Annamalai University 2008-2nd Year B.Sc Mathematics " 650 ALGEBRA AND SOLID GEOMETRY " ( ) ( - III ) 5233 - Question Paper
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(b) Find the equation of the plane which passes through the point (-1, 3, 2) and perpendicular to the two planes
x + 2y + 2z = 5 ;
3x + 3y + 2z = 8.
8. (a) Find the image of the point (2, 3, 4) in the plane
x - 2y + 5z = 6.
(b) Show that the lines :
x+1 y + 1 z+1
2 3 4
x-1 y - 2 z-3
and
8 -7
are coplanar. Find also, the equation of the plane containing them.
9. (a) Find the equation of the sphere
passing through the four points
(2, 3, 1), (5, -1, 2), (2, 5, 3) and (4, 3, -1)
Name of the Candidate :
5 2 3 3 B.Sc. DEGREE EXAMINATION, 2008
(MATHEMATICS)
(SECOND YEAR)
(PART - III - A - MAIN)
(PAPER - III)
650. ALGEBRA AND SOLID GEOMETRY (Including Lateral Entry )
December ] [ Time : 3 Hours
Maximum : 100 Marks
Answer any FIVE questions.
All questions carry equal marks.
(5 x 20 = 100)
(c) Solve the equation
x3 - 12x2 + 39x - 28 = 0
whose roots are in arithematical progression.
2. (a) If a, (3, y are the roots of the equation
x3 + px2 + qx + r = 0, find the equation whose roots are a + (3, (3 + y, y + a.
(b) Solve the reciprocal equation
6x6 - 35x5 + 56x4 - 56x2 + 35x - 6 = 0.
3. (a) State and prove Lagranges theorem and
deduce Fermats theorem.
(b) Show that n5 - n is divisible by 30.
4. (a) Prove that a non void subset H of a
group G is a subgroup, if and only if,
a, b g H => ab-1 e H.
(b) State and prove the fundamental theorem of group homomorphisms.
5. (a) Show that the set of all complex numbers
of the form a + ib where a and b are integers is a commutative ring.
(b) Show that a finite integral domain is a field.
6. (a) If A and B are normal subgroup of a
group G, prove that A n B is also a normal subgroup of G.
(b) If H and K are finite-subgroups of a group G of orders 0(H) and 0(K) respectively, prove that
0(H) - 0(K)
O (HK) =
0(H n K)
(c) Show that every field is an integral domain.
7. (a) Show that (1, -1, 1) , (5, -5, 4),
(5, 0, 8) and (1, 4, 5) are the vertices of a rhombus.
(b) Show that the plane 2x -y -2z = 16 touches the sphere
x2 + y2 + z2 - 4x + 2y + 2z - 3 = 0
and find the point of contact.
10. (a) Show that
x2 - 2y2 + 3z2 - 4xy + 5yz - 6zx + 8x
- 19y - 2z = 20
represents a cone and find its vertex.
(a) Find the equation of the right circular one whose vertex is at the origin, whose axis is the line
x y z
1 = 2 3
and which has a vertical angle of 60.
(b) Show that the plane 2x -y -2z = 16 touches the sphere
x2 + y2 + z2 - 4x + 2y + 2z - 3 = 0
and find the point of contact.
10. (a) Show that
x2 - 2y2 + 3z2 - 4xy + 5yz - 6zx + 8x
- 19y - 2z = 20
represents a cone and find its vertex.
(a) Find the equation of the right circular one whose vertex is at the origin, whose axis is the line
x y z
1 = 2 3
and which has a vertical angle of 60.
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