# Recent activity by soujanyareddy13

1
Denote by the set all $n \times n$ complex matrices $A$ ($n\geq 2$ a natural number) having the property that $4$ is the only eigenvalue of $A$. Consider the following four statements. $\left ( A-4I \right )^{n}=0,$ $A^{n}=4^{n}I,$ $\left ( A^{2}-5A+4I \right )^{n}=0,$ $A^{n}=4nI.$ How many of the above statements are true for all $A \in$ ? $0$ $1$ $2$ $3$
2
Let $A$ be the set of all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following two properties: $f$ has derivatives of all orders, and for all $x,y \in \mathbb{R}$, $f\left ( x+y \right )-f\left ( y-x \right )=2x{f}'\left ( y \right ).$ ... less than or equal to $2$. There exists $f \in A$ which is not a polynomial. There exists $f \in A$ which is a polynomial of degree $4$.
3
Suppose $p$ is a degree $3$ polynomial such that $p\left ( 0 \right )=1,p\left ( 1 \right )=2,$ and $p\left ( 2 \right )=5$. Which of the following numbers cannot equal $p\left ( 3 \right )$ ? $0$ $2$ $6$ $10$.
4
How many zeroes does the function $f\left ( x \right )=e^{x}-3x^{2}$ have in $\mathbb{R}$? $0$ $1$ $2$ $3$
5
The set of real numbers in the open interval $(0,1)$ which have more than one decimal expansion is empty. non-empty but finite. countable infinite. uncountable.
6
The limit $\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$ is equal to $1$ $0$ $+\infty$ $1/2$
7
A stick of length $1$ is broken into two pieces by cutting at a randomly chosen point. What is the expected length of the smaller piece? $1/8$ $1/4$ $1/e$ $1/\pi$
8
Let $C^{\infty }\left ( 0,1 \right )$ stand for the set of all real-valued functions on $\left ( 0,1 \right )$ that have derivatives of all orders. Then the map $C^{\infty }\left ( 0,1 \right )\rightarrow C^{\infty }\left ( 0,1 \right )$ given by $f \mapsto f+\frac{df}{dx}$ is injective but not surjective surjective but not injective neither injective nor surjective both injective and surjective
9
Consider the different ways to colour the faces of a cube with six given colours, such that each face is given exactly one colour and all the six colours are used. Define two such colouring schemes to be equivalent if the resulting configurations can be obtained from one another by a rotation of the cube. Then the number of inequivalent colouring schemes is $15$ $24$ $30$ $48$
10
Let $X\subset \mathbb{R}^{2}$ be the subset $X=\left \{ \left ( x,y \right ) \left | x=0, \right |y \mid \leq 1\right \}\cup \left \{ \left ( x,y \right ) \mid 0 < x \leq 1, y=sin \frac{1}{x}\right \}.$ Consider the following statements: $X$ is compact $X$ is connected $X$ is path connected. How many of the statements (i)-(iii) is /are true? $0$ $1$ $2$ $3$
11
The number of ring homomorphisms from $\mathbb{Z}\left [ x,y \right ]$ to $\mathbb{F}_{2}\left [ x \right ]/\left ( x^{3}+x^{2}+x+1 \right )$ equals $2^{6}$ $2^{18}$ $1$ $2^{9}$
12
Consider functions $f:\mathbb{R}\rightarrow \mathbb{R}$ with the property that $\left | f\left ( x \right )-f\left ( y \right ) \right |\leq 4321\left | x-y \right |$ for all real numbers $x,y$. Then which one of the following statement is true? $f$ is always ... $\underset{n\rightarrow\infty }{lim}\left | \frac{f\left ( x_{n} \right )}{x_{n}} \right |\leq 10000$
13
Let the sequence $\left \{ x_{n} \right \}_{n\rightarrow 1}^{\infty }$ be defined by $x1=\sqrt{2}$ and $x_{n+1}=\left ( \sqrt{2} \right )^{x_{n}}$ for $n\geq 1$ ... increasing nor monotonically decreasing $\underset{n\rightarrow \infty }{lim}\:x_{n}$ does not exist $\underset{n\rightarrow \infty }{lim}\:x_{n}=\infty$
14
Let $\left \{ f_{n} \right \}_{n=1}^{\infty}$ be a sequence of functions from $\mathbb{R}$ to $\mathbb{R}$, defined by $f_{n}\left ( x \right )=\frac{1}{n}\:exp\left ( -n^{2} x^{2}\right ).$ Then which one of the ... converges pointwise but not uniformly on any interval containing the origin $\left \{{f}'_{n} \right \}$ converges pointwise but not uniformly on any interval containing the origin
15
