# Recent questions tagged tifrmaths2020

1
True/False Question : There exists no monotone function $f:\mathbb{R}\rightarrow \mathbb{R}$ which is discontinuous at every rational number.
2
True/False Question : Let $C\left ( \left [ 0,1 \right ] \right )$ denote the set of continuous real valued functions on $\left [ 0,1 \right ]$, and $\mathbb{R}^{\mathbb{N}}$ the set of all sequences of real numbers. Then there exists an injective map from $C\left ( \left [ 0,1 \right ] \right )$ to $\mathbb{R}^{\mathbb{N}}$ .
3
True/False Question : Let $\left \{ a_{n} \right \}^{\infty }_{n=1}$ be a bounded sequence of positive real numbers. Then: $\underset{n\rightarrow\infty }{lim sup}\:\frac{1}{a_{n}}=\frac{1}{\underset{n\rightarrow \infty }{lim\:inf \:a_{n}}}.$
4
True/False Question : Let $C\left ( \left [ 0,1 \right ] \right )$ denote the metric space of continuous real valued functions on $\left [ 0,1 \right ]$ under the supremum metric - i.e., the distance between $f$ and $g$ in $C\left ( \left [ 0,1 \right ] \right )$ ... in which the coefficient of $x ^{2}$ is $0$. Then $\text{Q}$ is dense in $C\left ( \left [ 0,1 \right ] \right )$ .
5
True/False Question : If $X$ is a metric space such that every continuous function $f:X\rightarrow \mathbb{R}$ is uniformly continuous, then $X$ is compact.
6
True/False Question : Let $X$ be a metric space, and let $C\left ( X \right )$ denote the $\mathbb{R}-$vector space of continuous real valued functions on $X$. Then $X$ is infinite if and only if $dim_{\mathbb{R}}C\left ( X \right )=\infty$.
7
True/False Question : Let $A$ be a countable union of lines in $\mathbb{R}^{3}$. Then $\mathbb{R}^{3} \setminus A$ is connected.
8
True/False Question : An invertible linear map from $\mathbb{R}^{2}$ to itself takes parallel lines to parallel lines.
9
True/False Question : For any matrix $C$ with entries in $\mathbb{C}$, let $m\left ( C \right )$ denote the minimal polynomial of $C$, and $p\left ( C \right )$ its characteristic polynomial. Then for any $n \in\mathbb{N}$, two matrices $A , B \in M_{n}\left ( \mathbb{C} \right )$ are similar if and only if $m\left ( A \right )=m\left ( B \right )$ and $p\left ( A \right ) = p\left ( B \right ).$
10
True/False Question : Let $A,B \in M_{3}\left ( \mathbb{R} \right ).$ Then $det\left ( AB -BA \right )=\frac{tr\left [ \left ( AB -BA \right )^{3} \right ]}{3}.$
11
True/False Question : There exist an integer $r\geq 1$ and a symmetric matrix $A \in M_{r}\left ( \mathbb{R} \right )$ such that for all $n \in \mathbb{N}$, we have: $2^{\sqrt{n}}\leq \left | tr\left ( A^{n} \right )\leq \right |2020 . 2^{\sqrt{n}}.$
12
True/False Question : The polynomial $1+x+\frac{x^{2}}{2!}+ \dots +\frac{x^{101}}{101!}$ is irreducible in $\mathbb{Q}\left [ x \right ]$ .
13
True/False Question : There exists an integer $n>3$ such that the group of units of the ring $\mathbb{Z} /2^{n}\mathbb{Z}$ is cyclic.
14
True/False Question : For every surjective ring homomorphism $\varphi :R\rightarrow S$, we have $\varphi \left ( R^{\times } \right )=S^{\times }$.
15
True/False Question : Let $G$ be a finite group and $P$ a $p$-Sylow subgroup of $G$, where $p$ is a prime number. Then for every subgroup $H$ of $G$, $H\cap P$ is a $p$-Sylow subgroup of $H$.
16
True/False Question : Let $G$ be an abelian group, with identity element $e$. If $\left \{ g \in G \mid g = e\:or\:g\:has\:infinite\:order \right \}$ is a subgroup of $G$, then either all elements of $G\setminus \left \{ e \right \}$ have infinite order, or all elements of $G$ have infinite order.
