DU SOL BA Programme 2nd Year Mathematics Question Paper 2018 B-155 G-I

[This question paper contains 5 printed pages.]

4346 yotrr Roil Ab. .... . .

B.A. Programme /II G.I

MATHEMATICS - Paper II

Paper Code : B-155

(Geometry, Differential Equations and Algebra)

Time : 3 Hours Muxintum lulark.s : 75

(l|trite your Roll lVo. on the top intntecliately
on receipt o./'this question paper.)

Note.- The mqximum marks printed on the qLte,sti,,
paper are applicable .for the .students o.f regular college.;
(Cat'A'). These marks will, hot+,et,er, be scalecl up
proportionately in respect of the sttrdents o.f SOL at the
time of posting o.f awards .for contpilation of result.

All questions are compulsory.

Attempt any two parts'.from each question

P.T.O.

4346

(a) Find'and sketch equation of the hyperbola with
vertices(0.+8)and asymptotes

A
!+

J

Also state the reflection property of hyperbola. (6Y')

(b) Describe the graph of the equation:

16rr + 9 yt - 64x - 54y + 1 : 0 (6%)

(c) Show that the locus of the middle point of the chords of
the parabola y2 : 4ax through the vertex is y2 : 2ax.
(6%)

(a) Find the equation of the sphere through the point
(1, -3, 4), (1, -5,2) and (1, -3, 0) and whose center lies
on the Plane x + ! + z:0. (6)

(b) Find the largest and the smallest distances between the
point P( I ,1 ,1 ) and the sphere

,t+yt*22-2y+62-6=0. (6)

(c) Find the area of the triangle with vertices A(1, 0. 1),
B(0,2,3) and C(2, 1, 0). Use the result to find the
:trp.
length of the aititude from the vertex C to the side
(6)

4346

3. (a) Solve

("t + yl + x)dx + xy dy:0 (6%)

(b) Solve the differential equation by the method of variation
of parameter:

qy
,; + 4, = +sec- Zx
dx-
(6Yr)
(c) Solve

)1., ,1,,
-.2u t t-."-Y ,1.._--3 (6Yr)
",t ---:--Z^-TLV
dx" dx
-i

A (a) Find the complete integral of the partial differential
anrrotinn.
vYusrrvrr.

::Px+qY+q2+P2. (6)

(b) Find whether the equation:

xyr - (r'- y')t -.xyt + py - qx:2(x2, - y2). (6)

is elliptic, parabolic or hyperbolic.

(c) Find the general integral of partial differential equation

(2xy- I)p+ (z -2x2) q = 2(x-yz). (6)

P.T.O.

4346 +,|

(a) Let G be a group and H a nonempty subset of G. Then
H is a subgroup of G if and only if

ab-r e tIYa, b e H (6Y,)

(b) Let
(r234s6i8eto)
(3 10 7 8 2 r 4 4 e s) be

permutation in S,o. Write'cx, as a product of disjoint
cycles and as a product of transpositions. Find it s inverse
and order. (6%)

(c) Discuss the clockwise rotation of a scluare and find all
the permutations obtained by the clockwise r otation. (6%)

o. (a) Find the matching or explain why none exists for the
following graph :

(6)

4346 f,

(b) Following is the cost matrix for a travelling salesperson
Problem. Solve it to find the
minimum cost.

To

I 39 7

From 2 3 -6 5

3 5 6 6

4 9
(6)

+) of order
(c) Show that the table for any finite group (G,
(6)
n is a Latin square of order n based on G.

(800)