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1281
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
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...
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120
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Sep 27, 2021
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1282
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
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...
soujanyareddy13
258
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soujanyareddy13
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Sep 27, 2021
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1283
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
soujanyareddy13
222
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Sep 27, 2021
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1284
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
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...
soujanyareddy13
137
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Sep 27, 2021
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1285
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:$$A=\{p(x)\in \mathbb{Z}[x]|p \text{ has degre...
soujanyareddy13
115
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Sep 27, 2021
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1286
TIFR2021-Maths-A: 19
Let $G$ be any finite group of order $2021$. For which of the following positive integers $m$ is the map $G\rightarrow G$, given by $g\mapsto g^m$, a bijection? $43$ $45$ $47$ none of the other three options
Let $G$ be any finite group of order $2021$. For which of the following positive integers $m$ is the map $G\rightarrow G$, given by $g\mapsto g^m$, a bijection?$43$$45$$4...
soujanyareddy13
217
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Sep 27, 2021
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1287
TIFR2021-Maths-A: 20
How many subgroups does $(\mathbb{Z}/13\mathbb{Z})\times (\mathbb{Z}/13\mathbb{Z})$ have? $13$ $16$ $4$ $25$
How many subgroups does $(\mathbb{Z}/13\mathbb{Z})\times (\mathbb{Z}/13\mathbb{Z})$ have?$13$$16$$4$$25$
soujanyareddy13
137
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Sep 27, 2021
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1288
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...
soujanyareddy13
160
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Sep 27, 2021
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1289
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...
soujanyareddy13
131
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Sep 27, 2021
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1290
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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149
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Sep 27, 2021
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1291
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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141
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Sep 27, 2021
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1292
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.
soujanyareddy13
116
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Sep 27, 2021
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1293
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)).$$The resulting metric space adm...
soujanyareddy13
96
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Sep 27, 2021
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1294
TIFR2021-Maths-B: 7
There exists a continuous function $f:[0,1]\rightarrow \{A\in M_2(\mathbb{R})|A^2=A\}$ such that $f(0)=0$ and $f(1)=\text{Id}$.
There exists a continuous function$$f:[0,1]\rightarrow \{A\in M_2(\mathbb{R})|A^2=A\}$$such that $f(0)=0$ and $f(1)=\text{Id}$.
soujanyareddy13
104
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Sep 27, 2021
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1295
TIFR2021-Maths-B: 8
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a monotone increasing (not necessarily continuous) function such that $f(0)>0$ and $f(1)<1$. Then there exists $x\in[0,1]$ such that $f(x)=x$.
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a monotone increasing (not necessarily continuous) function such that $f(0)>0$ and $f(1)<1$. Then there exists $x\in[0,1]$ such th...
soujanyareddy13
105
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Sep 27, 2021
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1296
TIFR2021-Maths-B: 9
The set $\{(x,y)\in \mathbb{N}\times\mathbb{N}| x^y \text{ divides } y^x,\:x\neq y,\:xy\neq0,\:x\neq1\}$ is finite.
The set$$\{(x,y)\in \mathbb{N}\times\mathbb{N}| x^y \text{ divides } y^x,\:x\neq y,\:xy\neq0,\:x\neq1\}$$is finite.
soujanyareddy13
91
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Sep 27, 2021
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1297
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...
soujanyareddy13
124
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soujanyareddy13
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Sep 27, 2021
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1298
TIFR2021-Maths-B: 11
The real vector space $M_n(\mathbb{R})$ cannot be spanned by nilpotent matrices, for any positive integer $n$.
The real vector space $M_n(\mathbb{R})$ cannot be spanned by nilpotent matrices, for any positive integer $n$.
soujanyareddy13
144
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Sep 27, 2021
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1299
TIFR2021-Maths-B: 12
Let $S \subseteq M_n(\mathbb{R})$ be a nonempty finite set closed under matrix multiplication. Then there exists $A\in S$ such that the trace of $A$ is an integer.
Let $S \subseteq M_n(\mathbb{R})$ be a nonempty finite set closed under matrix multiplication. Then there exists $A\in S$ such that the trace of $A$ is an integer.
soujanyareddy13
113
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Sep 27, 2021
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1300
TIFR2021-Maths-B: 13
Given a linear transformation $T:\mathbb{Q}^4\rightarrow\mathbb{Q}^4$, there exists a nonzero proper subspace $V$ of $\mathbb{Q}^4$ such that $T(V)\underline\subset V.$
Given a linear transformation $T:\mathbb{Q}^4\rightarrow\mathbb{Q}^4$, there exists a nonzero proper subspace $V$ of $\mathbb{Q}^4$ such that $T(V)\underline\subset V.$
soujanyareddy13
82
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Sep 27, 2021
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