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

[This question paper contains 5 printed pages.]

437 |

B.A. Programme /II G-II

MATHEMATICS - Paper II

Paper Code : B-155

(Geometry, Differential Equations and Algebra)

Time : 3 Hours Maximum Murks ; 75

(Write yotrr Roll No. on the top immetliately
on receipt of this question peper.)

Note:- The maximunt marks printetl on the question
paper are applicable ./br the stutlents of' (Cutegorv- ,A').
These marks will, howe,er, be scoletl up proportionutelv in
respect o.f the students of SOL at the tinte of posting of'
awarels .for compilation of results.

All questions are compulsory.

Attempt any two parts .from euch question.

P.T.O

437 | )

(a) Find an equation for the ellipse that passes through
(-3, 2) and ends of whose major axis are (0,t6). (6r/r)

(b) Describe the graph of the equation:

.y2 - 8x - 6y - 23 : o. (6Y,)

(c) Suppose the axes of an xy-coordinate system are rotated
through an angle of 0:45o to obtain an x'y'coordinates
system. Find the equation of the curve x2 * xy + y2 - 6
: 0 in .r'v' coordinates and also sketch it. (6/r)

2. (a) Find the area of the triangle that is determined by
Pt(2,.2.0), P, (-1,0,2) and P? (0,4,3) (6)

(b) Find k so that the vector from the point A(1,-1, 3) to the
point B(3,0,5) is orthogonal to the vector from A to
the point P(k, k, k). (6)

(c) Find the angle of intersection of the two planes:

2r-4v + 4z:6 and 6r+2v-32: 4 (6)

4371 3

(a) Show that e'and xe' are linearlY independent soiution
of the differential equation

dty *,,-o (6%)
dx--cd!
ax

(b)Solvethedifferentialequationbythemethodofvariatiorr
of parameters.
t .t
'4+
,tl
ay = 4tan.r. (6',/,)
dx-
(c) A body cools froni'60oC to 5OoC in 15 minutes in air
which is maintained at 30'C' How long will it take this
body to cool from 100'C to 80oC in air that is maintained
(6Yz)
at 50oC?

+. (a)Findthecompleteintegralofthepartialdifferential
equation:

q2 = zz72(r-P')' (6)
.
(b) Find whether the equation:

xzr-2xs+t+q:0.
(6)
is elliptic, parabolic or hyperbolic'

P.T.O.

4371 4

(c) Fincl the partial clifferential equation by eliminating
arbitrary function from the ecluation

.f(x'+ yz + 22,22 -ZxY)=
g (6)

5. (a) Let G be a group and H be a subgroup of G. Prove that
the order of the subgrouP H of G divides the order of
the grouP G. (6%)

(t 2 3 4 5 67 8 9 10)
(b)Writef=1.
\v) "'L\vr 4 l0 5 7 8269 tJ as a Product
\3 ^

of transposition. Find its inverse ahd order. (6'A)

(c) Prove that the set I of all integer is a group with resPect
to the binary operation * defined by a*b=a+b+1 V
a,beI. (6Y,)

6, (a) Find the matching or explain why none exists for the
following graPh :

(6)

5
431r
for the given
(b) Solve the travelling salesperson Problem
cost matrix.

2 3 4
To

From I
,
4
(6)

all sets of 2 vertices whose
(c) For the following graPh find
removal disconnects the graPh
of remaining vertices'

(6)

(700)