17
True/False Question : There exists a natural number $n$, with $1< n\leq 10$, such that $x^{n}$ and $x$ are conjugate for every element $x$ of $S_{7}$, the group of permutations of $\left \{ 1, \dots,7\right \}$.
18
True/False Question : Every noncommutative ring has at least $10$ elements.
19
True/False Question : Let $\left \{ a_{n} \right \}^{\infty }_{n=1}$ be a sequence of elements in $\left \{ 0,1 \right \}$ such that for all positive integers $n$, $\sum_{i=n}^{n+9}a_{i}$ is divisible by $3$. Then there exists a positive integer $k$ such that $a_{n+k}=a_{n}$ for all positive integers $n$.
20
True/false Question : The interior of any strip bounded by two parallel lines in $\mathbb{R}^{2}$, of width strictly greater than $1$, contains a point with integer coordinates.
21
Consider the sequences $\left \{ a_{n}\right \}_{n=1}^{\infty }$ and $\left \{ b_{n}\right \}_{n=1}^{\infty }$ defined by $a_{n}=\left ( 2^{n}+3^{n} \right )^{1/n}$ and $b_{n}=\frac{n}{\sum_{i=1}^{n}\frac{1}{a_{i}}}$. What is the limit of $\left \{ b_{n}\right \}_{n=1}^{\infty }$? $2$ $3$ $5$ The limit does not exist
22
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 $2$.
23
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined as: $f\left ( x \right )=\left\{\begin{matrix} x^{2}sin\frac{1}{x}, & if x\neq 0, & and \\0, & if x=0.& \end{matrix}\right.$ Which of the following statements is correct? $f$ is a surjective function. $f$ is bounded. The ... $f$ exists and is continuous on $\mathbb{R}$ . $\left \{ x\in\mathbb{R} |f\left ( x \right )=0\right \}$ is a finite set.
24
Let $\left \{ a_{n}\right \}_{n=1}^{\infty }$ be a strictly increasing bounded sequence of real numbers such that $\lim_{n\rightarrow \infty }a_{n}=A$. Let $f:\left [ a_{1},A \right ]\rightarrow \mathbb{R}$ be a continuous function such that for each positive ... $B$ is: necessarily $0$ at most $1$ possibly greater than $1$, but finite possibly infinite
25
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function that satisfies: $|f\left ( x \right )-f\left ( y \right )|\leq \left | x-y \right |\left | sin\left ( x-y \right ) \right |$, for all $x,y \in\mathbb{R}$. Which ... need not be uniformly continuous. $f$ is uniformly continuous but not necessarily differentiable. $f$ is differentiable, but its derivative may not be continuous. $f$ is constant.
26
Let $C = \{ f:\mathbb{R}\rightarrow \mathbb{R}| f$ is differentiable, and $\lim_{x\rightarrow \infty }\left ( 2f\left ( x \right ) +f{}'\left ( x \right )\right )=0\left \} \right.$. Which of the following statements is correct? For each $L$ with $0\neq L< \infty$, ... $f \in C$ such that $\lim_{x\rightarrow \infty }f\left ( x \right )\frac{1}{2}$
27
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 $\{a_{m}\} \infty_{m=1}$ ... than one limit point converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=\frac{1}{2}$log $2$ converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=0$
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that : $\left | f\left ( x \right ) -f\left ( y \right )\right |\geq log\left ( 1+\left | x-y \right | \right ),$ for all $x,y \in \mathbb{R}$. Then: $f$ is injective but not surjective $f$ is surjective but not injective $f$ is neither injective nor surjective $f$ is bijective
What is the greatest integer less than or equal to $\sum_{n=1}^{9999}\frac{1}{\sqrt[4]{n}}?$ $1332$ $1352$ $1372$ $1392$
Consider the following sentences: $\left ( I \right )$ For every connected subset $Y$ of a metric space $X$, its interior $Y^{\circ}$ is connected. $\left ( II \right )$ For every connected subset $Y$ of a metric space $X$, its boundary $\partial Y$ is connected. Which of the ... $\left ( II \right )$ are both true. $\left ( I \right )$and $\left ( II \right )$ are both true.