BANGALORE
UNIVERSITY
BSc,
IV SEMESTER MATHEMATICS
MODEL
QUESTION PAPER1
Time: 3 Hours Max. Marks: 90
I. Answer any fifteen
of the following
2x15=30
1.
Prove that every subgroup of an abelian
group is normal.
2.
Prove that intersection of two normal
subgroups of a group is also a normal subgroup.
3.
The center Z of a group G is a normal
subgroup of G.
4.
Define a homomorphism of groups.
5.
If G={x+y√2x,yєQ} and f:G→G is defined
by f(x+y√2)=xy√2, show that f is a homomorphism and find its kernel.
6.
Show that f(x,y)= √xy is not
differentiable at (0,0).
7.
Show that f(x,y)=tan ^{1}[y x)
at (1, 1) has limit.
8.
Prove that there is a minimum value at (0,0)
for the fuctions x^{3}+y^{3}3xy.
9.
Show that ^{π}_{о}∫^{2}cos^{10}
0d0=½β (11 1
2, 2)
10. Prove
that ┌(n+1)=n!
11. Prove
that _{о}∫^{∞2 }√xe^{x2} dx=½^{→3/4}
12. Find
the particular integral of y^{11}2y`+4y=e^{x} cos x.
13. Show
that x(2x+3)y^{11}+3(2x+1)y^{1}+2y=(x+1)e^{x} is exact.
14. Verify
the integrability condition for yzlogzdxzxlog zdy+xydz=0.
15. Reduce
x^{2} y^{11}2xy^{1}1+3y=x to a differential equation
with constant coefficients.
16. Find
∟[sin^{3t}].
17. Find
{∟^{1} (s+2) (s1)}
18. Define
convolution theorem for the functions f(t) and g(t).
19. Find
all basic solutions of the system of equations: 3x+2y+z=22, x++y+2z+9.
20. Solve
graphically x+y≤3, xy ≥3, Y ≥0, x≥1, x≤2.
II. Answer any
two of the following
2x5=10
1.
Prove that a subgroup H of a group G is
a normal subgroup of G if and only if the product of two right coses of H in G
is also a right coset of H in G.
2.
Prove that the product of two normal
subgroups of a group is a normal subgroup of the group.
3.
If G and G^{1} are groups and
F:G→G^{1} is a homomorphism with kernel K, prove that K is a normal
subgroup of G.
4.
If G=(Z6, +6), G^{1}=(Z2, +2)
and the function f: G→G^{1} is defined by f(x)=r where r is the
remainder obtained by dividing x by 2, then verify whether f is homomorphism.
If so, find its kernel. Is f an isomorphism?
III. Answer any
three of the following
3x5=15
1.
State and prove Taylor’s theorem for a
function of two variables.
2.
Find Maclaurin’s expansion of log
(1+xy).
3.
Find the stationary points of the
function f(x,y)=x^{3}y^{2}(12xy) satisfying the condition
x>0 and examine their nature.
4.
Show that _{о}∫^{∞}x^{4}(1+x^{5})
dx = 1
(1+x)^{15} 5005
OR
If n is a
positive integer, prove that ┌^{1}(m+1/2)= 1.3.5.......(2n1) √π
2^{n}
5. Show that _{о}∫^{π/2}
√sin0d0. 1 sin0 do + _{о}∫^{π/2 }1 2^{n}
√sino d0^{ }
OR
Evaluate _{о}∫^{∞ }dx
01+x^{4}◦
IV. Answer any
three of the following
3x5=15
1. Solve y^{11} ^{1}2y^{11}+4y=e^{x}
cosx.
2.
Solve x^{3}y^{111}+2x^{2}y+2y^{11}+10
(x+1/x).
3.
Solve d^{2}y (1+4e^{x}) dy + 3e^{2x}y=e^{2}(x+e^{x})
using changing the independent variable method.
dx^{2} dx
4.
Solve dx/dt+3x  y, dy/dt=x+y
5.
Solve dx x^{2}+y^{2}+yz
= dy/x^{2}+y^{2}zx = dz/z(x+y)
V. Answer any
two of the following
2x5=10
1.
Find (i) ∟{2sin st sin 5t/t (ii) ∟1log
s^{2}+1 s(s+1)}
2.
Verify convolution theorem for the
functions (t)=et and g(t)=cost.
3.
Solve y^{11}+2y^{1}=10sin
tsy given y(0)^{zo}, y(0)=1 using Laplace transform method.
VI. Answer any
two of the following: 2x5=10
1.
Find all the basic feasible solutions to
the LPP:
Maximize:
z=2x+3y+4z+7t
Subject to the
constrains: 2x+3yz+4t=8, x2y+6z7t=3 x,y,z,t >0.
2.
A quality engineer wants to determine
the quantity produced per month of products A and B.
Source

Product A

Product B

Available month

Material
Working hours
Assembly man
hours

60
8
3

120
5
4

12,000
600
500

Sale price

Rs. 30

Rs. 40



Find the product
mix that give maximum profit by graphical method.
3.
Using Simplex and method to maximize
f=5x+y+4z subject to x+z<8, y+z<3 x+y+z<5.