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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 $\pi/2$ $\pi/6$ $1/2$ $\infty$

soujanyareddy13 asked in Others Sep 27, 2021

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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 $0$ $1$ $2$ $\infty$

soujanyareddy13 asked in Others Sep 27, 2021

by soujanyareddy13

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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 $1/2$ $1$ $1/4$ $0$

soujanyareddy13 asked in Others Sep 27, 2021

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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

soujanyareddy13 asked in Others Sep 27, 2021

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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

soujanyareddy13 asked in Others Sep 27, 2021

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6

TIFR2021-Maths-A: 6
For a positive integer $n$, let $a_n$ denote the unique positive real root of $x^n+x^{n-1}+\dots+x-1=0.$ Then the sequence $\{a_n\}^{\infty}_{n=1}$ is unbounded $\displaystyle \lim_{n\rightarrow \infty} a_n=0$ $\displaystyle \lim_{n\rightarrow \infty} a_n=1/2$ $\displaystyle \lim_{n\rightarrow \infty} a_n$ does not exist

soujanyareddy13 asked in Others Sep 27, 2021

by soujanyareddy13

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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 $A=\{0,1\}$ $A=\{0,\frac{1}{2},1\}$ $A=\{0,\frac{1}{3},\frac{1}{2},\frac{2}{3},1\}$ $A=[0,1]$

soujanyareddy13 asked in Others Sep 27, 2021

by soujanyareddy13

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8

TIFR2021-Maths-A: 8
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 $f:X\rightarrow\mathbb{R}$. For each positive integer $n$ ... then $D$ has at most $7$ elements If each $D_n$ is uncountable, then $D$ is uncountable None of the other three statements is correct

soujanyareddy13 asked in Others Sep 27, 2021

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9

TIFR2021-Maths-A: 9
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an aritary function. Consider the following assertions: $f$ is continuous The set $ \text{Graph}(f)=\{(x,f(x))\in \mathbb{R}^2|x\in \mathbb{R}\}$ is a connected subset of $\mathbb{R}^2.$ Which one of the following statements is ... $(\text{I})$ $(\text{I})$ does not imply $(\text{II})$, and $(\text{II})$ does not imply $(\text{I})$

soujanyareddy13 asked in Others Sep 27, 2021

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10

TIFR2021-Maths-A: 10
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 relation on $\mathcal{C}$ defined as follows: two colorings are equivalent if and only if one of them can be obtained from the other by a ... $2^{62}+2^{30}+2^{15}$ $2^{64}-2^{32}+2^{16}$ $2^{63}-2^{31}+2^{15}$

soujanyareddy13 asked in Others Sep 27, 2021

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11

TIFR2021-Maths-A: 11
What is the number of surjective maps from the set $\{1,\dots,10\}$ to the set $\{1,2\}$? $90$ $1022$ $98$ $1024$

soujanyareddy13 asked in Others Sep 27, 2021

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12

TIFR2021-Maths-A: 12
Let $V$ be a vector space over a field $F$. Consider the following assertions: $V$ is finite dimensional For every linear transformation $T:V\rightarrow V$, there exists a nonzero polynomial $p(x)\in F[x]$ such that $p(T):V\rightarrow V$ is the zero map. Which one of the ... $(\text{I})$ does not imply $(\text{II})$, and $(\text{II})$ does not imply $(\text{I})$

soujanyareddy13 asked in Others Sep 27, 2021

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13

TIFR2021-Maths-A: 13
$T:\mathbb{C}[x]\rightarrow\mathbb{C}[x]$ be the $\mathbb{C}-$linear transformation defined on the complex vector space $\mathbb{C}[x]$ of one variable complex polynomials by $Tf(x)=f(x+1)$. How many eigenvalues does $T$ have? $1$ finite but more than $1$ countably infinite uncountable

soujanyareddy13 asked in Others Sep 27, 2021

by soujanyareddy13

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14

TIFR2021-Maths-A: 14
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 $T:\mathbb{R}^{\mathbb{N}}\rightarrow\mathbb{R}^{\mathbb{N}}$ ... space $\mathbb{R}^{\mathbb{N}}/T(\mathbb{R}^{\mathbb{N}})$ is infinite dimensional None of the other three statements is correct

soujanyareddy13 asked in Others Sep 27, 2021

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15

TIFR2021-Maths-A: 15
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 isomorphism $\mathbb{C}^2\rightarrow\mathbb{C}$ ... there exists a $\mathbb{Q}-$linear isomorphism $\mathbb{C}^2\rightarrow\mathbb{C}$ None of the other three statements is correct

soujanyareddy13 asked in Others Sep 27, 2021

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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

soujanyareddy13 asked in Others Sep 27, 2021

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17

TIFR2021-Maths-A: 17
Which of the following is a necessary and sufficient condition for two real $3\times 3$ matrices $A$ and $B$ to be similar $($i.e., $PAP^{-1}=B$ for an invertible real $3\times 3$ matrix $P)$? They have the same characteristic polynomial They have the same minimal polynomial They have the same minimal and characteristic polynomials None of the other three conditions

soujanyareddy13 asked in Others Sep 27, 2021

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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

soujanyareddy13 asked in Others Sep 27, 2021

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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

soujanyareddy13 asked in Others Sep 27, 2021

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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? $13$ $16$ $4$ $25$

soujanyareddy13 asked in Others Sep 27, 2021

by soujanyareddy13

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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 $\displaystyle\lim_{n\rightarrow \infty} f_n\left(\frac{1}{2}\right)=0.$

soujanyareddy13 asked in Others Sep 27, 2021

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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 $x\neq y$.

soujanyareddy13 asked in Others Sep 27, 2021

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23

TIFR2021-Maths-B: 3
Every infinite closed subset of $\mathbb{R}^n$ is the closure of a countable set.

soujanyareddy13 asked in Others Sep 27, 2021

by soujanyareddy13

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24

TIFR2021-Maths-B: 4
If $X$ is a compact metric space, there exists a surjective (not necessarily continuous) function $\mathbb{R}\rightarrow X$.

soujanyareddy13 asked in Others Sep 27, 2021

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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.

soujanyareddy13 asked in Others Sep 27, 2021

by soujanyareddy13

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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 $n$.

soujanyareddy13 asked in Others Sep 27, 2021

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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 $f(1)=\text{Id}$.

soujanyareddy13 asked in Others Sep 27, 2021

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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 $f(x)=x$.

soujanyareddy13 asked in Others Sep 27, 2021

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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.

soujanyareddy13 asked in Others Sep 27, 2021

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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.

soujanyareddy13 asked in Others Sep 27, 2021

by soujanyareddy13

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