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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 $H \cong G / H$.
If $\text{G}$ is a group of order $361$, then $\text{G}$ has a normal subgroup $\text{H}$ such that $H \cong G / H$.
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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 $2024$.
There exists a metric space $\text{X}$ such that the number of open subsets of $\text{X}$ is exactly $2024$.
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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.
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 $(\mathbb{R}, d)$ ...
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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.
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 $\mathrm{M}_{n}(\mathbb{R})$,...
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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.
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}(\m...
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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.
The complement of any countable union of lines in $\mathbb{R}^{3}$ is path connected.
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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.
The subsets $\left\{(x, y) \in \mathbb{R}^{2} \mid\left(y^{2}-x\right)\left(y^{2}-x-1\right)=0\right\}$ and $\left\{(x, y) \in \mathbb{R}^{2} \mid y^{2}-x^{2}=1\right\}$ ...
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TIFR Mathematics 2024 | Part B | Question: 8
$\mathbb{Q} \cap[0,1]$ is a compact subset of $\mathbb{Q}$.
$\mathbb{Q} \cap[0,1]$ is a compact subset of $\mathbb{Q}$.
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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.
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\lef...
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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.
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...
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TIFR Mathematics 2024 | Part B | Question: 11
There exists a function $f:[0,1] \rightarrow \mathbb{R}$, which is not Riemann integrable and satisfies \[ \sum_{i=1}^{n}\left|f\left(t_{i}\right)-f\left(t_{i-1}\right)\right|^{2}<1 \]
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)\rig...
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TIFR Mathematics 2024 | Part B | Question: 12
Let $E \subset[0,1]$ be the subset consisting of numbers that have a decimal expansion which does not contain the digit 8 . Then $E$ is dense in $[0,1]$.
Let $E \subset[0,1]$ be the subset consisting of numbers that have a decimal expansion which does not contain the digit 8 . Then $E$ is dense in $[0,1]$.
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TIFR Mathematics 2024 | Part B | Question: 13
Let $\text{G}$ be a proper subgroup of $(\mathbb{R},+)$ which is closed as a subset of $\mathbb{R}$. Then $G$ is generated by a single element.
Let $\text{G}$ be a proper subgroup of $(\mathbb{R},+)$ which is closed as a subset of $\mathbb{R}$. Then $G$ is generated by a single element.
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TIFR Mathematics 2024 | Part B | Question: 14
There exists a unique function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is continuous at $x=0$, and such that for all $x \in \mathbb{R}$ \[ f(x)+f\left(\frac{x}{2}\right)=x . \]
There exists a unique function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is continuous at $x=0$, and such that for all $x \in \mathbb{R}$\[f(x)+f\left(\frac{x}...
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TIFR Mathematics 2024 | Part B | Question: 15
A map $f: V \rightarrow W$ between finite dimensional vector spaces over $\mathbb{Q}$ is a linear transformation if and only if $f(x)=f(x-a)+f(x-b)-f(x-a-b)$, for all $x, a, b \in V$.
A map $f: V \rightarrow W$ between finite dimensional vector spaces over $\mathbb{Q}$ is a linear transformation if and only if $f(x)=f(x-a)+f(x-b)-f(x-a-b)$, for all $x,...
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TIFR Mathematics 2024 | Part B | Question: 16
Let $R$ be the ring $\mathbb{C}[x] /\left(x^{2}\right)$ obtained as the quotient of the polynomial ring $\mathbb{C}[x]$ by its ideal generated by $x^{2}$. Let $R^{\times}$be the multiplicative group of units of this ring. Then there is an injective group homomorphism from $(\mathbb{Z} / 2 \mathbb{Z}) \times(\mathbb{Z} / 2 \mathbb{Z})$ into $R^{\times}$.
Let $R$ be the ring $\mathbb{C}[x] /\left(x^{2}\right)$ obtained as the quotient of the polynomial ring $\mathbb{C}[x]$ by its ideal generated by $x^{2}$. Let $R^{\times}...
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TIFR Mathematics 2024 | Part B | Question: 17
Let $A \in \mathrm{M}_{2}(\mathbb{Z})$ be such that $\left|A_{i j}(n)\right| \leq 50$ for all $1 \leq n \leq 10^{50}$ and all $1 \leq i, j \leq 2$, where $A_{i j}(n)$ denotes the $(i, j)$-th entry of the $2 \times 2$ matrix $A^{n}$. Then $\left|A_{i j}(n)\right| \leq 50$ for all positive integers $n$.
Let $A \in \mathrm{M}_{2}(\mathbb{Z})$ be such that $\left|A_{i j}(n)\right| \leq 50$ for all $1 \leq n \leq 10^{50}$ and all $1 \leq i, j \leq 2$, where $A_{i j}(n)$ den...
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TIFR Mathematics 2024 | Part B | Question: 18
Let $A, B$ be subsets of $\{0, \ldots, 9\}$. It is given that, on choosing elements $a \in A$ and $b \in B$ at random, $a+b$ takes each of the values $0, \ldots, 9$ with equal probability. Then one of $A$ or $B$ is singleton.
Let $A, B$ be subsets of $\{0, \ldots, 9\}$. It is given that, on choosing elements $a \in A$ and $b \in B$ at random, $a+b$ takes each of the values $0, \ldots, 9$ with ...
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TIFR Mathematics 2024 | Part B | Question: 19
If $f: \mathbb{R} \rightarrow \mathbb{R}$ is uniformly continuous, then there exists $M>0$ such that for all $x \in \mathbb{R} \backslash[-M, M]$, we have $f(x) < x^{100}$.
If $f: \mathbb{R} \rightarrow \mathbb{R}$ is uniformly continuous, then there exists $M>0$ such that for all $x \in \mathbb{R} \backslash[-M, M]$, we have $f(x) < x^{100}...
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TIFR Mathematics 2024 | Part B | Question: 20
If a sequence $\left\{f_{n}\right\}$ of continuous functions from $[0,1]$ to $\mathbb{R}$ converges uniformly on $(0,1)$ to a continuous function $f:[0,1] \rightarrow \mathbb{R}$, then $\left\{f_{n}\right\}$ converges uniformly on $[0,1]$ to $f$.
If a sequence $\left\{f_{n}\right\}$ of continuous functions from $[0,1]$ to $\mathbb{R}$ converges uniformly on $(0,1)$ to a continuous function $f:[0,1] \rightarrow \ma...
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