Annamalai University question paper december 2014 Total No. of Pages: 2 5277

Register Number:

Name of the Candidate:

B.Sc. DEGREE EXAMINATION December 2014

(MATHEMATICS)

(THIRD YEAR)

(PART - III)

710: VECTOR CALCULUS AND LINEAR ALGEBRA

Time: Three hours Maximum: 100 marks

Answer any FIVE questions (5× 20=100)

1. a) A particle moves along the curve x=1–t3, y=1+t2, z=2t–5. Determine its velocity

and acceleration.

b) Prove that div curl f =0 and curl grad ϕ=0.

2. a) Prove that f =(x2–yz)i

r

+ (y–zx) j

r

+ (z2–xy) k

r

is irrotational.

b) Find div r and curl r

c) If f ax y z i x y z j x y z k

r r r

= ( + 3 + 4 ) + ( − 3 + 3 ) + 3( + 2 − ) is solenoidal, find the value of a

3. a) Evaluate ∫

c

f . rd where f x y i x y j

r r

( ) ( )

2 2 2 2

= + + − and C is the curve y=x2 joining

(0,0) and (1,1).

b) Verify Gauss divergence theorem for the vector function f x yz i x jy k

r r r

( ) 2 2

2 2

= − − +

over the cube bounded by x=0, y=0, z=0, x=a, y=a, z=a.

4. a) Show that ( )( )( )( )

1

1

1

2 3

2 3

2 3

a b b c c a bc ac ab

c c

b b

a a

= − − − + +

b) Prove that 3

2 2 2

2 2 2

2 2 2

2 ( )

( )

( )

( )

abc a b c

c c a b

b c a b

b c a a

= + +

+

+

+

5. Verify Cayley-Hamilton theorem for the matrix

− −

=

7 2 3

2 1 1

2 2 0

A

6. a) Find the inverse of a matrix

−

−

2 2 1

0 1 4

1 2 3

b) Define an orthogonal matrix and prove that the matrix

0 0 1

1 0 0

0 1 0

is orthogonal.

7. a) Find the rank of the matrix

=

2 1 0 7

6 3 4 7

4 2 1 3

A

b) Show that the equations x+y+z=6; x+2y+3z=14; x+4y+7z=30 are consistent and

solve them.

8. a) Define a vector space. Give an example.

b) Prove that the intersection of two subspaces of a vector space is a subspace.

c) Define a linear transformation. Give an example.

9. a)

Prove that

W

V

is a vector space over F.

b) Show that in v3(R), the vectors (1, 0, 0), (0, 1, 0), (1, 1, 1) are linearly independent.

10. a) Let V be vector space over a field F. Let s={v1, v2.......vn} span V and let S′={w1,

w2,......wm} be a linearly independent set of vectors in V. Prove that m≤n.

b) Let V be a finite dimensional vector space over a field F. Let W be a subspace of V.

Prove that dim

W

V

=dim V – dim W.

--------------------

Register Number:

Name of the Candidate:

B.Sc. DEGREE EXAMINATION December 2014

(MATHEMATICS)

(THIRD YEAR)

(PART - III)

710: VECTOR CALCULUS AND LINEAR ALGEBRA

Time: Three hours Maximum: 100 marks

Answer any FIVE questions (5× 20=100)

1. a) A particle moves along the curve x=1–t3, y=1+t2, z=2t–5. Determine its velocity

and acceleration.

b) Prove that div curl f =0 and curl grad ϕ=0.

2. a) Prove that f =(x2–yz)i

r

+ (y–zx) j

r

+ (z2–xy) k

r

is irrotational.

b) Find div r and curl r

c) If f ax y z i x y z j x y z k

r r r

= ( + 3 + 4 ) + ( − 3 + 3 ) + 3( + 2 − ) is solenoidal, find the value of a

3. a) Evaluate ∫

c

f . rd where f x y i x y j

r r

( ) ( )

2 2 2 2

= + + − and C is the curve y=x2 joining

(0,0) and (1,1).

b) Verify Gauss divergence theorem for the vector function f x yz i x jy k

r r r

( ) 2 2

2 2

= − − +

over the cube bounded by x=0, y=0, z=0, x=a, y=a, z=a.

4. a) Show that ( )( )( )( )

1

1

1

2 3

2 3

2 3

a b b c c a bc ac ab

c c

b b

a a

= − − − + +

b) Prove that 3

2 2 2

2 2 2

2 2 2

2 ( )

( )

( )

( )

abc a b c

c c a b

b c a b

b c a a

= + +

+

+

+

5. Verify Cayley-Hamilton theorem for the matrix

− −

=

7 2 3

2 1 1

2 2 0

A

6. a) Find the inverse of a matrix

−

−

2 2 1

0 1 4

1 2 3

b) Define an orthogonal matrix and prove that the matrix

0 0 1

1 0 0

0 1 0

is orthogonal.

7. a) Find the rank of the matrix

=

2 1 0 7

6 3 4 7

4 2 1 3

A

b) Show that the equations x+y+z=6; x+2y+3z=14; x+4y+7z=30 are consistent and

solve them.

8. a) Define a vector space. Give an example.

b) Prove that the intersection of two subspaces of a vector space is a subspace.

c) Define a linear transformation. Give an example.

9. a)

Prove that

W

V

is a vector space over F.

b) Show that in v3(R), the vectors (1, 0, 0), (0, 1, 0), (1, 1, 1) are linearly independent.

10. a) Let V be vector space over a field F. Let s={v1, v2.......vn} span V and let S′={w1,

w2,......wm} be a linearly independent set of vectors in V. Prove that m≤n.

b) Let V be a finite dimensional vector space over a field F. Let W be a subspace of V.

Prove that dim

W

V

=dim V – dim W.

--------------------

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