University of Delhi 2010-2nd Year B.Sc PMCS (Physics, Mathematics, Computer Science) Prog MATHEMATICS-II-ALGEBRA DIFFERENTIAL EQUATIONS UNIVERSITY - Question Paper
This question paper contains 4 pnnted pages]
Your Roll No
B.Sc. (Prog.) / II J
MA-202 : MATHEMATICS - II - Algebra & Differential Equations (For Physical Sciences/Applied Physical Sciences)
(Admissions of 2008 and onwards)
Time : 3 Hours Maximum Marks : 112
(Write your Roll No on the top immediately on receipt of this question paper )
Attempt two parts from each question All questions are compulsory
UNIT-I
1. (a) Let G = {(a, b) j a, b e IR, b 0} Define a binary operation O on G by (a, b) 0 (c, d) =
(a + be, bd) Show that (G, 0) is a non-abehaji group. 7j
|
(b) Let H be a subgroup of G. Show that C(H) = {x e G | xh = hx for all h e H} is a subgroup of |
 |
* \ '(c) Is U(0 cyclic group 9 Justify your answer 7f
(a) If H is a subgroup of a cyclic group, show that H is also cyclic l\
(b) If H is a subgroup of G, show that Ha n Hb = <f)
or Ha = Hb. 7
(c) Let G be a group. Let a, b e G [f ab = ba and g.c d. (0(a), 0(b)) = 1, show that
0(ab) = 0(a) 0(b). * l\
(a) Let H be a subgroup of G such that index of
H m G is 2. Show that H is normal in G 7
(b) Let N be a normal subgroup of G such that
/Q\
o(n) m H is a subgroup of G such that 0(H) = n and gc d (m, n) = 1, then show that HcN lj
(c) Let a~12 1 3 5 4 -6
f 1 2 3 4 5 6
1 2 4 3 5 Compute P"1 a P and find its order.
UNIT-n
(a) Solve
(l) (x2y - 2xy2)dx - (x3 - 3x2yjdy = 0 ,
(n) p\x + 2y) + 3p2(x + y) + (y+- 2x) p = 0,
(b) Solve by the method of vanation of parameters:
dx2 dx u2
(c) Prove that the Wronskian of two solutions of the second order homogenous linear differential equation
0 + a,(x) + a2(x)y = 0
where a0, aj, a2 are continuous real valued functions of x defined on (a, b) and aQ(x) 0 for any x m (a, b), is either identically zero or never zero on (a, b). 11
(a) Solve *
dt+ 2dt-2* + 2y = 3e
++2jc+y=4e2t 11
(b) Solve.
yz(l + 4;tz)dx - xz(l + 2xz)dy - xy dz = 0. 11
(c) An 8 lb weight is attached to the lower end
of a coil spring suspended from a fixed
support The weight comes to rest in its
equilibrium position, thereby stretching the
spring 6 m The weight is then pulled down
9 m below its equilibrium position and
released at t = 0. The medium offers a
resistance in pounds numerically equal to
fa . dx 4, where is the instantaneous velocity
in feet per second Determine the displacement of the weight as a function of the time 11
UNIT-III
6 (a) Find the general integral of
(z2 - 2yz - y2)p + (*y + xz)q = xy - xz,
, dz dz
where p = &.q = Sy 11
(b) Find the complete integral of
7? pq xy
u fa * where p = = gy 11
(c) Reduce the equation
cfiz 2cPz dx2 + dy2
to the canonical form 11
5187 4 2,000
|
Attachment: |
| Earning: Approval pending. |
