# Recent questions and answers in Exam Queries

1
The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.
2
Let $V$ be the vector space of consisting polynomials of $\mathbb{R}\left [ t \right ]$ deg$\leq 2$. The map $T:V\rightarrow V$ sending $f\left ( t \right )$ to $f\left ( t \right )+{f}'\left ( t \right )$ is invertible.
3
The automorphism group $Aut\left ( \mathbb{Z}/2\times \mathbb{Z}/2 \right )$ is abelian
4
The polynomials $\left ( t-1 \right )\left ( t-2 \right ),\left ( t-2 \right )\left ( t-3 \right ),\left ( t-3 \right )\left ( t-4 \right ),\left ( t-4 \right )\left ( t-6 \right )\in \mathbb{R}\left [ t \right ]$ are linearly independent.
5
The matrix $\begin{pmatrix} 1 & \pi &3 \\ 0& 2&4 \\ 0&0 &3 \end{pmatrix}$ is diagonalisable
6
If $z_{1},z_{2},z_{3},z_{4}\in \mathbb{C}$ satisfy $z_{1}+z_{2}+z_{3}+z_{4}=0$ and $\left | z_{1} \right |^{2}+\left | z_{2} \right |^{2}+\left | z_{3} \right |^{2}+\left | z_{4} \right |^{2}=1$, then the least value of $\left | z_{1} -z_{2}\right |^{2}+\left | z_{1} -z_{4}\right |^{2}+\left | z_{2}-z_{3} \right |^{2}+\left | z_{3} -z_{4}\right |^{2}$ is $2$.
1 vote
7
$f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.
8
True/False Question : The matrices $\begin{pmatrix} x &0 \\ 0 & y \end{pmatrix} and \begin{pmatrix} x &1 \\ 0 & y \end{pmatrix}, x\neq y,$ for any $x,y \in \mathbb{R}$ are conjugate in $M_{2}\left ( \mathbb{R} \right )$ .
9
True/False Question : Let $y\left ( t \right )$ be a real valued function defined on the real line such that ${y}'=y \left ( 1-y \right )$, with $y\left ( 0\right ) \in \left [ 0,1 \right ]$. Then $\lim_{t\rightarrow \infty }y\left ( t \right )=1$ .
10
True/False Question : In the vector space $\left \{ f \mid f : \left [ 0,1 \right ] \rightarrow \mathbb{R}\right \}$ of real-valued function on the closed interval $\left [ 0,1 \right ]$, the set $S=\left \{ sin\left ( x \right ) , cos\left ( x \right ),tan\left ( x \right )\right \}$ is linearly independent.
11
Let $S=\left \{ x \in\mathbb{R} \mid x=Trace\:(A) \:for\:some\:A \in M_{4} (\mathbb{R}) such\:that\:A^{2}=A \right\}.$ Then which of the following describes $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
12
True/False Question : If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
13
What are the last $3$ digits of $2^{2017}$? $072$ $472$ $512$ $912.$
14
Consider the set of continuous functions $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ that satisfy: $\int_{0}^{1}f\left ( x \right )\left ( 1-f\left ( x \right ) \right )dx=\frac{1}{4}.$ Then the cardinality of this set is: $0$. $1$. $2$. more than $2$.
15
If a rectangle $R:=\left \{ \left ( x,y \right ) \in \mathbb{R}^{2}\mid A\leq x\leq B,C\leq y\leq D\right \}$ can be covered (allowing overlaps ) by $25$ discs of radius $1$ then it can also be covered by $101$ dics of radius $\frac{1}{2}.$
16
Given any integer $n\geq 2$, we can always finds an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,\dots,m+n$are composite.
17
The $10 \times 10$ matrix $\begin{pmatrix} v_{1}w_{1} & \cdots&v_{1}w_{10} \\ v_{2}w_{2}& \cdots & v_{2}w_{10}\\ v_{10}w_{1}&\cdots & v_{10}w_{10} \end{pmatrix}$has rank $2$, where $v_{i},w_{i}\in \mathbb{C}.$
18
If every continuous function on $X\subset \mathbb{R}^{2}$ is bounded, then $X$ is compact.
