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271
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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272
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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273
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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3 votes
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274
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 $G \times H$...
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275
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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276
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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277
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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278
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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282
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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284
TIFR Mathematics 2023 | Part A | Question: 6
For every positive integer $n,$ define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by $f_{n}(x)=\dfrac{\sin \left(n^{2} x\right)+\cos \left(e^{n} x\right)}{1+n^{2} x^{2}}$. Then \[ \lim _{n \rightarrow \infty} \int_{0}^{1-\sin (1 / n)} f_{n}(x) d x \] equals $1$. $0.$ $\infty$. $1 / 2$.
For every positive integer $n,$ define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by $f_{n}(x)=\dfrac{\sin \left(n^{2} x\right)+\cos \left(e^{n} x\right)}{1+n^{2} x^{2}}$. Then...
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287
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...
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