Recent questions tagged tifrmaths2021

Let $M$ be an $n \times m$ real matrix. Consider the following:- Let $k_1$ be the smallest number such that $M$ can be factorized as $A \cdot B$, where $A$ is an $n \time...

The number of bijective maps $g:\mathbb{N}\rightarrow\mathbb{N}$ such that$$\sum_{n=1}^\infty\frac{g(n)}{n^2}<\infty$$is$0$$1$$2$$\infty$

The dimension of the real vector space$V=\{f:(-1,1)\rightarrow\mathbb{R}|f$ is infinitely differentiable on $(-1,1)$ and $f^{(n)}(0)=0$ for all $n\geq 0\}$is$0$$1$greater...

For a positive integer $n$, let $a_n$ denote the unique positive real root of $x^n+x^{n-1}+\dots+x-1=0.$ Thenthe sequence $\{a_n\}^{\infty}_{n=1}$ is unbounded$\displayst...

Let $X\subseteq \mathbb{R}$ be a subset. Let $\{f_n\}^{\infty}_{n=1}$ be a sequence of functions $f_n:X\rightarrow \mathbb{R}$, that converges uniformly to a function $f:...

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an aritary function. Consider the following assertions:$f$ is continuousThe set $$ \text{Graph}(f)=\{(x,f(x))\in \mathbb{R}^2|x...

Let $\mathcal{C}$ denote the set of colorings of an $8\times 8$ chessboard, where each square is colored either black or white. Let $\thicksim$ denote the equivalence rel...

Let $V$ be a vector space over a field $F$. Consider the following assertions:$V$ is finite dimensionalFor every linear transformation $T:V\rightarrow V$, there exists a ...

$T:\mathbb{C}[x]\rightarrow\mathbb{C}[x]$ be the $\mathbb{C}-$linear transformation defined on the complex vector space $\mathbb{C}[x]$ of one variable complex polynomial...

Let $\mathbb{R}^{\mathbb{N}}$ denote the real vector space of sequences $(x_0,x_1,x_2,\dots)$ of real numbers. Define a linear transformation $T:\mathbb{R}^{\mathbb{N}}\r...

Which one of the following statements is correct?There Exists a $\mathbb{C}-$linear isomorphism $\mathbb{C}^2\rightarrow\mathbb{C}$There exists no $\mathbb{C}-$linear iso...

Consider the following two subgroups $A,B$ of the group $\mathbb{Q}[x]$ of one variable rational polynomials under addition:$$A=\{p(x)\in \mathbb{Z}[x]|p \text{ has degre...

Let $G$ be any finite group of order $2021$. For which of the following positive integers $m$ is the map $G\rightarrow G$, given by $g\mapsto g^m$, a bijection?$43$$45$$4...

How many subgroups does $(\mathbb{Z}/13\mathbb{Z})\times (\mathbb{Z}/13\mathbb{Z})$ have?$13$$16$$4$$25$

Let $f_n:[0,1]\rightarrow \mathbb{R}$ be a continuous function for each positive integer $n$. If $$\displaystyle\lim_{n\rightarrow \infty} \displaystyle \int_0^1 f_n(x)^2...

Let $(X,d)$ be an infinite compact metric space. Then there exists no function $f:X\rightarrow X$, continuous or otherwise, with the property that $d(f(x),f(y))>d(x,y)$ f...

Every infinite closed subset of $\mathbb{R}^n$ is the closure of a countable set.

If $X$ is a compact metric space, there exists a surjective (not necessarily continuous) function $\mathbb{R}\rightarrow X$.

If $X$ is a compact metric space, then every isometry $f:X\rightarrow X$ is surjective.

Define a metric on the set of finite subsets of $\mathbb{Z}$ as ollows:$$d(A,B)=\text{the cardinality of } (A\cup B \backslash (A\cap B)).$$The resulting metric space adm...

Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a monotone increasing (not necessarily continuous) function such that $f(0)>0$ and $f(1)<1$. Then there exists $x\in[0,1]$ such th...

The set$$\{(x,y)\in \mathbb{N}\times\mathbb{N}| x^y \text{ divides } y^x,\:x\neq y,\:xy\neq0,\:x\neq1\}$$is finite.