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TIFR Mathematics 2024 | Part B | Question: 1
If $\text{G}$ is a group of order $361$, then $\text{G}$ has a normal subgroup $\text{H}$ such that $H \cong G / H$.

admin asked in Others Jan 19

by admin

67 views

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1 answer

2

TIFR Mathematics 2024 | Part B | Question: 2
There exists a metric space $\text{X}$ such that the number of open subsets of $\text{X}$ is exactly $2024$.

admin asked in Others Jan 19

by admin

91 views

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3

TIFR Mathematics 2024 | Part B | Question: 3
The function $d: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ given by $d(x, y)=\left|e^{x}-e^{y}\right|$ defines a metric on $\mathbb{R}$, and $(\mathbb{R}, d)$ is a complete metric space.

admin asked in Others Jan 19

by admin

51 views

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4

TIFR Mathematics 2024 | Part B | Question: 4
Let $n$ be a positive integer, and $A$ an $n \times n$ matrix over $\mathbb{R}$ such that $A^{3}=\mathrm{Id}$. Then $A$ is diagonalizable in $\mathrm{M}_{n}(\mathbb{R})$, i.e., there exists $P \in \mathrm{M}_{n}(\mathbb{R})$ such that $P$ is invertible and $P A P^{-1}$ is a diagonal matrix.

admin asked in Others Jan 19

by admin

54 views

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TIFR Mathematics 2024 | Part B | Question: 5
If $A \in \mathrm{M}_{n}(\mathbb{Q})$ is such that the characteristic polynomial of $A$ is irreducible over $\mathbb{Q}$, then $A$ is diagonalizable in $\mathrm{M}_{n}(\mathbb{C})$, i.e., there exists $P \in \mathrm{M}_{n}(\mathbb{C})$ such that $P$ is invertible and $P A P^{-1}$ is a diagonal matrix.

admin asked in Others Jan 19

by admin

62 views

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TIFR Mathematics 2024 | Part B | Question: 6
The complement of any countable union of lines in $\mathbb{R}^{3}$ is path connected.

admin asked in Others Jan 19

by admin

40 views

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7

TIFR Mathematics 2024 | Part B | Question: 7
The subsets $\left\{(x, y) \in \mathbb{R}^{2} \mid\left(y^{2}-x\right)\left(y^{2}-x-1\right)=0\right\}$ and $\left\{(x, y) \in \mathbb{R}^{2} \mid y^{2}-x^{2}=1\right\}$ of $\mathbb{R}^{2}$ (with the induced metric) are homeomorphic.

admin asked in Others Jan 19

by admin

50 views

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8

TIFR Mathematics 2024 | Part B | Question: 8
$\mathbb{Q} \cap[0,1]$ is a compact subset of $\mathbb{Q}$.

admin asked in Others Jan 19

by admin

42 views

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TIFR Mathematics 2024 | Part B | Question: 9
Suppose $f: X \rightarrow Y$ is a function between metric spaces, such that whenever a sequence $\left\{x_{n}\right\}$ converges to $x$ in $X$, the sequence $\left\{f\left(x_{n}\right)\right\}$ converges in $Y$ (but it is not given that the limit of $\left\{f\left(x_{n}\right)\right\}$ is $\left.f(x)\right)$. Then $f$ is continuous.

admin asked in Others Jan 19

by admin

51 views

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10

TIFR Mathematics 2024 | Part B | Question: 10
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable, and assume that $\left|f^{\prime}(x)\right| \geq 1$ for all $x \in \mathbb{R}$. Then for each compact set $C \subset \mathbb{R}$, the set $f^{-1}(C)$ is compact.

admin asked in Others Jan 19

by admin

52 views

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11

TIFR Mathematics 2024 | Part B | Question: 11
There exists a function $f:[0,1] \rightarrow \mathbb{R}$, which is not Riemann integrable and satisfies \[ \sum_{i=1}^{n}\left|f\left(t_{i}\right)-f\left(t_{i-1}\right)\right|^{2}<1 \]

admin asked in Others Jan 19

by admin

46 views

1 vote

1 answer

12

MADE EASY TEST SERIES
Unique not null is equivalent to primary key. Relational Algebra and SQL has same expressive power. Which of the above statements are False?

