## Friday, January 04, 2019

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)

1. (a) What will be the equation of the line when the origin is transferred to the point    1
(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
1. Show that an infinite cyclic group has precisely two generators.
2. Let H, K be subgroups of G. Show that HK is a subgroup of G if and only if HK = KH.
3. Let G be a group. Show that
where n is an integer and
7. Answer any two questions: 5x2=10
1. Prove that the set of matrices
where is a real number, forms a group under matrix multiplication.

1. Show that if G is a group of order 10, then it must have a subgroup of order 5.
2. 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 -
1. ;
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
1. Show that a subgroup H of a group G is normal in G if and only if
1. Show that every group is isomorphic to a permutation group.

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