## 28 Jul 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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#### Latest posts by Gonit Sora (see all)

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

Thank you