Sajal Mallick asked in Databases Nov 5, 2023

by Sajal Mallick

349 views

2 votes

1 answer

13

TIFR Mathematics 2023 | Part B | Question: 1
Answer whether the following statements are True or False. Let $\alpha$ be a positive real number, and let $f:(0,1) \rightarrow \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq$ $|x-y|^{\alpha}$ for all $x, y \in(0,1)$. Then $f$ can be extended to a continuous function $[0,1] \rightarrow \mathbb{R}$.

admin asked in Others Mar 14, 2023

by admin

541 views

2 votes

1 answer

14

TIFR Mathematics 2023 | Part B | Question: 2
Answer whether the following statements are True or False. Suppose $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions such that $f^{2}+g^{2}$ is uniformly continuous. Then at least one of the two functions $f$ and $g$ is uniformly continuous.

admin asked in Others Mar 14, 2023

by admin

337 views

2 votes

0 answers

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TIFR Mathematics 2023 | Part B | Question: 3
Answer whether the following statements are True or False. Let $\left\{f_{n}\right\}_{n}$ be a sequence of (not necessarily continuous) functions from $[0,1]$ to $\mathbb{R}$. Let $f:[0,1] \rightarrow \mathbb{R}$ ... $\displaystyle{}\lim _{n \rightarrow \infty} f_{n}\left(x_{n}\right)=f(x)$. Then $f$ is continuous.

admin asked in Others Mar 14, 2023

by admin

272 views

2 votes

1 answer

16

TIFR Mathematics 2023 | Part B | Question: 4
Answer whether the following statements are True or False. Let $A, B \in \mathrm{M}_{2}(\mathbb{Z} / 2 \mathbb{Z})$ be such that $\operatorname{tr}(A)=\operatorname{tr}(B)$ and $\operatorname{tr}\left(A^{2}\right)=\operatorname{tr}\left(B^{2}\right)$. Then $A$ and $B$ have the same eigenvalues.

admin asked in Others Mar 14, 2023

by admin

244 views

2 votes

1 answer

17

TIFR Mathematics 2023 | Part B | Question: 5
Answer whether the following statements are True or False. Let $v_{1}, v_{2}, w_{1}, w_{2}$ be nonzero vectors in $\mathbb{R}^{2}$. Then there exists a $2 \times 2$ real matrix $A$ such that $A v_{1}=v_{2}$ and $A w_{1}=w_{2}.$

admin asked in Others Mar 14, 2023

by admin

255 views

2 votes

1 answer

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TIFR Mathematics 2023 | Part B | Question: 6
Answer whether the following statements are True or False. Let $A=\left(a_{i j}\right) \in \mathrm{M}_{n}(\mathbb{R})$ be such that $a_{i j} \geq 0$ for all $1 \leq i, j \leq n$. Assume that $\displaystyle{}\lim _{m \rightarrow \infty} A^{m}$ exists, and ... by $B=\left(b_{i j}\right)$. Then, for all $1 \leq i, j \leq n$, we have $b_{i j} \in\{0,1\}$.

admin asked in Others Mar 14, 2023

by admin

229 views

2 votes

0 answers

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TIFR Mathematics 2023 | Part B | Question: 7
Answer whether the following statements are True or False. Given any monic polynomial $f(x) \in \mathbb{R}[x]$ of degree $n,$ there exists a matrix $A \in \mathrm{M}_{n}(\mathbb{R})$ such that its characteristic polynomial equals $f$.

