Categories: 1st and 2nd Semester

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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