# Questions by soujanyareddy13

1 vote
1
$f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.
2
The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.
3
The matrix $\begin{pmatrix} 1 & \pi &3 \\ 0& 2&4 \\ 0&0 &3 \end{pmatrix}$ is diagonalisable
4
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 $\frac{1}{2}.$
5
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.
6
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 $v_{i},w_{i}\in \mathbb{C}.$
7
If every continuous function on $X\subset \mathbb{R}^{2}$ is bounded, then $X$ is compact.
8
The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
9
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 $2$.
10
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.
11
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.
12
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 $\int_{0}^{1}f^{2}\left ( x \right )dx=0.$
13
Suppose that $f \in \mathfrak{L}^{2} \left ( \mathbb{R} \right )$. Then $f \in \mathfrak{L}^{1} \left ( \mathbb{R} \right )$.
14
The Integral $\int_{-\infty }^{+\infty }\frac{e^{-x}}{1+x^{2}}\:dx$ is convergent.
15
If $A\subset \mathbb{R}$ and open then the interior of the closure $\overset{-0}{A}$is $A$.
16
If $f \in C^{\infty }$ and $f^{\left ( k \right )}\left ( 0 \right )=0$ for all integer $k\geq 0$, then $f\equiv 0$.
17
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 $\left [ 0,1 \right ]$.
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 $\mathbb{R}.$
The functions $f\left ( x \right )=x\left | x \right |$ and $x\left | sin\:x \right |$ are not differentiable at $x=0$.