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92
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 $A w_{1}=w_{2}.$
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 m...
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2 votes
0 answers
94
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 $f$.
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}(\...
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2 votes
1 answer
95
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 $\mathrm{M}_{4}(\mathbb{C})$.
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 d...
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2 votes
0 answers
96
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 $\lambda \in \mathbb{R}.$
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}(\...
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2 votes
0 answers
97
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 $x \in \mathbb{R}.$
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 \ma...
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2 votes
0 answers
98
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.
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^{...
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2 votes
0 answers
100
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.
Answer whether the following statements are True or False.A connected metric space with at least two points is uncountable.
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2 votes
0 answers
101
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 \[ \inf \{d(x, y) \mid x \in A, y \in B\} \neq 0. \]
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\} \ne...
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2 votes
0 answers
102
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 $X).$
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 spa...
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2 votes
0 answers
103
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.
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.
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2 votes
0 answers
104
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.
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.
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2 votes
0 answers
105
TIFR Mathematics 2023 | Part B | Question: 19
Answer whether the following statements are True or False. $\mathbb{R}[x] /\left(x^{4}+x^{2}+2023\right)$ is an integral domain.
Answer whether the following statements are True or False.$\mathbb{R}[x] /\left(x^{4}+x^{2}+2023\right)$ is an integral domain.
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2 votes
0 answers
106
TIFR Mathematics 2023 | Part B | Question: 20
Answer whether the following statements are True or False. Every finite group is isomorphic to a subgroup of a finite group generated by two elements.
Answer whether the following statements are True or False.Every finite group is isomorphic to a subgroup of a finite group generated by two elements.
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2 votes
0 answers
110
TIFR Mathematics 2023 | Part A | Question: 4
Consider the function $f:(0, \infty) \rightarrow(0, \infty)$ given by $f(x)=x e^{x}$. Let $L:(0, \infty) \rightarrow(0, \infty)$ ... . None of the remaining three options is correct.
Consider the function $f:(0, \infty) \rightarrow(0, \infty)$ given by $f(x)=x e^{x}$. Let $L:(0, \infty) \rightarrow(0, \infty)$ be its inverse function. Which of the fol...
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