-3 (a) For the basis ?? = {?? 1
, ?? 2
, ?? 3
} of ?? 3
where ?? 1
= (1, 1, 1), ?? 2
= (1, 1, 0) and ?? 3
=(1, 0, 0). Let ?? : ?? 3
? ?? 3
be a linear transformation such that (?? 1
) = (2, -1, 4),
?? (?? 2
) = (3, 0, 1), ?? (?? 3
) = (-1, 5, 1). Find a formula for ?? (?? 1
, ?? 2
, ?? 3
)and use it to find ?? (2, 4, -1).
7 (b) 1 (i) Find the Euclidean inner product u.v where
) 3 , 4 , 2 , 2 ( ), 5 , 4 , 1 , 3 ( ? ? ? ? ? v u (ii) For which values of k are ?? = (2, 1, 3) and ?? = (1, 7, ?? )
orthogonal?
4

2 For ?? = (2, 1, 3), ?? = (2, -1, 3) verify Cauchy-Schwarz inequality holds. 3
Q-4 (a) 1
Find eigen values and eigenvectors of the matrix ?? = [
3 1 4
0 2 6
0 0 5
]
4
2 Define Symmetric matrix and Skew-symmetric matrix by giving example. 3 (b) 1 Translate and rotate the coordinate axes, if necessary, to put the conic 9?? 2
–
4???? + 6?? 2
– 10?? – 20?? = 5 in standard position. Find the equation of the conic
in the final coordinate system.
4
2
Use Cayley-Hamilton theorem to find ?? -1
fro ?? = [
1 3 7
4 2 3
1 2 1
]
3

3
Q-5 (a) 1 Using Gram-Schmidt process, construct an orthonormal basis for ?? 3
, whose basis
is the set {(1, 1, 1), (1, -2, 1), (1, 2, 3)}
4
2 Show that ?? = (9, 2, 7) is a linear combination of the vectors ?? = (1, 2, -1) and
?? = (6, 4, 2) in ?? .
3 (b) 1 Find the least squares solution of the linear system , b Ax ? and find the orthogonal
projection of b onto the column space of A.
?? = [
2 -2
1 1
3 1
] , ?? = [
2
-1
1
]
4
2 Which of the following are subspaces of ?? 3
? (i) All the vectors of the form (?? , ?? , ?? ) where ?? = ?? + ?? (ii) All the vectors of the form (?? , ?? , ?? ) where ?? = ?? + ?? + 1
3
Q-6 (a) Verify Green?s theorem for the function ?? = (?? 2
+ ?? 2
)?? ^ – 2???? ?? ^ , where ?? is the
rectangle in the ???? -plane bounded by ?? = 0, ?? = ?? , ?? = 0 and ?? = ?? .
7 (b) 1 Verify Stoke?s theorem for ?? = (?? 2
– ?? 2
) ?? ^ + 2???? ?? ^ in the rectangular region ?? =
0, ?? = 0, ?? = ?? , ?? = ?? .
4
2
Find ?? -1
using Gauss-Jordan method if ?? = [
1 0 1
-1 1 1
0 1 0
]
3
Q-7 (a) 1 (i) Find ???????? (?) = log(?? 2
+ ?? 2
+ ?? 2
) at the point (1, 0, -2) (ii) Find a unit vector normal to the surface ?? 3
+ ?? 3
+ 3?????? = 3 at the
point (1, 2, -1)4

2 Find the directional derivative of the divergence of ?? ?( ?? , ?? , ?? ) = ???? ?? ^ + ?? ?? 2
?? ^ +
?? 2
?? ^
at the point (2, 1, 2) in the direction of the outer normal to the sphere ?? 2
+
?? 2
+ ?? 2
= 9.

3 1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER? 2
nd
? EXAMINATION ? SUMMER 2018
Subject Code:110015 Date: 17-05-2018
Subject Name: Vector Calculus and Linear Algebra
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) 1
If ?? = [
1 0 0
5 2 0
12 15 3
] then find the eigen values of ?? and hence find eigen values
of ?? 5
and ?? -1
.
4
2
Prove that ?? = [
1 3 + 4?? -2?? 3 – 4?? 2 9 – 7?? 2?? 9 + 7?? 3
] is a Hermitian matrix.
3 (b) 1 Solve the following system of equations
?? + 5?? = 2
11?? + ?? + 2?? = 3
?? + 5?? + 2?? = 1
Using Gauss elimination method
4
2 Determine whether or not vectors (1, -2, 1), (2, 1, -1), (7, -4, 1) in ?? 3
are
linearly independent.
3
Q-2 (a) 1 Investigate for what values of ? and ? the equations
?? + ?? + ?? = 6
?? + 2?? + 3?? = 10
x + 2y + ? z = ?
have (i) a unique solution (ii) no solution
4
2

