2014

(November)

MATHEMATICS

(Major)

Course: 302

(Coordinate Geometry and Algebra - I)

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions

GROUP – A

(Coordinate Geometry)

SECTION – I

(2-Dimension)

(Marks: 27)

1

(b) Choose a new origin (h, k), without changing the directions of the axes such that the equation

may reduce to the form 2

(c) Transform the equation

To rectangular axes inclined at angle to the old rectangular axes. 3

Or

Find the angle through which the axes be rotated so that the expression may become of the form

2. (a) Find the value of so that the equation 3

may represent a pair of lines.

(b) Find the equation of the pair of lines through the origin which represents the lines perpendicular to the pair of lines. 3

Or

Show that the angle between one of the lines given byand one of the lines is equal to the angle between the other two lines of the systems.

(c) Prove that the straight lines represented by the equation

Will be equidistant from the origin if

5

Or

If represents a pair of lines, prove that the area of the triangle formed by these lines and the x-axis is

3. (a) State True or False: 1

When the focus lies on the directrix, the conic section is a pair of lines.

(b) Find the equation of the chord of contact of tangents from a point to the conic 4

Or

Find the condition that the pair of lines may be conjugate diameters of the conic

(c) Reduce the following equation to the standard form:

5

Or

Prove that every Cartesian equation of the second degree, i.e.

represents a conic.

SECTION – II

(3-Dimension)

(Marks: 18)

4. (a) Write the direction cosines of the line joining the origin and the point . 1

(b) Find the equation of the plane which passes through the intersection of the planes and is perpendicular to the plane 4

(c) Find the distance of the point from the plane measured parallel to the line

5

Or

Show that the lines

and

intersect. Find the coordinates of the point of intersection.

5. (a) What is the shortest distance between two intersecting straight lines? 1

(b) Find the shortest distance between the line

2

and the z-axis.

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

5

Or

Prove that the shortest distance between the lines

is and the equation of shortest distance are

GROUP – B

(Algebra – I)

(Marks: 35)

6. (a) If a set A has n members, then state the number of binary compositions on A. 1

(b) The set of integers, with respect to usual multiplication does not form a group. Justify it. 2

(c) State True or False: A group of prime order is Abelian. 1

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

- Show that the centre of a group G is a subgroup of G.
- If G is a group is which for three consecutive integers and any a, b in G, then show that G is Abelian.
- Prove that an infinite cyclic group has precisely two generators.

7. Answer any two questions: 5x2=10

- Let G be a group. Suppose such that –

- .

- If H and K are finite subgroups of a group G, then prove that

- If a group has finite number of subgroups, then show that it is a finite group.

8. (a) Define Kernel of a group homomorphism. 1

(b) Prove that a subgroup H of a group G is normal in G iff for all 4

9. Answer any two questions: 5x2=10

- If G is a group such that is cyclic, where is centre of G, then show that G is Abelian?
- Prove that every quotient group of a cyclic group is cyclic.
- State and prove fundamental theorem of group homomorphism.

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