Here I am giving question paper for Jawaharlal Nehru University Pre PHD mathematics science examination in a file attached with it so you can get it easily.

Which of the following rings is a field?
(a) Z / 57Z
(b) (Z/3Z)x(Z/3Z)
(c) R[x] /(x2 -2)
(d) Q[x]+2+2)

2. For a finite group G
(a) there does not exist any group homomorphism 9: G -* Z
(b) there is a unique group homomorphism 9: G -> Z
(c) there are infinitely many group homomorphisms 9: G -> Z
(d) there are exactly [ G I group homomorphisms cp : G -3 Z

3. Let R be a subring of C containing Q. Suppose it, 4-3 e R. Which of the following is
not necessarily true?
(a) J/neR
(b) n/Je R
(c) [(n+1)2 - (n-1)21/(n15)e R
(d) (,/n2-7)/(-13- +1)eR

4. Let X be a set and let B and C be some fixed subsets of X. If for any subset A of X,
A c C implies A c B, which of the following statements is true?
(a) C * B
(b) Bc C
(c) Cc B
(d) B c C
/91 7 [ P.T.O.

5. Let f : X -> Y be a surjective map. Which of the following is necessarily true? (In the
following, Ids stands for the identity map on the set S)
(a) There exists g : Y -> X such that g o f = Idx
(b) There exists a unique g : Y -> X such that g o f = IdX
(c) There exists g : Y -a X such that f o g = Idy
(d) There exists a unique g : Y -* X such that f o g = Idy

6. Let - be some equivalence relation on R. We are told that under this relation , r - (r + 1)
for every r e R. We can now definitely conclude that
(a) the number of equivalence classes is infinite
the number of equivalence classes is finite
( 7T) 7C
(it -2)- (it +2)

7. Let V be a non-trivial inner product space over R. For vectors v, WE V we say v - w
if (v, w) = 0. Then the relation - is
(a) symmetric but neither reflexive nor transitive
(b) transitive but neither reflexive nor symmetric
(c) an equivalence relation (reflexive , symmetric and transitive)
(d) symmetric and transitive , but not reflexive
8. Let A e SL3 (lIt) be a matrix such that Av = v for some v * 0 in lit3. Which of the
following statements about A is necessarily true?
(a) A is a rotation
(b) A is the identity map
(c) A is diagonalizable
(d) None of the above
9. A box contains 4 blue and 3 green balls. Two balls are drawn out together at random
from the box. What is the probability that the two balls are of different colours?
(a) 5/7
(b) 4/7
(c) 3/7
(d) 2/7

Attached Files

JNU University Pre Ph.d Mathematics Exam Question Papers.PDF (200.2 KB, 85 views)

i want to ph.d. entrance paper