University of Hyderabad (UoH) 2007 Ph.D (Maths/Applied Maths) 2008-Entrance - Question Paper

(a) every point is an ordinary point.
(b) every point is a singular point.
(c) x = n¼ is a regular singular point.
(d) x = n¼ is an irregular singular point.

11. If in a group, an element a has order 65, then the order of a25 is
(a) 5. (b) 12. (c) 13. (d) 65.

12. The number of sub¯elds of F227 (distinct from F227 itself) is
(a) 1. (b) 2. (c) 3. (d) 4.

13. The number of Jordan canonical forms for a five £ five matrix with minimal polynomial
(x ¡ 2)2(x ¡ 3) is
(a) 1. (b) 2. (c) 3. (d) 4.

14. The number of degrees of freedom of a rigid cube moving in space is
(a) 1. (b) 3. (c) 5. (d) 6.

15. Let A ½ R be a measurable set. Then
(a) If A is dense then the Lebesgue measure of A is positive.
(b) If the Lebesgue measure of A is zero then A is nowhere dense.
(c) If the Lebesgue measure of A is positive then A contains a nontrivial interval.
(d) All of (a), (b), (c) are false.

16. The formula uxx + x2uyy = 0 is
(a) elliptic.
(b) elliptic everywhere other than on x = 0 axis.
(c) hyperbolic.
(d) hyperbolic everywhere other than on x = 0 axis.

17. The solution of the Laplace formula in spherical polar co-ordinates (r; µ; Á) is
(a) log(r). (b) r. (c) 1=r. (d) r and 1=r.

18. A particle moves in a circular orbit in a force ¯eld F(r) = ¡K=r2, (K > 0). If K
reduces to half its original value then the particle's orbit
(a) is unchanged. (b) becomes parabolic.
(c) becomes elliptic. (d) becomes hyperbolic.

19. Let T : X ! Y be a linear map ranging from normed spaces over C. Then the minimum
requirement ensuring the continuity of T is
(a)X is ¯nite dimensional. (b)X and Y are ¯nite dimensional.
(c) Y = C. (d) Y is ¯nite dimensional.


20. Let H be a Hilbert space. Which of the subsequent is true?

Earning:   Approval pending.