2015

(November)

MATHEMATICS

(Major)

Course: 302

(Coordinate Geometry and Algebra - I)

Full Marks: 80

Pass Marks: 32 (Backlog)/24 (2014 onwards)

Time: 3 hours

The figures in the margin indicate full marks for the questions

GROUP – A

(Coordinate Geometry)

SECTION – I

(2-Dimension)

(Marks: 27)

(b) Transform, to axes inclined at 450 to the original axes, the equation. 2

(c) Show that the distance between the points and is unaltered by translation of axes. 4

Or

Find the angle through which the axes be rotated so that the expression

will be of the form

2. (a) Prove that the equation 4

Represents a pair of straight lines inclined to each other at 450.

(b) Find the value of k so that the equation 3

may represent a pair of straight lines.

Or

Show that the angle between one of the lines and one of the lines is equal to the angles between the other two lines of the system.

(c) Prove that the product of the perpendiculars from the point on the lines of is

3

Or

Show that the distance between the point of intersection of the lines represented by the equation

and the origin is

3. (a) Under which condition does a conic section represent a pair of straight lines? 2

(b) Show that the Cartesian equation

represents a parabola. 3

Or

Find the diameter of the conic

conjugate to the diameter .

(c) Find the condition that the line is a tangent to the conic 5

Or

Reduce the following equation to the standard form:

SECTION – II

(3-Dimension)

(Marks: 18)

4. (a) What are the direction cosines of a line equally inclined to the axes? 1

(b) Find the equation of the plane which passes through the point and is perpendicular to the line joining the points and 4

(c) Find the equation of the perpendicular from the point to the line

Find the coordinated of the foot of the perpendicular. 5

Or

Show that the lines

are coplanar. Find their point of intersection and the equation of the plane in which they lie.

5. (a) Write down the nature of the non-intersecting lines if their shortest distance be zero. 1

(b) Find the shortest distance between the lines

and the z-axis. 2

(c) Find the length and the equations of the shortest distance between the lines

5

Or

Find the length of the shortest distance between the lines

Find also its equation and points where it intersects the lines.

GROUP – B

(Algebra – I)

(Marks: 35)

6. (a) Subtraction is not a binary composition on the set of natural numbers |N. Justify it. 1

(b) Does the set of all odd integers form a group with respect to addition? 1

(c) Show that if every element of a group G is its own inverse, then G is Abelian. 2

(d) Answer any two of the following questions: 3x2=6

- Prove that, if and only if G is an Abelian group.
- Prove that the identity element of a subgroup is the same as that of the group.
- Find the number of generators of a cyclic group of order 60.

7. Answer any two of the following questions: 5x2=10

- If for any elements (group), then prove that the equation and have unique solutions in G.
- Prove that the product of disjoint cycles is commutative.
- Prove that a non-empty subset H of a group G is a subgroup G if and only if –

whereis the inverse of a in G.

8. (a) What is the difference between a complex and a subgroup of a group? 1

(b) Prove that the kernel of a homomorphism of into is a normal subgroup of, where and are distinct groups. 4

9. Answer any two of the following questions: 5x2=10

- If is a finite group and is a normal subgroup of , then prove that
- Find the regular permutation groups isomorphic to the multiplicative group
- Prove that every subgroup of a cyclic group is cyclic.

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