admin asked in Others Mar 14, 2023

by admin

161 views

2 votes

1 answer

20

TIFR Mathematics 2023 | Part B | Question: 8
Answer whether the following statements are True or False. If $A \in \mathrm{M}_{4}(\mathbb{Q})$ is such that its characteristic polynomial equals $x^{4}+1,$ then $A$ is diagonalizable in $\mathrm{M}_{4}(\mathbb{C})$.

admin asked in Others Mar 14, 2023

by admin

256 views

2 votes

0 answers

21

TIFR Mathematics 2023 | Part B | Question: 9
Answer whether the following statements are True or False. If $A \in \mathrm{M}_{n}(\mathbb{R})$ is such that $A B=B A$ for all invertible matrices $B \in \mathrm{M}_{n}(\mathbb{R})$, then $A=\lambda \cdot \text{Id}$ for some $\lambda \in \mathbb{R}.$

admin asked in Others Mar 14, 2023

by admin

179 views

2 votes

0 answers

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TIFR Mathematics 2023 | Part B | Question: 10
Answer whether the following statements are True or False. There exists a homeomorphism $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(2 x)=3 f(x)$ for all $x \in \mathbb{R}.$

admin asked in Others Mar 14, 2023

by admin

174 views

2 votes

0 answers

23

TIFR Mathematics 2023 | Part B | Question: 11
Answer whether the following statements are True or False. There exists a continuous bijection from $[0,1] \times[0,1]$ to $\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leq 1\right\}$, which is not a homeomorphism.

admin asked in Others Mar 14, 2023

by admin

156 views

2 votes

0 answers

24

TIFR Mathematics 2023 | Part B | Question: 12
Answer whether the following statements are True or False. Let $f \in \mathbb{C}\left[z_{1}, \ldots, z_{n}\right]$ be a nonzero polynomial $(n \geq 1),$ and let \[ X=\left\{z \in \mathbb{C}^{n} \mid f(z)=0\right\} . \] Then $\mathbb{C}^{n} \backslash X$ is path connected.

admin asked in Others Mar 14, 2023

by admin

198 views

2 votes

0 answers

25

TIFR Mathematics 2023 | Part B | Question: 13
Answer whether the following statements are True or False. A connected metric space with at least two points is uncountable.

admin asked in Others Mar 14, 2023

by admin

278 views

2 votes

0 answers

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TIFR Mathematics 2023 | Part B | Question: 14
Answer whether the following statements are True or False. If $A$ and $B$ are disjoint subsets of a metric space $(X, d),$ then \[ \inf \{d(x, y) \mid x \in A, y \in B\} \neq 0. \]

admin asked in Others Mar 14, 2023

by admin

165 views

2 votes

0 answers

27

TIFR Mathematics 2023 | Part B | Question: 15
Answer whether the following statements are True or False. A countably infinite complete metric space has infinitely many isolated points $($an element $x$ of a metric space $X$ is said to be an isolated point if $\{x\}$ is an open subset of $X).$

admin asked in Others Mar 14, 2023

by admin

210 views

3 votes

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TIFR Mathematics 2023 | Part B | Question: 16
Answer whether the following statements are True or False. Suppose $G$ and $H$ are two countably infinite abelian groups such that every nontrivial element of $G \times H$ has order $7.$ Then $G$ is isomorphic to $H.$

admin asked in Others Mar 14, 2023

by admin

312 views

2 votes

0 answers

29

TIFR Mathematics 2023 | Part B | Question: 17
Answer whether the following statements are True or False. There exists a nonabelian group $G$ of order $26$ such that every proper subgroup of $G$ is abelian.

admin asked in Others Mar 14, 2023

by admin

184 views

2 votes

0 answers

30

TIFR Mathematics 2023 | Part B | Question: 18
Answer whether the following statements are True or False. Let $G$ be a group generated by two elements $x$ and $y,$ each of order $2$. Then $G$ is finite.

admin asked in Others Mar 14, 2023

by admin

212 views

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