[This question paper contains 6 printed pages']
4345 Your Roll No.
B.A. Programme /II G-r
Paper Code : B-155
MATHEMATICS - Paper II
(Geometry, Differential Equations and Algebra)
Time : 3 Hours 'Maximum Marks : 75
(Ittrrite your Roll No. on the top immetliately
on receipt oJ'this question paper.)
Note;- The maximurn marks printecl on the qucstion
paper are upplicable.for the students o.f the regttlur colleges
(Cat.*'A'). These marks v,ill, hov'ever, be sctt/etl up
proportionately in respect o/' the stuclents o/ SOL ut tlte
time o.f posting o.f awards .fbr compilation o.f result.
All questions are comptrlsorY.
Attempt any two parts Jrom each questir.tn
P.T.O
4345 )
I . (a) Find an equation for hyperbola whose vertices are
(11, 0) and asymptotes are ! = *2x (6yr)
(b) Sketch the hyperbola
4(y-3)2-9(x-2)2=36. (6%)
(c) Find an equation for the parabola which passes through
(3,2),and (2, -.1, ) and whose axis is y:0 and also
sketch.
By drawing the tangent at (3,2) illustrate the property
of reflection of parabola. (6Yz)
2. (a) Show that the lines
L,.: x: -2 + t, y:3 + 2t, z:4 - 5t,
L,:.r:3-t,y:4-2t,2=t
are parallel and find the plane they determine. (6)
(b) (i) Determine whether the vectors ri+ 1'l
ri : (1, -2,1), i: (3, 0,-2),lu: (5, -4, 0)
lie in the same plane
3
4345
Find the area of the triangle
with vertex P(2'0'-3)'
(ii)
Q0,4,5) and R(7 '2'9)'
that has (I'-2' 4) and
(c) Find the equation of the sphere
Also find its
(3,4, -12) as end points of a diameter' (6)
center and radius'
3. (a) Solve the equation
(5xY+ 4Y2+l)dx + (x2 + Zxy)d.y = o. 6Y')
the method of variation
(b) Solve the differential ecluation by
of Parameters'
!+ + 4), - sin- 2x (6Yz)
ax
(c) Solve
(6',/,)
-'#-z**,*2v=v3
partial differential
4. (a) (i) Find the complete integral of the
ecluation:
clz+pzzz:1
P.T.O.
4345 4
(ii) Eliminate the arbitrary function f from the equation:
z:x+y+f(xy). t-1-f1 I
(b) Find whether the equation:
is elliptic, parabolic or hyperbolic. (6)
(c) Find the general integral of partial differential equation
px(z - 2y') : (z - qy)(t - r, - r"t1
Also find the particular integral which passes through
the straight line x: -1 , z:7. (6)
5. (a) Compute the product ( 1425) (243) in Su. Hence or
otherwise, find its order and inverse. (6Yt)
(b) If H and K are subgroup of a group G, show that
H a K is also a subgroup of G. What can you say about
the union of two subgroups? (6yr)
(c) Let G: R -{1}, the set of reals exciuding 1. Define the
binary opreation * on G by a*b=a+b-ab.Provethat
G is group with respect to the *. (6Yr)
4345 5
exists for the
6. (a) Find the matching or explain whY none
following graPh :
(6)
matrlx
(b) Solve the travelling salesperson for the cost
I
, 6
From ) 8 9
(6)
P.T.O.
4345
(c) Does the graph given below have an interval model?
Justifv vour answer.
Also find a maximum independent set for the glven
graph. (6)
(300)