19
The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
20
Consider the differential equations (with $y$ is a function of $x$) $\begin{matrix} \frac{dy}{dx} & = & y\\ y\left ( 0 \right ) & = & 0 \end{matrix}$ $\begin{matrix} \frac{dy}{dx} & = & \left | y \right |^{\frac{1}{3}}\\ y\left ( 0 \right ) & = & 0. \end{matrix}$ Then $(1)$ has infinitely many solutions but $(2)$ has finite number of solutions.
21
Let $f : \mathbb{R}^{2}\rightarrow \mathbb{R}$ be a continuous function. Then the derivative $\frac{\partial ^{2}f}{\partial x\partial y}$ can exist without $\frac{\partial f}{\partial x}$ existing.
22
If $f$ is continuous on $\left [ 0,1 \right ]$ and if $\int_{0}^{1}f\left ( x \right )x^{n}dx=0$ for $n=1,2,3,\cdots .$ .Then $\int_{0}^{1}f^{2}\left ( x \right )dx=0.$
23
Suppose that $f \in \mathfrak{L}^{2} \left ( \mathbb{R} \right )$. Then $f \in \mathfrak{L}^{1} \left ( \mathbb{R} \right )$.
24
The Integral $\int_{-\infty }^{+\infty }\frac{e^{-x}}{1+x^{2}}\:dx$ is convergent.
25
If $A\subset \mathbb{R}$ and open then the interior of the closure $\overset{-0}{A}$is $A$.
26
If $f \in C^{\infty }$ and $f^{\left ( k \right )}\left ( 0 \right )=0$ for all integer $k\geq 0$, then $f\equiv 0$.
27
Let $f:\left [ 0,1 \right ]\rightarrow \left [ 0,1 \right ]$be continuous then $f$ assumes the value $\int_{0}^{1}f^{2}\left ( t \right )dt$ somewhere in $\left [ 0,1 \right ]$.
28
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function such that $\underset{h\rightarrow 0}{lim }\:\frac{f\left ( x+h \right )-f\left ( x-h \right )}{h}$ exists for all $x \in \mathbb{R}$. Then $f$ is differentiable in $\mathbb{R}.$
29
The functions $f\left ( x \right )=x\left | x \right |$ and $x\left | sin\:x \right |$ are not differentiable at $x=0$.
30
The composition of two uniformly continuous functions need not always be uniformly continuous.
31
$A\in M_{2}\left ( \mathbb{C} \right )$and $A$ is nilpotent then $A^{2}=0$.
32
Let $P$ be an $n \times n$ matrix whose row sums equal $1$. Then for any positive integer $m$ the row sums of the matrix $p^{m}$ equal $1$.
33
There is a non trivial group homomorphism from $C$ to $R$.
34
If the equation $xyz=1$ holds in a group $G$, does it follow that $yxz=1$.
35
Any $3 \times3$ and $5\times5$ skew-symmetric matrices have always zero determinants.
36
The rank of the matrix $\begin{bmatrix} 11 &12 &13 &14 \\ 21& 22 &23 & 24\\ 31& 32 &33 &34 \\ 41&42 & 43 & 44 \end{bmatrix}$ is $2$.
37
The number $2$ is a prime in $\mathbb{Z}\left [ i \right ]$
True/False Question : Let $A,B \in M_{3}\left ( \mathbb{R} \right ).$ Then $det\left ( AB -BA \right )=\frac{tr\left [ \left ( AB -BA \right )^{3} \right ]}{3}.$
Which of the following statements is correct for every linear transformation $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ such that $T^{3}-T^{2}-T+I=0$? $T$ is invertible as well as diagonalizable. $T$ is invertible, but not necessearily diagonalizable. $T$ is diagonalizable, but not necessary invertible. None of the other three statements.
Let $f\left ( x \right )=\frac{log\left ( 2+x \right )}{\sqrt{1+x}}$ for $x\geq 0$, and $a_{m}=\frac{1}{m}\int_{0}^{m}f\left ( t \right )dt$ for every positive integer $m$. Then the sequence diverges to.$+\infty$ has more than one limit point. converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=\frac{1}{2}$log $2$. converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=0$.