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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: 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: 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: 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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TIFR Mathematics 2024 | Part A | Question: 1
What is the number of even positive integers $n$ such that every group of order $n$ is abelian? $1$ $2$ Greater than $2$, but finite Infinite
What is the number of even positive integers $n$ such that every group of order $n$ is abelian?$1$$2$Greater than $2$, but finiteInfinite
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TIFR Mathematics 2024 | Part A | Question: 8
Consider the sequence $\left\{a_{n}\right\}$ for $n \geq 1$ ... $\lim _{n \rightarrow \infty} n^{2} a_{n}$ exists and equals 1
Consider the sequence $\left\{a_{n}\right\}$ for $n \geq 1$ defined by\[a_{n}=\lim _{N \rightarrow \infty} \sum_{k=n}^{N} \frac{1}{k^{2}} .\]Which of the following statem...
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TIFR Mathematics 2024 | Part A | Question: 10
Let $B$ denote the set of invertible upper triangular $2 \times 2$ matrices with entries in $\mathbb{C}$, viewed as a group under matrix multiplication. Which of the following subgroups of $B$ is the normalizer of itself in $\text{B}$ ...
Let $B$ denote the set of invertible upper triangular $2 \times 2$ matrices with entries in $\mathbb{C}$, viewed as a group under matrix multiplication. Which of the foll...