Categories: 1st and 2nd Semester

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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