2013

(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)

(b) Transform the equation referred to new axes through rotated through an angle 4

Or

Find the transformed equation of the curve when the two perpendicular lines and are taken as coordinate axes.

2. (a) Interpret the situation for the straight lines given by when 1

(b) Prove that the lines represented by have the same pair of bisectors for all values of. 2

(c) Show that the general equation of 2nd degreerepresents a pair of parallel straight lines if 4

Or

Find the condition that one of the lines given by may be perpendicular to one of the lines given by.

(d) If represents a pair of lines, then prove that the square of the distance of their point of intersection from the origin is 5

Or

If represents a pair of lines, then prove that the product of the perpendiculars from the origin on these lines is

3. (a) State True or False: A parabola has its centre at infinity. 1

(b) From the equation of the diameter of the conic conjugate to the diameter 3

(c) Define a conic section. Reduce the equationto the standard form. 1+5=6

Or

Find the equation of the polar of a given point with respect to the conic 6

SECTION – II

(3-Dimension)

4. (a) State the intercepts made on the axes by the plane 1

(b) Find the equation of the plane through the points and and parallel to the y-axis. 3

(c) Write the equation of the line through the point parallel to the z-plane. 1

(d) Put the equations of a line in the symmetrical form. 5

Or

Find the equation of the plane through the line parallel to the line

5. (a) Fill up the blank: 1

If the shortest distance between two lines is zero, then the lines are ____.

(b) Find the shortest distance between the y-axis and the line 2

(c) Find the shortest distance between the lines

and

And the equation of the line that represents shortest distance. 5

Or

Find the length and equations of the line of the shortest distance between the lines

and

GROUP – B

(Algebra – I)

(Marks: 35)

6. (a) State True or False: 1

“A map is invertible iff it is one-one into.”

(b) Give an example to show that a coset may not be a subgroup of a group. 1

(c) If be a group of prime order , then show that has no proper subgroup. 2

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

- Show that an infinite cyclic group has precisely two generators.
- Let H, K be subgroups of G. Show that HK is a subgroup of G if and only if HK = KH.
- Let G be a group. Show that

where n is an integer and

7. Answer any two questions: 5x2=10

- Prove that the set of matrices

where is a real number, forms a group under matrix multiplication.

- Show that if G is a group of order 10, then it must have a subgroup of order 5.
- State and prove Lagrange’s theorem.

8. (a) Define normal subgroup. 1

(b) If G is a finite group and N is a normal subgroup of G, then prove that 2

(c) If be a homomorphism, then prove that Ker is a normal subgroup of 3

9. Answer any one question: 4

(a) If is a homomorphism, then prove that -

- ;

an integer where and are identify elements of and respectively.

(b) If H and K are two subgroups which are not normal subgroups, then HK is a normal subgroup. Justify with an example.

10. Answer any one question: 5

- Show that a subgroup H of a group G is normal in G if and only if

- Show that every group is isomorphic to a permutation group.

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