NET/GATE Questions

Tick out the correct answers. More than one answer may be correct for a question. Tick out all.

  1. The number of maximal ideals in \mathbb{Z}/36\mathbb{Z} is

    1. 1

    2. 2

    3. 3

    4. 4.

  2. The number of subfields of \mathbb{F}_{2^{27}} (distinct from \mathbb{F}_{2^{27}} itself) is

    1. 1

    2. 2

    3. 3

    4. 4.

  3. Let G be a group of order 10. Then

    1. G is an abelian group

    2. G is a cyclic group

    3. there is a normal proper subgroup

    4. there is a subgroup of order 5 which is not normal.

  4. Let A be a 227\times 227 matrix with entries in \mathbb{Z}_{227}, such that all its eigenvalues are distinct. Then its trace is

    1. 0

    2. 226

    3. not definite

    4. 227^{227}.

  5. The number of roots of z^9+z^5+8z^3+2^z+1=0 between the circles |z|=1 and |z|=2 are

    1. 3

    2. 4

    3. 5

    4. 6.

  6. Let G be a group of order n. Which of the following conditions imply that G is abelian?

    1. n=15

    2. n=21

    3. n=36

    4. n=63.

  7. Let f:(\mathbb{Q},+)\rightarrow (\mathbb{Q},+) be a non-zero homomorphism. Then

    1. f is always one-one

    2. f is always onto

    3. f is always a bijection

    4. f need be neither one-one nor onto.

  8. Let R be the polynomial ring \mathbb{Z}_2[x] and write the elements of \mathbb{Z}_2 as \{0,1\}.

    Let (f(x)) denote the ideal generated by the element f(x)\in R. If f(x)=x^2+x+1, then the quotient ring \mathbb{R}/(f(x)) is

    1. a ring but not an integral domain

    2. an integral domain but not a field

    3. a finite field of order 4

    4. an infinite field.

  9. Let A be an n\times n matrix with complex entries which is not a diagonal matrix. Then A is diagonalizable if

    1. A is idempotent

    2. A is nilpotent

    3. A is unitary

    4. A is any arbitrary matrix.

  10. T:\mathbb{R}^5\rightarrow \mathbb{R}^5 is a linear transformation with a minimal polynomial (x^2+1)^2. Then

    1. there exists a vector v such that T(v)=v

    2. there exists a vector v such that T(v)=-v

    3. T must be singular

    4. such a linear transformation is not possible.

  11. Let f:\mathbb{R}^4\rightarrow \mathbb{R}^3 be given by

    f((a,b,c,d) )=(3a-2b+c+d,3a-7b-7c+8d,a+b+3c-2d) .


    1. f is onto but not one-one

    2. f is one-one but not onto

    3. f is both one-one and onto

    4. f is neither one-one nor onto.

  12. F(z-xy,x^2+y^2) =0 is the solution of the partial differential equation

    1. yz_x-xz_y=y^2-x^2

    2. yz_x+xz_y=y^2-x^2

    3. yz_x+xz_y=y^2+x^2

    4. yz_x-xz_y=y^2+x^2.

Gautam Kalita

Research Scholar, Tezpur University,

NET and GATE qualified.