Let $f$ be a continuous function on $\left [ 0,1 \right ]$. Then the limit $\underset{n\rightarrow \infty }{lim}\int ^{1}_{0}nx^{n} f\left ( x \right )dx$ is equal to $f(0)$ $f(1)$ $\underset{x\in\left [ 0,1 \right ]}{sup} f\left ( x\right )$ The limit need not exist
16
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 $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
17
Let $f:\left ( 0,\infty \right )\rightarrow \mathbb{R}$ be defined by $f\left ( x \right )=\frac{sin\left (x ^{3} \right )}{x}$ . Then $f$ is bounded and uniformly continuous bounded but not uniformly continuous not bounded but uniformly continuous not bounded and not uniformly continuous
18
Consider the following two statements: $(E)$ Continuous function on $[1,2]$ can be approximated uniformly by a sequence of even polynomials (i.e., polynomials $p\left ( x \right )\in\mathbb{R}\left [ x \right ]$ such that $p\left ( -x \right )=p\left ( x \right )$). $(O)$ Continuous ... both false $(E)$ and $(O)$ are both true $(E)$ is true but $(O)$ is false $(E)$ is false but $(O)$ is true
19
Let $A$ be an $n \times n$ matrix with rank $k$. Consider the following statements: If $A$ has real entries, then $AA^{t}$ necessarily has rank $k$ If $A$ has complex entries, then $AA^{t}$ necessarily has rank $k$. Then (i) and (ii) are true (i) and (ii) are false (i) is true and (ii) is false (i) is false and (ii) is true
20
Let $V$ be an n-dimensional vector space and let $T:V\rightarrow V$ be a linear transformation such that $Rank\:T \leq Rank\:T^{3}$. Then which one of the following statements is necessarily true? Null space$(T)$ = Range$(T)$ Null space$(T)$ $\cap$ Range$(T)$={$0$} There exists a nonzero subspace $W$ of $V$ such that Null space$(T)$ $\cap$ range$(T)$=$W$ Null space$(T)$ $\subseteq$ Range$(T)$
21
Consider differentiable functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have: $f\left ( b \right )-f\left ( a \right )=\left ( b-a \right ){f}'\left ( \frac{a+b}{2} \right )$ Then which one of the following sentence is true? Every ... $a,b \in \mathbb{R}$
22
Define a function: $f\left ( x \right )=\left\{\begin{matrix} x +x^{2} cos\left ( \frac{\pi}{x} \right ), & x\neq 0\\ 0,& x=0. \end{matrix}\right.$ Consider the following statements: ${f}'\left ( 0 \right )$ exists and is equal to $1$ $f$ is not increasing in any ... $f$ is increasing on $\mathbb{R}.$ How many of the above statements is/are true? $0$ $1$ $2$ $3$
23
For $n\geq 1$, the sequence $\left \{ x_{n} \right \}^{\infty }_{n=1},$ where: $x_{n}=1+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n}}-2\sqrt{n}$ is decreasing increasing constant oscillating
24
The following sum of numbers (expressed in decimal notation) $1+11+111+\cdots +\underset{n}{\underbrace{11\dots1}}$ is equal to $\left ( 10^{n+1}-10-9n \right )/81$ $\left ( 10^{n+1}-10+9n \right )/81$ $\left ( 10^{n+1}-10-n \right )/81$ $\left ( 10^{n+1}-10+n \right )/81$
25
True/False Question : Let $v:\mathbb{R}\rightarrow \mathbb{R}^{2}$ be $C^{\infty }$ (i.e., has derivatives of all orders). Then there exists $t_{0}\in \left ( 0,1 \right )$ such that $v\left ( 1 \right )-v\left ( 0 \right )$ is a scalar multiple of $\frac{\mathrm{dv} }{\mathrm{dt} }\mid _{t=t_{0}}$.
26
True/False Question : There exists a uniformly continuous function $f:\left ( 0,\infty \right )\rightarrow \left ( 0,\infty \right )$ such that $\sum_{n=1 }^{\infty }\frac{1}{f\left ( n \right )}$ converges.
True/False Question : Each solution of the differential equation ${y}''+e^{x}y=0$ remains bounded as $x\rightarrow \infty$.
True/False Question : If every proper subgroup of an infinite group $G$ is cyclic, then $G$ is cyclic.
True/False Question : Let $H,N$ be subgroups of a finite group $G$, with $N$ a normal subgroup of $G$. If the orders of $G/N$ and $H$ are relatively prime, then $H$ is necessarily contained in $N$.
True/False Question : Given any group $G$ of order $12$, and any $n$ that divides $12$, there exists a subgroup $H$ of $G$ of order $n$.