28 Jul NET/GATE Questions
Tick out the correct answers. More than one answer may be correct for a question. Tick out all.
 The number of maximal ideals in is
 1
 2
 3
 4.
 1
 The number of subfields of (distinct from itself) is
 1
 2
 3
 4.
 1
 Let be a group of order 10. Then
 is an abelian group
 is a cyclic group
 there is a normal proper subgroup
 there is a subgroup of order 5 which is not normal.
 is an abelian group
 Let be a matrix with entries in such that all its eigenvalues are distinct. Then its trace is
 0
 226
 not definite
 .
 0
 The number of roots of between the circles and are
 3
 4
 5
 6.
 3
 Let be a group of order Which of the following conditions imply that is abelian?



 .

 Let be a nonzero homomorphism. Then
 is always oneone
 is always onto
 is always a bijection
 need be neither oneone nor onto.
 is always oneone
 Let be the polynomial ring and write the elements of as .
Let denote the ideal generated by the element . If , then the quotient ring is
 a ring but not an integral domain
 an integral domain but not a field
 a finite field of order 4
 an infinite field.
 a ring but not an integral domain
 Let be an matrix with complex entries which is not a diagonal matrix. Then is diagonalizable if
 is idempotent
 is nilpotent
 is unitary
 is any arbitrary matrix.
 is idempotent
 is a linear transformation with a minimal polynomial . Then
 there exists a vector such that
 there exists a vector such that
 must be singular
 such a linear transformation is not possible.
 there exists a vector such that
 Let be given by
.
Then
 is onto but not oneone
 is oneone but not onto
 is both oneone and onto
 is neither oneone nor onto.
 is onto but not oneone
 is the solution of the partial differential equation



 .

Gautam Kalita
Research Scholar, Tezpur University,
NET and GATE qualified.