University of Hyderabad (UoH) 2005 Ph.D ENTRANCE , , Mathematics/ Applied Mathematics - Question Paper

e. none of the above.
7. Let, for every x two [0; 1), x = 0:x1x2x3:::: be the decimal expansion of x
not eventually all 9’s. describe f : [0; 1) ! IR by f(x) = x1, the 1st digit
in the expansion. Then R1
0 f(x) dx =
a. 41
2 b. 10 c. 0 d. one e. does not exist.
8. The function f(x; y) = ( xy
x2+y2 if(x; y) 6= (0; 0)
0 at (0; 0)
is
a. continuous at (0,0) but partial derivatives do not exist at (0,0).
b. continuous at (0,0) and partial derivatives exist at (0,0).
c. discontinuous at (0,0) and partial derivatives do not exist at (0,0).
d. discontinuous at (0,0) but partial derivatives exist at (0,0).
e. none of the above.
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9. If IR is provided the cofinite topology then
a. IR is compact and connected.
b. IR is connected but not compact.
c. IR is compact but not connected.
d. IR is neither compact nor connected.
e. IR has a countable base.
10. Let T = fÁ; IRgSf(x;1) j x two IRg. Then in the topological space (IR; T )
the set of integers ZZ is
a. an open set.
b. a closed set.
c. a dense set.
d. an uncountable set.
e. none of the above.
11. The number of homomorphisms from C2 £ C2 ! C2 is (Cn is the cyclic
group of order n)
a. five b. four c. three d. two e. 1
12. The number of zero-divisors in the ring of integers modulo 24 is
a. 20 b. 15 c. 12 d. eight e. none of the above.
13. If R is a unique factorization domain then
a. R is a Euclidean domain.
b. R is a principal ideal domain.
c. R[X] is a unique factorization domain.
d. R[X] is a principal ideal domain.
e. none of the above.
14. The number of proper subfields of F32 is
a. 16 b. eight c. four d. two e. 1.
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15. An example of a function on IR whose graph does not intersect the x-axis
is
a. f(x) = x3 ¡ 3x + 2
b. f(x) = x4 + x2 + 1
c. f(x) = x7 ¡ 2
d. f(x) = x11 ¡ x
2 + 1
e. none of the above.
16. I. Every Lebesgue measurable function on IR is continuous.
II. Every Lebesgue measurable subset of IR is Borel.

Earning:   Approval pending.