2
Obtain the reduced row echelon form of the matrix ?? = [
1 3 2 2
1 2 1 3
2 4 3 4
3 7 4 8
]
3
(b)
Prove that the set of all 2×2 matrices of the form [
?? 1
1 ?? ] with the operations
defined as
[
?? 1
1 ?? ] + [
?? 1
1 ?? ] = [
?? + ?? 1
1 ?? + ?? ] & k [
?? 1
1 ?? ] = [
???? 1
1 ????
] is a vector space.
7
Q-3 (a) For the basis ?? = {?? 1
, ?? 2
, ?? 3
} of ?? 3
where ?? 1
= (1, 1, 1), ?? 2
= (1, 1, 0) and ?? 3
=(1, 0, 0). Let ?? : ?? 3
? ?? 3
be a linear transformation such that (?? 1
) = (2, -1, 4),
?? (?? 2
) = (3, 0, 1), ?? (?? 3
) = (-1, 5, 1). Find a formula for ?? (?? 1
, ?? 2
, ?? 3
)and use it to find ?? (2, 4, -1).
7 (b) 1 (i) Find the Euclidean inner product u.v where
) 3 , 4 , 2 , 2 ( ), 5 , 4 , 1 , 3 ( ? ? ? ? ? v u (ii) For which values of k are ?? = (2, 1, 3) and ?? = (1, 7, ?? )
orthogonal?
4

2 For ?? = (2, 1, 3), ?? = (2, -1, 3) verify Cauchy-Schwarz inequality holds. 3
Q-4 (a) 1
Find eigen values and eigenvectors of the matrix ?? = [
3 1 4
0 2 6
0 0 5
]
4
2 Define Symmetric matrix and Skew-symmetric matrix by giving example. 3 (b) 1 Translate and rotate the coordinate axes, if necessary, to put the conic 9?? 2
–
4???? + 6?? 2
– 10?? – 20?? = 5 in standard position. Find the equation of the conic
in the final coordinate system.
4
2
Use Cayley-Hamilton theorem to find ?? -1
fro ?? = [
1 3 7
4 2 3
1 2 1
]
3

3
Q-5 (a) 1 Using Gram-Schmidt process, construct an orthonormal basis for ?? 3
, whose basis
is the set {(1, 1, 1), (1, -2, 1), (1, 2, 3)}
4
2 Show that ?? = (9, 2, 7) is a linear combination of the vectors ?? = (1, 2, -1) and
?? = (6, 4, 2) in ?? .
3 (b) 1 Find the least squares solution of the linear system , b Ax ? and find the orthogonal
projection of b onto the column space of A.
?? = [
2 -2
1 1
3 1
] , ?? = [
2
-1
1
]
4
2 Which of the following are subspaces of ?? 3
? (i) All the vectors of the form (?? , ?? , ?? ) where ?? = ?? + ?? (ii) All the vectors of the form (?? , ?? , ?? ) where ?? = ?? + ?? + 1
3
Q-6 (a) Verify Green?s theorem for the function ?? = (?? 2
+ ?? 2
)?? ^ – 2???? ?? ^ , where ?? is the
rectangle in the ???? -plane bounded by ?? = 0, ?? = ?? , ?? = 0 and ?? = ?? .
7 (b) 1 Verify Stoke?s theorem for ?? = (?? 2
– ?? 2
) ?? ^ + 2???? ?? ^ in the rectangular region ?? =
0, ?? = 0, ?? = ?? , ?? = ?? .
4
2
Find ?? -1
using Gauss-Jordan method if ?? = [
1 0 1
-1 1 1
0 1 0
]
3
Q-7 (a) 1 (i) Find ???????? (?) = log(?? 2
+ ?? 2
+ ?? 2
) at the point (1, 0, -2) (ii) Find a unit vector normal to the surface ?? 3
+ ?? 3
+ 3?????? = 3 at the
point (1, 2, -1)4

2 Find the directional derivative of the divergence of ?? ?( ?? , ?? , ?? ) = ???? ?? ^ + ?? ?? 2
?? ^ +
?? 2
?? ^
at the point (2, 1, 2) in the direction of the outer normal to the sphere ?? 2
+
?? 2
+ ?? 2
= 9.

3
4
(b) 1
Show that ?? ?
= (?? 2
– ?? 2
+ 3???? – 2?? )?? ^ + (3???? + 2???? )?? ^ + (3???? – 2???? + 2?? )?? ^
is
both solenoidal and irrotational.
4
2
Evaluate ? ?? ?
?? . ?? ?? ? , where ?? ?
= (2?? + 3?? )?? ^ – (???? + ?? )?? ^ + (?? 2
+ 2?? )?? ^
and ?? is
the surface of the sphere having center at (3, -1, 2) and radius 3.
3

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