﻿ NET/GATE Questions - Gonit Sora

## 28 Jul NET/GATE Questions

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# 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

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.

Managing Editor of the English Section, Gonit Sora and Research Fellow, Faculty of Mathematics, University of Vienna.

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• ##### Pushpanjali singh
Posted at 23:56h, 18 May Reply

Thank you

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