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Recent questions tagged tifrmaths2021

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1
TIFR2021-Maths-A: 1
For each positive integer $n$, let $s_n=\frac{1}{\sqrt{4n^2-1^2}}+\frac{1}{\sqrt{4n^2-2^2}}+\dots+\frac{1}{\sqrt{4n^2-n^2}}$ Then the $\displaystyle \lim_{n\rightarrow \infty}s_n$ equals $\pi/2$ $\pi/6$ $1/2$ $\infty$
For each positive integer $n$, let$$s_n=\frac{1}{\sqrt{4n^2-1^2}}+\frac{1}{\sqrt{4n^2-2^2}}+\dots+\frac{1}{\sqrt{4n^2-n^2}}$$Then the $\displaystyle \lim_{n\rightarrow \i...
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2
TIFR2021-Maths-A: 2
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 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$$\infty$
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3
TIFR2021-Maths-A: 3
The value of $\displaystyle\lim_{n\rightarrow\infty}\prod_{k=2}^{n}\left(1-\frac{1}{k^2}\right)$ is $1/2$ $1$ $1/4$ $0$
The value of $$\displaystyle\lim_{n\rightarrow\infty}\prod_{k=2}^{n}\left(1-\frac{1}{k^2}\right)$$is$1/2$$1$$1/4$$0$
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4
TIFR2021-Maths-A: 4
The set $S=\{x\in \mathbb{R}|x>0\text{ and } (1+x^2) \tan(2x)=x\}$ is empty nonempty but finite countably infinite uncountable
The set $$S=\{x\in \mathbb{R}|x>0\text{ and } (1+x^2) \tan(2x)=x\}$$isemptynonempty but finitecountably infiniteuncountable
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5
TIFR2021-Maths-A: 5
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 than one, but finite infinite
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$$1$greater...
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7
TIFR2021-Maths-A: 7
Let $A$ be the set of all real numbers $\lambda \in [0,1]$ such that $\displaystyle\lim_{p\rightarrow 0}\frac{\log(\lambda2^p+(1-\lambda)3^p)}{p}=\lambda \log2+(1-\lambda)\log3$ Then $A=\{0,1\}$ $A=\{0,\frac{1}{2},1\}$ $A=\{0,\frac{1}{3},\frac{1}{2},\frac{2}{3},1\}$ $A=[0,1]$
Let $A$ be the set of all real numbers $\lambda \in [0,1]$ such that$$\displaystyle\lim_{p\rightarrow 0}\frac{\log(\lambda2^p+(1-\lambda)3^p)}{p}=\lambda \log2+(1-\lambda...
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2 answers
11
TIFR2021-Maths-A: 11
What is the number of surjective maps from the set $\{1,\dots,10\}$ to the set $\{1,2\}$? $90$ $1022$ $98$ $1024$
What is the number of surjective maps from the set $\{1,\dots,10\}$ to the set $\{1,2\}$?$90$$1022$$98$$1024$
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16
TIFR2021-Maths-A: 16
The matrix $\begin{pmatrix} 4 & -3 & -3\\3 & -2 & -3\\ -1 & 1& 2 \end{pmatrix}$ is diagonalizable nilpotent idempotent none of the other three options
The matrix$$\begin{pmatrix} 4 & -3 & -3\\3 & -2 & -3\\ -1 & 1& 2 \end{pmatrix}$$isdiagonalizablenilpotentidempotentnone of the other three options
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18
TIFR2021-Maths-A: 18
Consider the following two subgroups $A,B$ of the group $\mathbb{Q}[x]$ of one variable rational polynomials under addition: $A=\{p(x)\in \mathbb{Z}[x]|p \text{ has degree at most } 2\}, \text{ and} $ ... $[B:A]$ of $A$ in $B$ equals $1$ $2$ $4$ none of the other three options
Consider the following two subgroups $A,B$ of the group $\mathbb{Q}[x]$ of one variable rational polynomials under addition:$$A=\{p(x)\in \mathbb{Z}[x]|p \text{ has degre...
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19
TIFR2021-Maths-A: 19
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? $43$ $45$ $47$ none of the other three options
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...
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20
TIFR2021-Maths-A: 20
How many subgroups does $(\mathbb{Z}/13\mathbb{Z})\times (\mathbb{Z}/13\mathbb{Z})$ have? $13$ $16$ $4$ $25$
How many subgroups does $(\mathbb{Z}/13\mathbb{Z})\times (\mathbb{Z}/13\mathbb{Z})$ have?$13$$16$$4$$25$
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21
TIFR2021-Maths-B: 1
Let $f_n:[0,1]\rightarrow \mathbb{R}$ be a continuous function for each positive integer $n$. If $\displaystyle\lim_{n\rightarrow \infty} \displaystyle \int_0^1 f_n(x)^2 dx=0,$ then $\displaystyle\lim_{n\rightarrow \infty} f_n\left(\frac{1}{2}\right)=0.$
Let $f_n:[0,1]\rightarrow \mathbb{R}$ be a continuous function for each positive integer $n$. If $$\displaystyle\lim_{n\rightarrow \infty} \displaystyle \int_0^1 f_n(x)^2...
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22
TIFR2021-Maths-B: 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 $d(f(x),f(y))>d(x,y)$ for all $x\neq y$.
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...
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23
TIFR2021-Maths-B: 3
Every infinite closed subset of $\mathbb{R}^n$ is the closure of a countable set.
Every infinite closed subset of $\mathbb{R}^n$ is the closure of a countable set.
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24
TIFR2021-Maths-B: 4
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, there exists a surjective (not necessarily continuous) function $\mathbb{R}\rightarrow X$.
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25
TIFR2021-Maths-B: 5
If $X$ is a compact metric space, then every isometry $f:X\rightarrow X$ is surjective.
If $X$ is a compact metric space, then every isometry $f:X\rightarrow X$ is surjective.
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26
TIFR2021-Maths-B: 6
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 admits an isometry into $\mathbb{R}^n,$ for some positive integer $n$.
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...
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27
TIFR2021-Maths-B: 7
There exists a continuous function $f:[0,1]\rightarrow \{A\in M_2(\mathbb{R})|A^2=A\}$ such that $f(0)=0$ and $f(1)=\text{Id}$.
There exists a continuous function$$f:[0,1]\rightarrow \{A\in M_2(\mathbb{R})|A^2=A\}$$such that $f(0)=0$ and $f(1)=\text{Id}$.
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28
TIFR2021-Maths-B: 8
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 that $f(x)=x$.
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...
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29
TIFR2021-Maths-B: 9
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.
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.
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30
TIFR2021-Maths-B: 10
Suppose a line segment of a fixed length $L$ is given. It is possible to construct a triangle of perimeter $L$, whose angles are $105^{\circ},\: 45^{\circ} \text{ and } 30^{\circ}$, using only a straight edge and a compass.
Suppose a line segment of a fixed length $L$ is given. It is possible to construct a triangle of perimeter $L$, whose angles are $105^{\circ},\: 45^{\circ} \text{ and } 3...
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