## 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 $\mathbb{Z}/36\mathbb{Z}$ 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

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

Then

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.

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

Thank you