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 frac{Z}{36Z} is

A) 1

B) 2

C) 3

D) 4.

 

2) The number of subfields of F_{2^{27}} (distinct from F_{2^{27}} itself) is

A) 1

B) 2

C) 3

D) 4.

 

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

A) G is an abelian group

B) G is a cyclic group

C) there is a normal proper subgroup

D) there is a subgroup of order 5 which is not normal.

 

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

A) 0

B) 226

C) not definite

D) 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

A) 3

B) 4

C) 5

D) 6.

 

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

A) n=15

B) n=21

C) n=36

D) n=63.

 

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

A) f is always one-one

B) f is always onto

C) f is always a bijection

D) f need be neither one-one nor onto.

 

8) Let R be the polynomial ring Z_2[x] and write the elements of 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 R/(f(x)) is

A) a ring but not an integral domain

B) an integral domain but not a field

C) a finite field of order 4

D) an infinite field.

 

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

A) A is idempotent

B) A is nilpotent

C) A is unitary

D) A is any arbitrary matrix.

 

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

A) there exists a vector v such that T(v)=v

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

C) T must be singular

D) such a linear transformation is not possible.

 

11) Let f:R^4rightarrow R^3 be given by

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

Then

A) f is onto but not one-one

B) f is one-one but not onto

READ:   Hypotheses Non Fingo - 1

C) f is both one-one and onto

D) f is neither one-one nor onto.

 

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

A) yz_x-xz_y=y^2-x^2

B) yz_x+xz_y=y^2-x^2

C) yz_x+xz_y=y^2+x^2

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

 

Gautam Kalita

Research Scholar, Tezpur University,

NET and GATE qualified.

 

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1Comment
  • Pushpanjali singh
    Posted at 23:56h, 18 May

    Thank you