Register

Recent questions tagged tifrmaths2023

2 votes
1 answer
2
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.
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 ...
User Avatar
2 votes
1 answer
5
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...
User Avatar
2 votes
0 answers
7
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}(\...
User Avatar
2 votes
1 answer
8
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...
User Avatar
2 votes
0 answers
9
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}(\...
User Avatar
2 votes
0 answers
10
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...
User Avatar
2 votes
0 answers
11
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^{...
User Avatar
2 votes
0 answers
13
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.
User Avatar
2 votes
0 answers
14
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...
User Avatar
2 votes
0 answers
15
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...
User Avatar
3 votes
0 answers
16
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 $H.$
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 $G \times H$...
User Avatar
2 votes
0 answers
17
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.
User Avatar
2 votes
0 answers
18
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.
User Avatar
2 votes
0 answers
19
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.
User Avatar
2 votes
0 answers
20
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.
User Avatar
2 votes
0 answers
24
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...
User Avatar
2 votes
0 answers
29
TIFR Mathematics 2023 | Part A | Question: 9
Let $T: \mathrm{M}_{3}(\mathbb{R}) \rightarrow \mathbb{R}^{3}$ be the linear map defined by $T(A)=A\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$. Then the dimension of the kernel of $T$ equals $2$. $8$. $1$. None of the remaining three options.
Let $T: \mathrm{M}_{3}(\mathbb{R}) \rightarrow \mathbb{R}^{3}$ be the linear map defined by $T(A)=A\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$. Then the dimensi...
User Avatar
Page: