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Recent questions tagged tifrmaths2022

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1
TIFR Mathematics 2022 | Part B | Question: 1
Answer whether the following statements are True or False. $\mathbb{R}^{2} \backslash \mathbb{Q}^{2}$ is connected but not path-connected.
Answer whether the following statements are True or False.$\mathbb{R}^{2} \backslash \mathbb{Q}^{2}$ is connected but not path-connected.
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2
TIFR Mathematics 2022 | Part B | Question: 2
Answer whether the following statements are True or False. If $X$ is a connected metric space, and $F$ is a subring of $C(X, \mathbb{R})$ that is a field, then every element of $C(X, \mathbb{R})$ that belongs to $F$ is a constant function.
Answer whether the following statements are True or False.If $X$ is a connected metric space, and $F$ is a subring of $C(X, \mathbb{R})$ that is a field, then every eleme...
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3
TIFR Mathematics 2022 | Part B | Question: 3
Answer whether the following statements are True or False. Let $K \subseteq[0,1]$ be the Cantor set. Then there exists no injective ring homomorphism $C([0,1], \mathbb{R}) \rightarrow C(K, \mathbb{R})$.
Answer whether the following statements are True or False.Let $K \subseteq[0,1]$ be the Cantor set. Then there exists no injective ring homomorphism $C([0,1], \mathbb{R})...
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TIFR Mathematics 2022 | Part B | Question: 4
Answer whether the following statements are True or False. There exists a metric space $(X, d)$ such that the group of isometries of $X$ is isomorphic to $\mathbb{Z}$.
Answer whether the following statements are True or False.There exists a metric space $(X, d)$ such that the group of isometries of $X$ is isomorphic to $\mathbb{Z}$.
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5
TIFR Mathematics 2022 | Part B | Question: 5
Answer whether the following statements are True or False. Let $A \subset \mathbb{R}^{2}$ be a nonempty subset such that any continuous function $f: A \rightarrow \mathbb{R}$ is constant. Then $A$ is a singleton set.
Answer whether the following statements are True or False.Let $A \subset \mathbb{R}^{2}$ be a nonempty subset such that any continuous function $f: A \rightarrow \mathbb{...
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TIFR Mathematics 2022 | Part B | Question: 7
Answer whether the following statements are True or False. There exists $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in \mathrm{M}_{2}(\mathbb{R})$, with $A^{2}=A \neq 0$, such that $|a|+|b|<1 \quad \text{and} \quad|c|+|d|<1.$
Answer whether the following statements are True or False.There exists $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in \mathrm{M}_{2}(\mathbb{R})$, with $A...
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TIFR Mathematics 2022 | Part B | Question: 8
Answer whether the following statements are True or False. If $A \in \mathrm{M}_{3}(\mathbb{C})$ is such that $A^{i}$ has trace zero for all positive integers $i$, then $A$ is nilpotent.
Answer whether the following statements are True or False.If $A \in \mathrm{M}_{3}(\mathbb{C})$ is such that $A^{i}$ has trace zero for all positive integers $i$, then $A...
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TIFR Mathematics 2022 | Part B | Question: 9
Answer whether the following statements are True or False. For any finite cyclic group $G$, there exists a prime power $q$ such that $G$ is a subgroup of $\mathbb{F}_{q}^{\times}.$
Answer whether the following statements are True or False.For any finite cyclic group $G$, there exists a prime power $q$ such that $G$ is a subgroup of $\mathbb{F}_{q}^{...
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10
TIFR Mathematics 2022 | Part B | Question: 10
Answer whether the following statements are True or False. There are only finitely many isomorphism classes of finite nonabelian groups, all of whose proper subgroups are abelian.
Answer whether the following statements are True or False.There are only finitely many isomorphism classes of finite nonabelian groups, all of whose proper subgroups are ...
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TIFR Mathematics 2022 | Part B | Question: 11
Answer whether the following statements are True or False. Every subring of a unique factorization domain is a unique factorization domain.
Answer whether the following statements are True or False.Every subring of a unique factorization domain is a unique factorization domain.
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TIFR Mathematics 2022 | Part B | Question: 13
Answer whether the following statements are True or False. There exists a finite abelian group $G$ such that the group Aut$(G)$ of automorphisms of $G$ is isomorphic to $\mathbb{Z} / 7 \mathbb{Z}$.
Answer whether the following statements are True or False.There exists a finite abelian group $G$ such that the group Aut$(G)$ of automorphisms of $G$ is isomorphic to $\...
