Annamalai University December 2014 question paper

B.Sc. DEGREE EXAMINATION December 2014

(MATHEMATICS)

(SECOND YEAR)

(PART - III)

GROUP – A : Main

650: ALGEBRA AND SOLID GEOMETRY

Time: Three hours Maximum: 100 marks

Answer any FIVE questions (5× 20=100)

1. a) Solve the equation x4–5x3+4x2+8x–8=0, given that one of the roots in 1– 5 .

b) Solve the equation x3–6x2+13x–10=0 whose roots are in arithmetic progression.

2. a) If α, β, γ, δ are the roots of the equation x4+3x3–2x2+4x+1=0 then find ∑α2.

b) Increase by 7 the roots of the equation 3x4+7x3–15x2+x–2=0

c) Solve the equation 6x5–x4–43x3+43x2+x–6=0

3. a) Define Euler’s function ϕ(n) and evaluate ϕ(720) and ϕ(729).

b) State and prove Wilson’s theorem.

4. a) Define an equivalence relation. Show that intersection two equivalence relations is

an equivalence relation.

b) Prove that f:Z Z defined by f(x)=5x–3 is not a bijection.

5. a) Show that a non-empty subset H of a group G is a subgroup of G if and only if a,

b∈H implies ab–1∈H.

b) State and prove the fundamental theorem of homomorphism on groups.

6. a) Prove that Zn is an integral domain if and only if n is prime.

b) Show that any finite integral domain is a field.

c) Prove that any commutative ring without zero divisors is a field.

7. a) Find the equation of the plane passing through the points (3, 1, 2), (3, 4, 4) and

perpendicular to the plane 5x+y+4z=0.

b) Find the equation of the plane through the point (1, –2, 3) and the intersection of

the planes 2x–y+4z=7 and x+2y–3z+8=0.

8. a)

Prove that the lines

2

1

8

10

3

1 −

=

+

=

−

x + y z

and

1

4

7

1

4

3 −

=

+

=

−

x + y z

are coplanar. Find

their point of intersection and the plane through them.

b) Find the shortest distance between the lines

1

2

2

4

1

3 +

=

−

=

−

x − y z

and

2

2

3

7

1

1 +

=

+

=

x − y z

.

9. a) Find the equation of the sphere having the circle x2+y2+z2–2x+4y–6z+7=0;

2x–y+2z=5 as a great circle.

b) Prove that the plane 2x–2y+z+12=0 touches the sphere x2+y2+z2–2x–4y+2z–3=0.

Find the point of contact.

10. a) Find the equation of a sphere which touches the sphere x2+y2+z2–6x+2z+1=0 at

the point (2, –2, 1) and passes through the origin.

b) Find the equation of a right circular cone whose vertex is the origin, axis is the

z-axis is the z-axis and semi-vertical angle is 30º

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

B.Sc. DEGREE EXAMINATION December 2014

(MATHEMATICS)

(SECOND YEAR)

(PART - III)

GROUP – A : Main

650: ALGEBRA AND SOLID GEOMETRY

Time: Three hours Maximum: 100 marks

Answer any FIVE questions (5× 20=100)

1. a) Solve the equation x4–5x3+4x2+8x–8=0, given that one of the roots in 1– 5 .

b) Solve the equation x3–6x2+13x–10=0 whose roots are in arithmetic progression.

2. a) If α, β, γ, δ are the roots of the equation x4+3x3–2x2+4x+1=0 then find ∑α2.

b) Increase by 7 the roots of the equation 3x4+7x3–15x2+x–2=0

c) Solve the equation 6x5–x4–43x3+43x2+x–6=0

3. a) Define Euler’s function ϕ(n) and evaluate ϕ(720) and ϕ(729).

b) State and prove Wilson’s theorem.

4. a) Define an equivalence relation. Show that intersection two equivalence relations is

an equivalence relation.

b) Prove that f:Z Z defined by f(x)=5x–3 is not a bijection.

5. a) Show that a non-empty subset H of a group G is a subgroup of G if and only if a,

b∈H implies ab–1∈H.

b) State and prove the fundamental theorem of homomorphism on groups.

6. a) Prove that Zn is an integral domain if and only if n is prime.

b) Show that any finite integral domain is a field.

c) Prove that any commutative ring without zero divisors is a field.

7. a) Find the equation of the plane passing through the points (3, 1, 2), (3, 4, 4) and

perpendicular to the plane 5x+y+4z=0.

b) Find the equation of the plane through the point (1, –2, 3) and the intersection of

the planes 2x–y+4z=7 and x+2y–3z+8=0.

8. a)

Prove that the lines

2

1

8

10

3

1 −

=

+

=

−

x + y z

and

1

4

7

1

4

3 −

=

+

=

−

x + y z

are coplanar. Find

their point of intersection and the plane through them.

b) Find the shortest distance between the lines

1

2

2

4

1

3 +

=

−

=

−

x − y z

and

2

2

3

7

1

1 +

=

+

=

x − y z

.

9. a) Find the equation of the sphere having the circle x2+y2+z2–2x+4y–6z+7=0;

2x–y+2z=5 as a great circle.

b) Prove that the plane 2x–2y+z+12=0 touches the sphere x2+y2+z2–2x–4y+2z–3=0.

Find the point of contact.

10. a) Find the equation of a sphere which touches the sphere x2+y2+z2–6x+2z+1=0 at

the point (2, –2, 1) and passes through the origin.

b) Find the equation of a right circular cone whose vertex is the origin, axis is the

z-axis is the z-axis and semi-vertical angle is 30º

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

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