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TIFR Mathematics 2022 | Part B | Question: 14
Answer whether the following statements are True or False. There exists an integral domain $R$ and a surjective homomorphism $R \rightarrow R$ of rings that is not injective.
Answer whether the following statements are True or False.There exists an integral domain $R$ and a surjective homomorphism $R \rightarrow R$ of rings that is not injecti...
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TIFR Mathematics 2022 | Part B | Question: 15
Answer whether the following statements are True or False. There exists $f \in C([0,1], \mathbb{R})$ satisfying the following two conditions: $\displaystyle{}\int_{0}^{1} f(x) d x=1;$ and $\displaystyle{}\lim _{n \rightarrow \infty} \int_{0}^{1} f(x)^{n} d x=0$.
Answer whether the following statements are True or False.There exists $f \in C([0,1], \mathbb{R})$ satisfying the following two conditions:$\displaystyle{}\int_{0}^{1} f...
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TIFR Mathematics 2022 | Part B | Question: 16
Answer whether the following statements are True or False. Let $a_{n} \geq 0$ for each positive integer $n$. If the series $\sum_{n=1}^{\infty} \sqrt{a_{n}}$ converges, then so does the series $\sum_{n=1}^{\infty} \frac{a_{n}}{n^{1 / 4}}$.
Answer whether the following statements are True or False.Let $a_{n} \geq 0$ for each positive integer $n$. If the series $\sum_{n=1}^{\infty} \sqrt{a_{n}}$ converges, th...
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TIFR Mathematics 2022 | Part B | Question: 17
Answer whether the following statements are True or False. There exists a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ \lim _{x \rightarrow \infty} f(x)=2 \quad \text { and } \quad \lim _{x \rightarrow \infty} f^{\prime}(x)=1 .\]
Answer whether the following statements are True or False.There exists a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that\[ \lim _{x \rightarrow \...
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TIFR Mathematics 2022 | Part B | Question: 20
Answer whether the following statements are True or False. The set $\{f(x) \in \mathbb{R}[x] \mid f(n) \in \mathbb{Z}$ for all $n \in \mathbb{Z}\}$ is uncountable.
Answer whether the following statements are True or False.The set $\{f(x) \in \mathbb{R}[x] \mid f(n) \in \mathbb{Z}$ for all $n \in \mathbb{Z}\}$ is uncountable.
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TIFR Mathematics 2022 | Part A | Question: 3
What is the number of solutions of: \[ x=\frac{x^{2}}{50}-\cos \frac{x}{2}+2 \] in $[0,10]?$ $0$ $1$ $2$ $\infty$
What is the number of solutions of: \[ x=\frac{x^{2}}{50}-\cos \frac{x}{2}+2 \] in $[0,10]?$ $0$$1$$2$$\infty$
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TIFR Mathematics 2022 | Part A | Question: 4
Let $A$ be an element of $\mathrm{M}_{4}(\mathbb{R})$ with characteristic polynomial $t^{4}-t$. What is the characteristic polynomial of $A^{2}$ ? $t^{4}-t$ $t^{4}-2 t^{3}+t^{2}$ $t^{4}-t^{2}$ None of the other three options
Let $A$ be an element of $\mathrm{M}_{4}(\mathbb{R})$ with characteristic polynomial $t^{4}-t$. What is the characteristic polynomial of $A^{2}$ ?$t^{4}-t$$t^{4}-2 t^{3}+...
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TIFR Mathematics 2022 | Part A | Question: 5
Let $n$ be a positive integer, and let $V=\{f \in \mathbb{R}[x] \mid \operatorname{deg} f \leq n\}$ be the real vector space of real polynomials of degree at most $n$. Let $\operatorname{End}_{\mathbb{R}}(V)$ denote the real vector space of linear ... $1$ $n$ $n+1$ $n^{2}$
Let $n$ be a positive integer, and let $V=\{f \in \mathbb{R}[x] \mid \operatorname{deg} f \leq n\}$ be the real vector space of real polynomials of degree at most $n$. Le...
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TIFR Mathematics 2022 | Part A | Question: 10
Let $p$ be a prime number. What is the number of elements in the group $\mathbb{Q} / \mathbb{Z}$ that have order exactly $p?$ $0$ $p-1$ $p$ $\infty$
Let $p$ be a prime number. What is the number of elements in the group $\mathbb{Q} / \mathbb{Z}$ that have order exactly $p?$$0$$p-1$$p$$\infty$
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